Structural, Electronic and Magnetic Properties of Transition Metal Clusters [PDF]

Zitiervorschau

Institut f¨ur Materialphysik Center for Computational Materials Science

Structural, Electronic and Magnetic Properties of Transition Metal Clusters Dissertation zur Erlangung des akademischen Grades Doktor der Naturwissenschaften an der Fakult¨at f¨ ur Physik der Universit¨at Wien

eingereicht von

Mag. Tanja Futschek April 2005 Gutachter: O. Univ. Prof. Dr. J¨ urgen Hafner Ao. Univ. Prof. Dr. Raimund Podloucky

Acknowledgements A Brief History of My Time at the Materials Physics Institute We all have the same sky, but do we all have the same horizon? (based on Konrad Adenauer)

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ever again solid-state physics I explained to my parents in the late summer 2001 after my last solid-state physics examination I took with Prof. J¨ urgen Hafner. Therefore my first thanks apply for my parents, not only because they are responsible for my existence, for propping me up in bad times or sharing my happiness in good ones, but also for supporting me in my decisions (for example to study physics), even if I changed them completely within a short time like the preconception concerning solid-state physics. Under the slogan ”never say never again” it came - as expected - differently than meant - you surely will recognize this reading the title of this PhD-thesis. In the early winter 2001 - I was on the way finishing my master degree - I tried to find a position as a PhD student. It turned out not to be as easy as finding a diploma position, which I got, because I was NOT able to read the university calendar carefully enough (greetings from the ”Pisa”-study). However at this time too many students were searching for a PhD position and too little positions were available. But my friend and study colleague Christina Forster introduced me into a group of physicists, who had already finished or had been finishing their PhD thesis in the group of Prof. Hafner. Thanks to Tina! I want to mention them in the order of their appearance, beginning with Thomas Demuth, CPG winner 2002, and Robin Hirschl, who graduated sub auspiciis 2002 and received the CPG-prize 2003 (that’s only for your information what outstanding scientist are working in this group). Thomas, who was actually responsible for making my mind up to attend the Materials Science group and Robin, who introduced me into this group and was always available, especially at the beginning, as it was not as simple as I thought. First of all the Linux problem! As a ”Windows-User” I had no idea of Unix/Linux. The only thing I got to know from the very first beginning was that this ”Linux” penguin i

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Acknowledgements

was more congenial than those ”microsoft” window (”We suck more!”). I couldn’t believe, what kind of problems one can have, for example ”how to create directories” or ”how to list files”. Even my DOS knowledge couldn’t help me solve even ”easy” problems I had at the beginning. But it became even worse, being confronted with VASP (Vienna Ab-initio Simulation Package, see Sec. 2.5 for further information) for the first time. It is a Fortran based code. I had already appropriated knowledge in Fortran77 during my diploma thesis and therefore asked Robin to tell me, where I could find the main files. I - naively as I was - was very enthusiastic being able to understand what’s going on only be reading the code. Far from it! This project failed because of the fact, that the code consist of more than 92.000 lines. However Robin was always around, if I had problems and was completely lost, whether it was a matter of technical or physical questions. At this point of time I would like to thank my supervisor, Prof. J¨ urgen Hafner, who always demanded and promoted me scientifically, but also supported me in activities in addition to physics. Special thanks apply also for George Kresse, Andreas Eichler, Florian Mittendorfer and to the coauthor of my paper, see therefore Chapter 4, Martijn Marsman. As you can see, the most members in our group are male. Until October 2003 I was the only female member on board of this crew (beside the secretary). Therefore a special ”thank you” to my dearest room colleague, Doris Vogtenhuber. Scientific discussions ran off a little more relaxed with her. As the VASP-master of our group she found solutions for most problems, the others turned out to be nearly unsolvable. She was always there for me, even if I had problems beyond physics. In addition to that I would like to thank my two other room colleagues, Judith Harl and Samuel Dennler - who cheered me up with humor - and all the other members of the Materials Physics Institute, I haven’t mentioned by name. Beside the Materials Physics Institute I would like to thank my friend Petra and her boyfriend Franz for fruitful discussions, regardless talking about physical or routine problems. Last but not least I would like to thank the person, who supported me mentally most of all, who always brought me up, when things went bad, understood my outbursts of fury and due to its humor could raise a smile from my lips. Thank you for being there, Boris, my ”little” brother!

Abstract

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he investigation of the unique physico-chemical properties of small clusters is a very active field of research, as such microscopic particles represent an intermediate state of matter between single atoms and solid material. Metallic clusters, for example, are known to possess unique catalytic properties. In addition to that transition metals that are non-magnetic in the bulk may become magnetic when the dimensionality is reduced. Over the past years, numerous theoretical methods have been used to study clusters, but the central problem until now is the determination of the ground state geometry - as with increasing clustersize the number of conceivable configurations increases tremendously. Experimental information, on the other side, is only indirect and in most cases not sufficient to determine the structure precisely. The first part of this thesis will give a brief introduction into the history of quantum mechanics and an overview about the most common methods, which are used for solving physical and/or chemical problems, followed by a short insight into the mystery of Density Functional Theory (DFT) as the VASP code (Vienna Ab-initio Simulation Package), which has been written and is further developed in our group, is based on this method. The final chapter of this part is devoted to cluster science. In the second part we present a comprehensive investigation of the structural, electronic, and magnetic properties of PdN and RhN clusters with up to N = 13 atoms. The novel aspects of our investigation are: (i) the structural optimization of the cluster by a symmetry-unconstrained static total-energy minimization by a search of the ground-state-structure by dynamical simulated annealing and (ii) the spin-polarized calculations, which have been performed in a fixed-moment mode. This allows to study coexisting magnetic isomers and leads to a deeper insight into the importance of magneto-structural effects. In part three a more complete documentation can be found concerning magnetic and structural information about all calculated structures and isomers.

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Zusammenfassung

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ie Studie der einzigartigen physikalisch-chemischen Eigenschaften kleiner Cluster ist derzeit Gegenstand intensiver Forschung, da solche Mikropartikel das Bindeglied zwischen einzelnen Atomen und dem Festk¨orper bilden. Metallische Cluster, beispielsweise, besitzen einzigartige katalytische Eigenschaften. ¨ Zus¨atzlich dazu, k¨onnen Ubergangsmetalle, die im Bulk nicht magnetisch sind, magnetisch werden, wenn ihre Dimensionalit¨at verringert wird. In den letzten Jahren sind zahlreiche theoretische Methoden zur Erforschung kleiner Cluster verwendet worden. Das zentrale Problem ist bis dato aber immer noch die Ermittlung der Grundzustandgeometrie, da sich mit der Zunahme der Clustergr¨oße auch die Zahl der m¨oglichen Konfigurationen enorm erh¨oht. Ergebnisse experimenteller Untersuchungen, andererseits sind nur indirekt und in den meisten F¨allen nicht ausreichend, um die Struktur genau festzustellen. Der erste Teil dieser Dissertation gibt eine kurze Einf¨ uhrung in die Geschichte ¨ der Quantenmechanik und einen Uberblick u ¨ber die verschiedene Methoden, die zum L¨osen physikalischer und/oder chemischer Probleme verwendet werden, gefolgt von einem kurzen Einblick in das Geheimnis der Dichte-Funktionstheorie (DFT), da das VASP Programm (Vienna Ab-initio Simulation Package), das in unserer Gruppe entwickelt worden ist und immer noch benutzt wird, auf dieser Methode basiert. Das letzte Kapitel des ersten Abschnittes ist der Cluster-Forschung gewidmet. Im zweiten Teil pr¨asentieren wir eine ausf¨ uhrliche Studie struktureller, elektronischer und magnetischer Eigenschaften von PdN und RhN Cluster mit bis zu N = 13 Atomen. Die neuen Aspekte unserer Untersuchung umfassen: (i) die strukturelle Optimierung der Cluster durch eine symmetrieungezwungene, statische Gesamtenergieminimierung - mittels Grundzustandsstruktursuche durch ”simuliertes Aufheizen” - und (ii) Ergebnisse spin-polarisierten Berechnungen, die in einem o¨rtlich festgelegten Moment-Modus durchgef¨ uhrt worden sind. Das erlaubt die Studie von nebeneinander bestehenden magnetischen Isomeren und f¨ uhrt zu einem tieferen Einblick in die Bedeutung magneto-struktureller Effekte. Im dritten Teil findet sich eine komplette Dokumentation hinsichtlich der magnetischen und strukturellen Informationen aller berechneter Strukturen und Isomere. v

Table of Contents I

Methods

1 Historical Background 1.1 Milestones . . . . . . . . . . 1.2 Overview . . . . . . . . . . . 1.2.1 Classical Methods . . 1.2.2 Quantum-mechanical

1 . . . . . . . . . . . . . . . Methods

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2 Electronic structure methods 2.1 The Schr¨odinger Equation . . . . . . . . . . . . . . 2.1.1 Born - Oppenheimer Approximation . . . . 2.2 Density Functional Theory . . . . . . . . . . . . . . 2.2.1 Introduction . . . . . . . . . . . . . . . . . . 2.2.2 Pair Density . . . . . . . . . . . . . . . . . . 2.2.3 The Energy Functional . . . . . . . . . . . . 2.2.4 The Thomas-Fermi model . . . . . . . . . . 2.2.5 The Slater Method . . . . . . . . . . . . . . 2.2.6 Hohenberg - Kohn Theorem . . . . . . . . . 2.2.7 Kohn - Sham Equations . . . . . . . . . . . 2.3 Exchange-Correlation Functionals . . . . . . . . . . 2.3.1 Local Density Approximation (LDA) . . . . 2.3.2 The Exchange-Correlation Hole . . . . . . . 2.3.3 Local Spin-Density Approximation (LSDA) . 2.3.4 Interpretation of the L(S)DA . . . . . . . . 2.3.5 Gradient Expansion Approximation (GEA) . 2.3.6 Generalized Gradient Approximation (GGA) 2.3.7 Meta-GGA . . . . . . . . . . . . . . . . . . 2.3.8 Hybrid Functionals . . . . . . . . . . . . . . 2.4 Plane Waves and Pseudopotentials . . . . . . . . . 2.4.1 Plane Waves . . . . . . . . . . . . . . . . . . 2.4.2 Pseudopotentials . . . . . . . . . . . . . . . 2.5 The Vienna Ab-Initio Simulation Package (VASP) . vii

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TABLE OF CONTENTS

3 Cluster and Nanostructures 3.1 Introduction . . . . . . . . . . . . . . . . . . . . 3.1.1 Clusters between Atom and Bulk . . . . 3.2 Structures of Clusters . . . . . . . . . . . . . . . 3.2.1 Shell Structure of Simple Metal Clusters 3.2.2 Shapes . . . . . . . . . . . . . . . . . . . 3.3 Production of Clusters . . . . . . . . . . . . . . 3.3.1 Supersonic Jets . . . . . . . . . . . . . . 3.3.2 Gas aggregation sources . . . . . . . . . 3.3.3 Surface Sources . . . . . . . . . . . . . . 3.3.4 Pick-up Sources . . . . . . . . . . . . . . 3.3.5 Embedded and deposit clusters . . . . . 3.4 Experimental Measurements . . . . . . . . . . . 3.4.1 Mass Spectrometers . . . . . . . . . . . . 3.4.2 Optical Response . . . . . . . . . . . . . 3.4.3 Vibrational Spectra . . . . . . . . . . . . 3.4.4 Photoelectron Spectroscopy . . . . . . . 3.5 Theoretical Developments . . . . . . . . . . . . 3.6 Magnetism of Free Clusters . . . . . . . . . . . 3.6.1 Common Ideas on Magnetism . . . . . . 3.6.2 Implications for Clusters . . . . . . . . . 3.6.3 Stern-Gerlach Experiments . . . . . . . .

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Applications

4 Pd 4.1 4.2 4.3

and Rh clusters Introduction . . . . . . . . . . . . . . . . . . . . Computational Method . . . . . . . . . . . . . . Trends in binding energies, geometries, magnetic 4.3.1 Binding energy . . . . . . . . . . . . . . 4.3.2 Cluster geometry . . . . . . . . . . . . . 4.3.3 Magnetic moment . . . . . . . . . . . . . 4.3.4 Electronic properties . . . . . . . . . . . 4.4 Structure, magnetic and electronic properties ... 4.4.1 Pd2 and Rh2 . . . . . . . . . . . . . . . . 4.4.2 Pd3 and Rh3 . . . . . . . . . . . . . . . . 4.4.3 Pd4 and Rh4 . . . . . . . . . . . . . . . . 4.4.4 Pd5 and Rh5 . . . . . . . . . . . . . . . . 4.4.5 Pd6 and Rh6 . . . . . . . . . . . . . . . . 4.4.6 Pd7 and Rh7 . . . . . . . . . . . . . . . . 4.4.7 Pd8 and Rh8 . . . . . . . . . . . . . . . . 4.4.8 Pd9 and Rh9 . . . . . . . . . . . . . . . .

45 45 45 48 50 50 53 53 54 54 55 55 56 56 57 58 63 66 67 67 73 74

79 . . . . . . . . . . . . . . moments ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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81 81 83 84 92 94 96 96 99 99 99 101 106 107 109 112 113

TABLE OF CONTENTS

4.5

III

4.4.9 Pd10 and Rh10 . . . 4.4.10 Pd11 and Rh11 . . . 4.4.11 Pd12 and Rh12 . . . 4.4.12 Pd13 and Rh13 . . . Summary and Conclusions

Appendices

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114 116 117 118 121

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A Pd-Cluster

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B Rh-Cluster

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Bibliography

207

List of Figures 1.1 2.1 2.2

2.3

2.4

2.5 2.6 2.7

2.8 2.9 3.1

3.2

3.3

Overview over various methods and models for solving physical and/or chemical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representations of the electron density of the water molecule. . . . . A cartoon representing the relationship between the ”real” many body system (left hand side) and the non-interacting system of Kohn Sham density functional theory (right hand side). . . . . . . . . . . . The exchange hole, nx (r, r0 ), for a neon atom, comparing the exact result (continuous line) to that within the LD-Approximation (dashed line). The top panel is for r = 0.09a0 and the lower for r = 0.4a0 . See [21] for reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . The spherical average of the exchange hole in a neon atom, comparing the exact result (continuous line) with that computed within the LDApproximation (dashed line). The top panel is for r = 0.09a0 and the lower for r = 0.4a0 . See [21] for reference. . . . . . . . . . . . . . The local density approximation. From Ref. [1] . . . . . . . . . . . . Supercell geometry for an isolated molecule (icosahedron). The dashed line encloses the periodic supercell. . . . . . . . . . . . . . . . . . . . A schematic illustration of all-electron (dashed lines) and pseudo(solid lines) potentials and their corresponding wave-functions. The radius at which all-electron and pseudopotential values match is rc . From Ref. [50], redrawn by S. Dennler [54]. . . . . . . . . . . . . . . . Decomposition of the all-electron wave-function in the PAW method . Typical flow-chart of VASP. . . . . . . . . . . . . . . . . . . . . . . .

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38 41 44

Total spin-polarized differential density of states for Pd2 (top panel) and for Pd13 (mid panel). Local density of states of Pd (100)(bottom panel). Solid blue line represent surface Pd and black one subsubsurface Pd, which is estimated to be bulk-like in this context. . . . . . . 47 Equidensity plot of the electronic density of N a4 computed with LDA. Plotted is the density in the x-y plane integrated over all z. The ionic positions are marked by small diamonds. The x and y scales are in units of a0 . See [79] for Ref. . . . . . . . . . . . . . . . . . . . . . . . 49 Experimental abundances of Na clusters, after [81]. . . . . . . . . . . 50 xi

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LIST OF FIGURES 3.4

3.5

3.6

3.7 3.8

3.9 3.10 3.11

3.12 3.13 3.14

3.15

3.16

4.1

Structure of Ar clusters for various sizes. The larger clusters in the lower part represent atomic shell closures for icosahedral symmetry. See [78] for reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . Single electron energies vs. deformation α20 for the axially symmetric deformed harmonic oscillator. The numbers indicate (deformed) shell closures, up to N = 20. The shaded areas indicate bands of deformed magic shells. Typical shapes associated with given deformation α20 are indicated an top. See [79] for reference. . . . . . . . . . . . . . . . Basic layout of typical cluster sources. A seeded supersonic nozzle source is represented in (a), a gas aggregation source in (b) and surface sources, namely laser evaporation source in (c) and pulsed arc cluster ion source in (d). See [87] for reference. . . . . . . . . . . . . . . . . . Schematic representation of basic mass spectrometers: (a) RadioFrequency (RF) quadrupole and (b) Wien filter. See [87] for reference. Schematic view of absorption and depletion spectroscopy. The left panel describes the various situations in the spectral representation. The right panel shows the corresponding observed strengths. Several situations are considered: typical ionization potential (IP) measurement (top), access to excited (non-dissociative) level, and access to excited (dissociative) level (bottom). From [79]. . . . . . . . . . . . . Schematic view of a RAIRS spectrometer. . . . . . . . . . . . . . . . Schematic view of an EELS spectrometer . . . . . . . . . . . . . . . . Typical example of a spectra recorded via high resolution electron energy spectroscopy (HREELS). In this special case: HREELS spectra in 5 off-specular geometry before and after removal of the chemisorbed oxygen for two different Pd (111)-O doses (a) and (b). From Ref. [92]. Single particle picture of photoemission. . . . . . . . . . . . . . . . . All-electron picture of the photoemission process. . . . . . . . . . . . Approximate relation of 1 eV per Bohr magneton between magnetic exchange splitting 4Exc and local magnetic moment µ for 3d electrons. See [113] for Ref. . . . . . . . . . . . . . . . . . . . . . . . . . Stoner criterion for ferromagnetism. The two important factors are (a) the density of states at the Fermi level, D(EF ), and (b) an atomic exchange integral, I. (c) Their product has to be larger than unity to drive the transition to ferromagnetism . See [115] for Ref. . . . . . . . Perspective view of the quadrupol sector pole faces used at the University of Virginia. This configuration offers optimal field gradient homogeneity. Observable parameters correspond to equation 3.8. See [98] for reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

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Final symmetric structures and structures produced by dynamical simulated annealing (labelled by an asterisk) for clusters with 2 to 13 atoms: XN with X=Pd or Rh and N = 2 − 13. . . . . . . . . . . . . . 86

LIST OF FIGURES 4.2

4.3

4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11

4.12

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4.16 4.17 4.18

Binding energy, average coordination number, magnetic moment/atom, average bond length and HOMO/LUMO gap of energetically preferred PdN and RhN clusters with N=2-13. . . . . . . . . . . . . . . . Total spin-polarized differential (left scale) and integrated (right scale) density of states for Pd2 , Pd3 , tetrahedral Pd4 , trigonal-bipyramidal Pd5 and octahedral Pd6 , calculated for different spin isomers. . . . . . Isosurfaces of the magnetization densities for M=0µB (left) and M=2µB (right) for the Pd3 triangle. . . . . . . . . . . . . . . . . . . . . . . . Isosurface plots of the magnetization densities of the S = 0 (left) and S = 1 (right) spin-isomers of Pd4 . . . . . . . . . . . . . . . . . . . . . Local density of states at sites 1 and 4 of the antiferromagnetic S = 0 P d4 tetrahedron, carrying negative moments of - 0.26 µB . . . . . . . . Isosurface plots of the magnetization densities of the S = 1 (left), S = 2 (center) and S = 3 (right) spin-isomers of Rh4 . . . . . . . . . . Total spin-polarized differential and integrated DOS for tetrahedral Rh4 cluster with S = 0 to S = 3 (top to bottom) . . . . . . . . . . . Isosurface plots of the magnetization densities in trigonal bipyramids of Pd5 for the S = 0 (left) and S = 1 (right) spin-isomers. . . . . . . . Isosurface plots of the magnetization densities of the S = 1 to S = 3 (from left to right) spin-isomers of a (nearly) octahedral Rh6 cluster. Isosurface plots of the magnetization densities of the S = 0 (left) and S = 1 (right) spin-isomers of a Pd7 cluster forming a slightly distorted pentagonal bipyramid. . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-polarized electronic density of states and iso-surface plots of the electronic density distributions of eigenstates in the vicinity of the Fermi-level, as calculated for the stable (S = 1) spin-isomer of a Pd7 cluster forming a slightly distorted pentagonal bipyramid. . . . . . . . Isosurfaces of the magnetization densities for M = 0 µB to M = 4 µB for the Pd10 tetragonal antiprism with capped square faces (from left to right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local magnetic moments and isosurfaces of the magnetization densities in tetrahedral Pd10 clusters with S = 1 (distorted C2v symmetry) and S = 2 (T symmetry). . . . . . . . . . . . . . . . . . . . . . . . . Isosurfaces of the magnetization densities in Rh10 clusters with the structure of a capped tetragonal antiprism, for S = 3 to S = 8 (top left to bottom right). . . . . . . . . . . . . . . . . . . . . . . . . . . . Isosurface plots of the magnetization densities of the S = 0 and S = 4 spin-isomers of a (nearly) icosahedral Pd13 cluster. . . . . . . . . . . . Structure of the stable Pd13 cluster with local magnetic moments (left), side-view of the relaxed cluster (right). Cf. text. . . . . . . . . Isosurfaces of the magnetization densities for Pd13 clusters with a polyoctahedral structure, for S = 1 to S = 3 (from left to right). Cf text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

93

98 100 102 103 104 105 106 108

109

111

114

115

116 118 119

120

xiv

LIST OF FIGURES

4.19 Isosurface plots of the magnetization densities of the S = 7/2 and S = 21/2 isomers of an icosahedral Rh13 cluster. . . . . . . . . . . . . 121 4.20 Structure of the stable Rh13 cluster with local magnetic moments (left), side-view of the relaxed cluster (right). Cf. text. . . . . . . . . 121 4.21 Probability PN =6 and PN =9 to find a P d6 and Rh9 cluster with Spin S, respective to its isomeric structure I, with respective to temperature [K], see Section 3.6.3 for theory. . . . . . . . . . . . . . . . . . . . . 123 4.22 Thermally averaged magnetic moment/atom [µB ] for Pd and Rh clusters with respective to temperature [K], see Section 3.6.3 for theory. 125

List of Tables 3.1

3.2

4.1

4.2

4.3

Schematic classification of clusters according to the number N of atoms. As a complement the diameter d for Na clusters is given (second row), together with an estimate of the ratio of surface to volume atoms f (surface fraction, in third row). After [77]. . . . . . . 46 Classification of binding in clusters. For each one of the four types of bonding, examples of clusters (second column), the nature (third column) and typical binding energies (last column) are given. See [78] for Ref. [77]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Final structure (notation according to Fig. 4.1), point group symmetry PGS, total magnetic moment M (in µB ), average coordination number NC , average nearest-neighbor distances d (in ˚ A), HOMOLUMO gap Eg (in eV), and binding energies (in eV/atom) for structural and magnetic isomers of PdN clusters with N = 2 to 13. . . . . 87 Final structure (notation according to Fig. 4.1), point group symmetry PGS, total magnetic moment M (in µB ), average coordination number NC , average nearest-neighbor distances d (in ˚ A), HOMOLUMO gap Eg (in eV), and binding energies (in eV/atom) for structural and magnetic isomers of RhN clusters with N = 2 to 13. . . . . 89 Energies of eigenstates for spin-up and spin-down states and exchange splitting (∆Eex ) of the stable Pd7 isomer (pentagonal bipyramid with (S = 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

xv

Part I Methods

1

Chapter Historical Background

In physics, you don’t have to go around making trouble for yourself - nature does it for you. (Franz Wilczek)

Q

uantum Mechanics is now over 100 years old, and is one of the most successful scientific theories ever created. Only by looking at phenomena at very short length and time scales we can see quantum behavior. By the 1890’s, classical physics - Newtonian mechanics plus Maxwell’s electromagnetic theory and Boltzmann’s statistical mechanics - seemed capable of explaining virtually all physical phenomena. But a number of seemingly minor puzzles proved to be gaps that would completely overthrow the classical structure of physics. Density functional theory (DFT) with its early developments in the late 1920’s (Thomas and Fermi) is the first successful method avoiding the most fundamental difficulty in condensed matter theory - the many body problem. It is until now one of the most popular approaches to quantum mechanical many-body electronic structure calculations of molecular and condensed matter systems.

1.1

Milestones

1897: Joseph John Thomson discovered the electron. 1859: Gustav Kirchhoff proved a theorem about blackbody radiation. He stated that the emitted energy, E, depends only on the temperature T and the frequency ν of the emitted energy, i.e. E = J(T, ν). The new challenge for physicists, namely finding an expression for the function J, was born. 1879: Josef Stefan proposed, on experimental grounds, that the total energy emitted by a hot body was proportional to the fourth power of the temperature. 1884: Ludwig Boltzmann reached the same statement for blackbody radiation, this time from theoretical considerations using thermodynamics and Maxwell’s electromagnetic theory. The result, now known as the Stefan-Boltzmann law , does not fully answer Kirchhoff’s challenge since it is not wavelength-specific. 1896: Wilhelm Wien proposed a solution of Kirchhoff’s formula. However although 3

4

1 Historical Background

his solution matches experimental observations closely for small values of the wavelength, it was shown to break down in the far infrared by Rubens and Kurlbaum. 1900: Max Planck introduced the idea that energy is quantized, in order to derive the formula of Kirchhoff’s function J for the observed frequency dependence of the energy emitted by a black body. 1918 Planck got the Nobel Prize for Physics for this work. 1905: Albert Einstein explained the photoelectric effect by postulating that light energy comes in quanta called photons. He had correctly ”guessed” that energy changes occur in a quantum material oscillator in jumps which are multiples of ~ν, where ~ is Planck’s reduced constant and ν is the frequency. 1921 Einstein received the Nobel Prize for Physics for this work on the photoelectric effect. 1913: Niels Bohr wrote a revolutionary paper on the hydrogen atom. He discovered the major laws of the spectral lines. This work earned Bohr the 1922 Nobel Prize for Physics. 1923: Arthur Compton derived relativistic kinematics for the scattering of a photon (a light quantum) off an electron. 1924: Louis de Broglie put forward his theory of matter waves. 1925: Werner Heisenberg developed matrix mechanics and Erwin Schr¨odinger invented wave mechanics and the Schr¨odinger equation - modern quantum mechanics was born. Wolfgang Pauli formulated the exclusion principle. 1927: Werner Heisenberg formulated his uncertainty principle and the Copenhagen interpretation took shape at about the same time. Paul Dirac unified quantum mechanics with special relativity. He also pioneered the use of operator theory, including the influential bra-ket notation. Also in 1927 Niels Bohr stated that space-time coordinates and causality are complementary. 1928: Paul Dirac gave the first solution of the problem of expressing quantum theory in a form which was invariant under the Lorentz group of transformations of special relativity. He expressed d’Alembert’s wave equation in terms of operator algebra. John von Neumann last but not least formulated the rigorous mathematical basis for quantum mechanics - setting up the theory with the help of operator algebra.

Further developments towards DFT: 1927: Llewellen Hilleth Thomas (1926) and Enrico Fermi (1928) introduced the idea of expressing the total energy of a system as a functional of the total electron density. Although this was an important first step, the Thomas-Fermi equation’s accuracy was limited because it did not attempt to represent the exchange energy of an atom. 1930: Paul Dirac developed an approximation for the exchange interaction based on the homogeneous electron gas. However, the Thomas-Fermi-Dirac theory remained rather inaccurate for most applications, because it is difficult to represent kinetic energy with a density functional, it neglects electron correlation entirely.

1.1 Milestones

5

1935: Carl Friedrich von Weizs¨acker formulated the first gradient-corrected kinetic energy functional. 1936: Hans Jahn and Edward Teller present their systematic study of the symmetry types for which the Jahn-Teller effect is expected. ∼ 1938: 1937 Hans Gustav Adolf Hellmann and 1939 Richard Feynman find the Hellmann-Feynman theorem. 1951: Slater formulated its ”Xα method” - the predecessor of the modern chemical approaches to the DFT. This method was developed as an approximate solution to the HF equations. The exchange energy E xα is given here as a functional of the density and contains an adjustable parameter α, which was empirically optimized for each atom of the periodic table. Its value was between 0.7 − 0.8 for most atoms. 1954: R. G´asp´ar determined this parameter α for a special case, namely the homogenous electron gas. Its value is exactly 2/3. 1964: Pierre Hohenberg and Walter Kohn formulated two theorems that constitutes the theoretical basis of DFT. 1965: Walter Kohn and Lu Jeu Sham introduced a method which treated the majority of the kinetic energy exactly. The Kohn-Sham implementation is based upon an orbital-density description of DFT which removes the necessity of knowing the exact form of the universal functional of the kinetic energy.

6

1 Historical Background

1.2

Overview

For solving physical and/or chemical problems there are two main methods. On the one hand classical models and on the other hand techniques, based on computational quantum mechanics. The latest one is primarily concerned with the numerical computation of molecular electronic structures by ab-initio 1 and semi-empirical techniques. The following figure will give an short overview over typical methods scientist use for their calculations.

Methods classical

quantum-mechanical wf-based semi-empirical

•Force Field •Molecular Mechanics

• (Extended) Hückel =Tight Binding • Pariser-Parr-Pople • Complete Neglect of Differential Overlap • (Modified) Intermediate NDO • Modified NDO

density-based Ab-initio

• Hartree Fock • Post-HF CI CC MPn

• LDA • GGA • Meta-GGA

Figure 1.1: Overview over various methods and models for solving physical and/or chemical problems

1.2.1

Classical Methods

Models, based on this theory, like Force Field and Molecular Mechanics, use classical physics to explain and interpret the behavior of atoms and molecules. Advantages: They + rely on force-field with embedded empirical parameters. + are computationally least intensive - fast and useful with limited computer resources. + can be used for bio-molecules as large as enzymes (thousands of atoms).

1

ab-initio: Latin for ”from scratch”

1.2 Overview

7

Disadvantages: They - are applicable only for a limited class of molecules. - do not calculate electronic properties. - require experimental data (or data from ab-initio) for parameters. - calculate systems or processes with no breaking or forming of bonds. Models: MMI-4, AMBER, CHARMM, GROMOS, UFF

1.2.2

Quantum-mechanical Methods

As mentioned before, there are two main types of quantum mechanical methods, the semi-empirical techniques and the ab-initio methods, which are either based on the wave-function method, like Hartree-Fock and Post-Hartree-Fock methods, or density-functional based using the electron-density as central quantity instead of the wave-function, like the Density Functional Theory (DFT).

1.2.2.1

Semi-empirical techniques

Methods based on this theory use approximations from empirical (experimental) data to provide the input into the mathematical models. Advantages: They + are faster than ab-initio methods. + are successful in organic chemistry. Disadvantages: They - are capable of calculating transition states and excited states. - require experimental data (or data from ab-initio) for parameters. - are less rigorous than ab-initio methods. - are only useful for medium-sized systems (hundreds of atoms). Models: • TBEH: tight-binding extended H¨ uckel (extended H¨ uckel tight-binding) • PPP: Pariser-Parr-Pople • CNDO: Complete Neglect of Differential Overlap

8

1 Historical Background • INDO, MINDO (Modified) Intermediate NDO • MNDO: Modified Neglect of Differential Overlap • AM1: Austin Model 1 • PM3: Parametric Method 3

1.2.2.2

Ab-initio methods

With such models molecular structures within a self consistent field procedure can be calculated using nothing but the Schr¨odinger equation. This technique is most often used for systems involving electronic transitions and requiring rigorous accuracy. Advantages: They + are useful for a broad range of systems. + do not depend on experimental data. + allow to determine accurate forces and stresses via Hellmann-Feynman theorem. + allow to determine accurate total energies and energy differences. Disadvantages: They - are computationally expensive. - are only useful for small systems (tens of atoms). - are within the DFT strictly applicable to ground state only. Models: • HF: Hartree Fock and Post-HF • Post-HF: MPn (Møller-Plesset n-th order), CI (Configuration Interaction) and CC (Coupled Cluster) • KS-DFT: Kohn Sham Density Functional Theory

2

Chapter

Electronic structure methods I don’t like it, and I’m sorry I ever had anything to do with it. (Erwin Schr¨odinger on quantum mechanics)

T

he use of Density Functional Theory (DFT) is an inevitable tool in modern ab-initio computational physics for the description of ground state properties of metals, semiconductors, and insulators. With the help of DFT, the structure and dynamics of molecules, clusters and solids can be studied. In this chapter we will give a short insight into the most important concepts of DFT. Section 2.1 introduces into the mystery of the Schr¨odinger Equation. Section 2.2 will go into deeper details for the density functional theory, as the VASP code (Vienna Ab-initio Simulation Package), which has been written and is further developed in our group, is based on this method. In Section 2.3 various exchange-correlation functionals will be discussed and Section 2.4 will short summarizes the necessity of plane waves and pseudopotentials. Fur further information concerning VASP, see Section 2.5.

2.1

¨ dinger Equation The Schro

The ultimate goal of the most quantum-mechanical approaches, is the - approximate - solution of the multi-particle, in this case time-independent and non relativistic, Schr¨ odinger equation, formulated by Erwin Schr¨odinger (1887-1964) in the year 1926. It can be written for a molecular system consisting of M nuclei and N electrons as: ˆ i (r1 , r2 , ..., rN , R1 , R2 , ..., RM ) = Ei ψi (r1 , r2 , ..., rN , R1 , R2 , ..., RM ), Hψ

(2.1)

ˆ is the many-body Hamiltonian and the wave-function ψ is a function of where H all the electronic and nuclear coordinates, denoted by ri and Rα , respectively. A solid typically contains of the order of 1025 electrons which are all interacting and moving in the electromagnetic field of ∼ 1024 positively charged ion cores. Under the estimation of having non-relativistic effects, meaning neither the electrons nor ion 9

10

2 Electronic structure methods

cores move at velocities near the speed of light (vi ,Vα ¿ c), the Hamiltonian can be written as the sum of the non-relativistic kinetic energies and Coulomb interaction of the electrons and ion cores: ˆ = H

N M N X M X X X p2i P2α Zα + − 2m α=1 2Mα |ri − Rα | i=1 i=1 α=1

+

N X N X i=1 j>i

M

(2.2)

M

X X Zα Zβ 1 + , |ri − rj | α=1 β>α |Rα − Rβ |

where pi and Pα are the momenta of the electrons and ion cores, respectively. Mα denotes the mass of the nuclei at the position Rα and Zα the charge of that nuclei. The sums over i and j run over all the electrons and the sums over α and β run over all the ion cores. The first two parts describe the kinetic energy of the electrons (Tˆe ) and nuclei (TˆN ), respectively. The remaining three terms define the potential part of the Hamiltonian and represent the attractive Coulomb interaction between the nuclei and the electrons (VˆN e ) and the repulsive Coulomb potential energy due to the electron-electron (Vˆee ) and nucleus-nucleus interaction (VˆN N ), respectively. Therefore equation (2.2) can be rewritten as: ˆ = Tˆe + TˆN + VˆN e + Vˆee + VˆN N . H

(2.3)

In this thesis, the unit mass system has been used, which means that physical quantities are expressed as multiples of fundamental constants and, if necessary, as combination of such constants. The mass of the electron, me , the modulus of its charge, |e|, Planck’s constant h divided by 2π, ~, and 4πε0 , the permittivity of the vacuum, are all set to unity. Mass and charge are then expressed as multiples of these constants, which can be therefore dropped from all equations. The unit of energy, 1 Hartree, corresponds to twice the ionization energy of a hydrogen atom, or, equivalently, that the exact total energy of an H atom equals −0.5 Eh . Thus, 1 Hartree corresponds to 27.211 eV or 627.51 kcal/mol2 .

2.1.1

Born - Oppenheimer Approximation

The Schr¨odinger equation, equation (2.1), within the Hamiltonian, equation (2.2), can further be simplified, if we take into account, that there are significant differences between the masses of nuclei and electrons. Even the lightest nuclei, the 1 H atom weighs roughly 1.800 times more than one electron. Therefore the ratio of the electron mass to the mass of the nucleus, m/Mα is ordinarily less than 10−5 (for atoms heavier than Calcium). Thus, the nuclei move much slower than the electrons. In a dynamical sense, the electrons can be regarded as particles that follow the nuclear motion adiabatically, meaning that they are ”dragged” along with the nuclei

¨ dinger Equation 2.1 The Schro

11

without requiring a finite relaxation time. Thus the electrons are considered to be in the adiabatic equilibrium and in their ground-state with respect to the position of the nuclei at all times. This, of course, is an approximation, since there could be non-adiabatic effects that do not allow the electrons to follow in this ”instantaneous” manner, however, in many systems, the adiabatic separation between electrons and nuclei is an excellent approximation. This is the famous Born - Oppenheimer or adiabatic approximation, formulated by Max Born (1882-1970) and J. Robert Oppenheimer (1904-1967) in the year 1927. The subsequently consequence is, that the wave-function for the system can be decoupled, meaning: ψ(ri , Rα ) = ψelec (ri , Rα )ψnuc (Rα ),

(2.4)

where ψnuc is a nuclear wave-function and ψelec denotes the wave-function of the electrons that depends only parametrically on the position of the nuclei. The total energy of a system can then be written as a function of the ions’ position in phase space: ˙ α ) = T N (R ˙ α ) + VN N (Rα ) + Eelec (Rα ) Etot (Rα , R (2.5) including the kinetic as well as the Coulomb interaction energy of the nuclei and the ground-state energy of the electrons, Eelec , which consists of the kinetic energy of the electrons, (Te ), the electron-electron interaction, (Vee ), and the potential energy of the electrons in the field of the nuclei, (VN e ). It can be obtained from the Schr¨odinger equation of the electrons for fixed ionic positions: ˆ elec ψelec (ri , Rα ) = Eelec (Rα )ψelec (ri , Rα ), H

(2.6)

ˆ elec is the so-called electronic Hamiltonian: where H ˆ elec = H

M N N N X N X X X X Zα 1 p2i − + = Tˆ + VˆN e + Vˆee . 2m |r − R | |r − r | i α i j i=1 i=1 α=1 i=1 j>i

(2.7)

The attractive potential exerted on the electrons due to the nuclei - the expectation value of the second operator, VˆN e , in equation (2.7) - is often termed the ”external potential”, Vˆext , in density functional theory, even though the external potential is not necessarily limited to the nuclear field but may include external magnetic or electric fields. From now we will consider exclusively the electronic problem, and therefore drop the subscript ”elec”. The wave-function ψ itself is not observable. It can only become physically ”useful”, if the we take the square of the wave-function, |ψ(r1 , r2 , ..., rN )|2 dr1 dr2 ...drN ,

(2.8)

which represents the probability that electrons 1, 2, ..., N are found simultaneously in volume elements dr1 dr2 ...drN . Since electrons are indistinguishable, this probability

12

2 Electronic structure methods

must not change if the coordinates of any two electrons (in this case i and j) are switched: |ψ(r1 , r2 , ..., ri , rj , ..., rN )|2 = |ψ(r1 , r2 , ..., rj , ri , ..., rN )|2 . (2.9) Thus, the two wave-functions can at most differ by a unimodular complex number eiφ . It can be shown that the only possibilities occurring in nature are that either the two functions are identical (symmetric wave-function, applies to particles called bosons, which have integer spin, including zero) or that the interchange leads to a sign change (asymmetric wave-function, applies to fermions, whose spin is half-integral). Electrons are fermions with spin = 1/2 and ψ must therefore be antisymmetric with respect to interchange of the spatial and spin coordinates of any two electrons: ψ(r1 , r2 , ..., ri , rj , ..., rN ) = −ψ(r1 , r2 , ..., rj , ri , ..., rN ).

(2.10)

This represents the quantum-mechanical generalization of Pauli’s exclusion principle (”no two electrons can occupy the same state”). A consequence of the probability interpretation of the wave-function is that the integral of equation (2.8) over the full range of variables equals one, meaning that the probability of finding N electrons anywhere in space must be exactly unity, Z Z ... |ψ(r1 , r2 , ..., rN )|2 dr1 dr2 ...drN = 1. (2.11) A wave-function, which satisfies equation (2.11), is said to be normalized.

2.2

Density Functional Theory

Since the formulation of quantum mechanics in the 1920s, two major approaches have emerged for the computation of the properties of atoms, molecules and solids: Hartree-Fock theory and Density Functional Theory (DFT). The Hartree-Fock and related methods are most popular in the quantum chemistry community, while density functional theory is the common method in the quantum physics community. In this chapter the basic concepts of density functional theory are presented, beginning with an introduction into electron density, discussing the Hohenberg-Kohn Theorems and formulating the Kohn-Sham Equations.

2.2.1

Introduction

The probability interpretation of the wave-function, equation (2.8), leads directly to the central quantity of this chapter and the reason, how this theory got it’s name, the electron density n(r). It is defined as the multiple integral over the spin coordinates

2.2 Density Functional Theory over all electrons and over all but one of the spatial coordinates : Z Z n(r1 ) = N ... |ψ(r1 , r2 , ..., rN )|2 ds1 dr2 ...drN ,

13

(2.12)

where n(r) determines the probability of finding any of the N electrons within the volume element dr1 , but with arbitrary spin while the other N-1 electrons have arbitrary positions and spin in the state presented by ψ. Strictly, n(r) is a probability density, but calling it electron density is common practice. It should be noted, that the multiple integral as such represents the probability that one particular electron is with in the volume element dr1 . However, since electrons are indistinguishable, the probability of finding any electron at this position is just N times the probability for one particular electron. Clearly, n(r) is a non-negative function of only the three spatial variables, which vanishes at infinity and integrates the total number of electrons: n(r → ∞) = 0 (2.13) Z n(r)dr = N. (2.14)

Unlike the wave-function, the electron density is an observable and can be measured experimentally, e.g. by X-ray diffraction. As a typical example for the illustration of the electron density, see Fig. 2.1.

Figure 2.1: Representations of the electron density of the water molecule: (a) relief map showing values of n(r) projected onto the plane, which contains the nuclei (large values near the oxygen atom are cut out); (b) three dimensional molecular shape represented by an envelope of constant electron density (0.001 a.u.). From Ref. [1]

14

2.2.2

2 Electronic structure methods

Pair Density

The Hamilton operator, equation (2.7), consists of single electron and bi-electronic interactions. In order to compute the total energy we do not need to know the 3N dimensional wave-function, because the knowledge of the probability of finding an electron at r1 and an electron at r2 (=two-particle probability density) is sufficient. A quantity of great use in analyzing the energy expression is the second order density matrix, which is defined as: Z N (N − 1) 0 0 P2 (r1 , r2 ; r1 , r2 ) = ψ ∗ (r01 , r02 , ..., r0N )ψ(r1 , r2 , ..., rN )dr3 , ..., rN . (2.15) 2 The diagonal elements of P2 , often referred to as the two-particle density matrix or pair density, are: P2 (r1 , r2 ) = P2 (r1 , r2 ; r1 , r2 ). (2.16) This is the required two electron probability function and completely determines all two particle operators. The first order density matrix is defined in a similar manner and may be written in terms of P2 as: Z 2 0 P2 (r01 , r02 ; r1 , r2 )dr2 . (2.17) P1 (r1 ; r1 ) = N −1 If P1 and P2 are given, the total energy can be determined exactly as: ! # Z "Ã Nat X Z 1 α 2 0 ˆ Pˆ ) = E = tr(H P1 (r1 , r1 ) − ∇i − dr1 + 2 |r1 − Rα | 0 α r1 =r1 Z 1 P2 (r1 , r2 )dr1 dr2 . |r1 − r2 |

(2.18)

The diagonal elements of the first and second order density matrices determine completely the total energy. Once again, the solution of the full Schr¨odinger equation for ψ is not required, it is sufficient to determine P1 and P2 . The problem in a space of 3N coordinates has been reduced to a problem in a six-dimensional space. The problem models have, which are based on the direct minimization of E(P1 , P2 ), is, that the density matrices must be constructible from an antisymmetric ψ. This is a non-trivial problem which remains currently unsolved, too [2, 3]. To conclude that: Equation (2.18) does not lead immediately to a reliable method for computing the total energy without calculating the many body wave-function. But within density functional theory we do not even require P2 to find E, the ground-state energy can be completely determined by the diagonal elements of the first order density matrix, the charge density.

2.2 Density Functional Theory

2.2.3

15

The Energy Functional

The energy of a many-electron system given by the expectation value of the Hamiltonian can be written as a functional of the electron density, ˆ = E[n(r)], E = hHi

(2.19)

ˆ denotes the electronic Hamiltonian as defined in equation (2.7) or written where H within the expression of an external potential as:

ˆ = Tˆ + VˆN e (= Vˆext ) + Vˆee H ¶ X N X N N µ X 1 1 2 . = − ∇i + Vext (ri ) + 2 |r − r | i j i=1 j>i i=1

(2.20)

The strategy for finding the ground state energy is to minimize the functional E[n(r)] as follows: E0 = min E[n(r)] = minhψ|Tˆ + VˆN e + Vˆee |ψi, ψ,n

ψ,n

(2.21)

where minψ|n from equation (2.21) denotes a minimization with respect to the density n(r). The density which gives the lowest energy will be n0 (r) and the energy will be the true ground state energy E0 . Thus the ground-state energy can be found by minimization of the energy functional: ¯ ∂E[n(r)] ¯¯ = 0. (2.22) ∂n(r) ¯n0

This provides an enormous conceptual simplification to the problem of solving equation (2.7), because it reduces the number of degrees of freedom from 3N , where N ∼ 1024 , to the degrees of freedom of a scalar function in three dimensional space, i.e. 3. The idea of using the electron density as a fundamental quantity in the quantum theory of atoms, molecules, and solids originated in the early days of quantum mechanics with the work of Thomas [4] and Fermi [5], see following section for further information.

2.2.4

The Thomas-Fermi model

The early thinking that lead to practical implementations of density functional theory was dominated by one particular system for which near exact results could be obtained: the homogeneous electron gas. In this system the electrons are subject to a constant external potential and thus the charge density is constant. The system

16

2 Electronic structure methods

is thus specified by a single number - the value of the constant electron density n = N/V . L. H. Thomas and E. Fermi (1901-1954) studied the homogeneous electron gas in the early 1920’s [4, 5]. The orbitals of the system are, by symmetry, plane waves. If the electron-electron interaction is approximated by the classical Coulomb potential - meaning that exchange and correlation effects are neglected - the total energy functional can be readily computed [4, 5]. Under these conditions the dependence of the kinetic and exchange energy on the density of the electron gas can be extracted (Dirac [6–8]) and expressed in terms of a local functions of the density. This suggests that in the inhomogeneous system we might approximate the functional as an integral over a local functional of the charge density. Using the kinetic and exchange energy densities of the non-interacting homogeneous electron gas, this leads to: Z Z 3 2 2/3 5/3 TT F [n(r)] = (3π ) n (r)dr = C n5/3 (r)dr (2.23) 10 for the kinetic energy. If this is combined with the classical expression for the nuclear-electron attractive potential and the electron-electron repulsive potential we have the famous Thomas-Fermi expression for the energy of an atom, Z Z Z Z nr n(r)n(r0 ) 1 3 2 2/3 5/3 dr + drdr0 (2.24) n (r)dr − Z TT F [n(r)] = (3π ) 0 10 r 2 |r − r | Thus, we have the first example of a genuine density functional for the energy! In other words, equation (2.27) is a prescription for how to map a density n(r) onto an energy E without any additional information required. In particular no recourse to the wave-function is taken. The Thomas-Fermi model employs the variational principle. It is assumed that the ground state of the system is connected to the electron density for R which the energy according to equation (2.27) is minimized under the constraint of n(r1 )dr1 = N . Note, at this point we do not know either whether expressing the energy as a functional of n(r) is physically justified or whether a procedure employing the variational principle on the density is really allowed in this context. Thus, for the time being the only right of existence of the Thomas-Fermi model is that it seems reasonable!

2.2.5

The Slater Method

Another early attempt was made by John C. Slater [9] in 1951, where the electron density is exploited as the central quantity. This approach was originally constructed not with density functional theory in mind, but as an approximation to the non-local and complicated exchange contribution of the Hartree-Fock scheme. The exchange contribution stemming from the antisymmetry of the wave-function can be expressed as the interaction between the charge density of spin σ and the Fermi hole of the

2.2 Density Functional Theory

17

same spin Z Z n(r1 )hx (r1 ; r2 ) 1 dr1 dr2 (2.25) Ex = 2 r1 2 Slater’s idea was to assume that the exchange hole is spherically symmetric and centered around the reference electron at r1 and that within the sphere the exchange hole density is constant, having minus the value of n(r1 ), while outside it is zero. Since the Fermi hole is known to contain exactly one elementary charge, the radius of this sphere is given by: rs =

µ

3 4π

¶1/3

n(r1 )−1/3 .

(2.26)

The radius rs is sometimes called the ”Wigner-Seitz radius” and can be interpreted to a first approximation as the average distance between two electrons in the particular system. Regions of high density are characterized by small values of rs and vice versa. From standard electrostatics it is known that the potential of a uniformly charged sphere with radius rs is proportional to 1/rs , or, equivalently, to n(r1 )1/3 . Thus for the exchange energy functional one can write: Z (2.27) Ex [n] = Cx n(r1 )4/3 (r1 )dr1 , where Cx is a numerical constant. The non-local and complicated exchange term of Hartree-Fock theory, given in equation (2.25), has been replaced by a simple approximate expression which depends only on the local values of the electron density. Thus, this expression represents a density functional for the exchange energy. To improve the quality of this approximation an adjustable, semi-empirical parameter α was introduced into the pre-factor Cx which leads to the Xα or Hartree-FockSlater (HFS) method , 9 Exα [n] = − 8

µ ¶1/3 Z 3 α n(r1 )4/3 dr1 . π

(2.28)

Typical values of α are between 2/3 and 1. R. G´asp´ar [10] determined this parameter α for a special case in 1954, namely the homogenous electron gas. Its value is exactly 2/3. Slater’s method enjoyed a significant amount of popularity among physicists, but never had much impact in chemistry.

2.2.6

Hohenberg - Kohn Theorem

In the year 1964, P. Hohenberg and W. Kohn (1923-) published a Physical Review Paper [11], where they stated two fundamental theorems which gave birth to modern density functional theory, an alternative approach to deal with many body problem in electronic structure theory.

18

2 Electronic structure methods

The first theorem may be stated as follows: The external potential Vext (r) is (to within a constant) a unique functional of the ˆ we see that the full many particle density n(r); since, in turn Vext (r) fixes H ground-state energy is a unique functional of n(r), which means: Vext (r) ⇔ n(r).

(2.29)

It’s mapping is bijective. If this statement is true, then it immediately follows that the electron density uniquely determines the Hamiltonian operator - equation (2.20). This follows as the Hamiltonian is specified by the external potential and the total number of electrons, N , which can be computed from the density simply by integration over all space. Thus, in principle, given the charge density, the Hamiltonian operator could be uniquely determined and this the wave-functions ψ (of all states) and all material properties computed. Hohenberg and Kohn [11] gave a straightforward proof of this theorem, which was generalized to include systems with degenerate states by Levy in 1979 [12–14]. Therefore, the first theorem may be summarized by saying that: the energy is a unique functional of the density E[n(r)]. The second theorem establishes the variational principle: Given any trial density nt (r) of a N electron system in an external potential Vext (r), the total energy calculated with this density will always be larger or equal than the ground-state energy E0 , E0 ≤ E[nt (r)] = T [nt (r)] + Vext [nt (r)] + Vee [nt (r)].

(2.30)

The equal sign is only valid if the trial electron density nt (r) is equal to the real electron density n0 (r) in the ground-state. The proof of this theorem is straightforward. From the first theorem we know ˆ t , and thus the that the trial density nt (r) determines a unique trial Hamiltonian, H ˆ wave-function ψt . E[nt (r)] = hψt |H|ψt i ≥ E0 follows immediately from the variational theorem of the Schr¨odinger equation (2.22). This theorem restricts density functional theory to studies of the ground-state. The two theorems lead to the fundamental statement of density functional theory: · ¸ Z δ E[n(r)] − µ(

n(r)dr − N ) = 0.

(2.31)

The ground-state energy and density correspond to the minimum of some functional E[n(r)] subject to the constraint that the density contains the correct number of electrons. The Lagrange multiplier of this constraint is the electronic chemical potential

2.2 Density Functional Theory

19

µ. The above discussion establishes that there is a universal functional E[n(r)] (i.e. it does not depend on the external potential which represents the particular system of interest), which, if we knew its form, could be inserted into the above equation and minimized to obtain the exact ground-state density and energy.

2.2.7

Kohn - Sham Equations

From the Hohenberg-Kohn theorem using the electron density as the fundamental quantity we now know, that the ground state energy of an atomic or molecular system can be written as: µ ¶ Z n(r)VN e dr , (2.32) E0 = min F [n(r)] + n→N

where the universal functional F [n] contains the individual contributions of the kinetic energy, T [n(r)], the classical Coulomb interaction, J[n(r)], and the nonclassical portion due to self-interaction correction, exchange (i. e., antisymmetry), and electron correlation effects, Z E[n(r)] = F [n(r)] + n(r)VN e dr (2.33) Z = T [n(r)] + n(r)Vext dr + J[n(r)] + Encl [n(r)]. The only known part of this equation, is J[n(r)], the classical Coulomb interaction, also known as Hartree energy: Z Z 1 n(r)n(r0 ) J[n] = drdr0 , (2.34) 0 2 |r − r | while the explicit form of the other two contributions remain a mystery. If good approximations to these functionals could be found, direct minimization of the energy would be possible.

Kohn and Sham proposed the following approach to approximate the kinetic and electron-electron functionals [15]. They postulated an auxiliary set of one-electron orbitals, the Kohn-Sham orbitals, ϕi , of a (fictitious) system of N non-interacting electrons (that is ”electrons” which behave as uncharged fermions and therefore do not interact with each other via Coulomb interaction), moving in an effective potential Veff (see Fig. 2.2).

20

2 Electronic structure methods

In such a system the exact kinetic energy of non-interacting fermions is known from Hartree-Fock theory: N

1X hϕi |∇2 |ϕi i, TKS [n] = − 2 i=1

(2.35)

where the subscript ”KS” emphasizes that this is not the true, but the ”KohnSham” kinetic energy. Nevertheless TKS describes the a main part of the true kinetic energy (TKS < T [n]). The orbitals, ϕi , from equation (2.35) are chosen such that the expectation value attains its minimum. The connection of this artificial system to the one we are really interested in is now established by choosing the effective potential Veff such that the density resulting from the summation of the moduli of the squared orbitals ϕi exactly equals the ground state density of our real target system of interacting electrons: nS (r) =

N X i=1

|ϕi |2 = n0 (r).

(2.36)

The energy functional from equation (2.33) within the Kohn-Sham approach can be rewritten as: Z E[n] = TKS [n(r)] + n(r)Vext dr + J[n(r)] + Exc [n(r)], (2.37) where Exc [n] denotes the exchange-correlation functional: Exc [n] = (T [n] − TKS [n]) + (Eee [n] − J[n]).

(2.38)

Exc is simply the sum of the error made in using a non-interacting kinetic energy and the error made in treating the electron-electron interaction classically. Writing the functional - equation (2.38) - explicitly in terms of the density built from non-interacting orbitals, equation (2.36), and applying the variational theorem, equation (2.31), we find that the orbitals, which minimize the energy, satisfy the following set of equations: · ¸ Z 0 1 2 0 n(r ) − ∇ + νext (r) + dr + νxc (r) ϕi (r) = εi ϕi (r), (2.39) 2 |r − r0 | in which we have introduced a local multiplicative potential which is the functional derivative of the exchange correlation energy with respect to the density: νxc (r) =

δExc [n(r)] . δn(r)

(2.40)

2.2 Density Functional Theory

21

This set of non-linear equations - the Kohn-Sham equations - describes the behavior of non-interacting ”electrons” in an effective local potential. For the exact functional, and thus exact local potential, the ”orbitals” yield the exact ground-state density via equation (2.36) and exact ground-state energy via equation (2.38). The Kohn-Sham approach achieves an exact correspondence of the density and ground-state energy of a system consisting of non-interacting fermions and the ”real” many body system described by the Schr¨odinger equation. The following figure, Fig. 2.2, represents the relationship.

E,n Kohn-Sham

Figure 2.2: A cartoon representing the relationship between the ”real” many body system (left hand side) and the non-interacting system of Kohn Sham density functional theory (right hand side).

The correspondence of the charge density and energy of the many-body and the non-interacting system is exact only if the exact functional is known. In this sense Kohn-Sham density functional theory is an empirical methodology - we do not know (and have no way of systematically approaching) the exact functional. However, the functional is universal - it does not depend on the materials being studied. For any particular system we could, in principle, solve the Schr¨odinger equation exactly and determine the energy functional and its associated potential. This, of course, involves a greater effort than a direct solution for the energy. Nevertheless, the ability to determine exact properties of the universal functional in a number of systems allows excellent approximations to the functional to be developed and used in unbiased and thus predictive studies of a wide range of materials - a property usually associated with an ab-initio theory. For this reason the approximations to density functional theory, discussed in Section 2.3, are often referred to as ab-initio or first principles methods. The computational cost of solving the Kohn Sham equations, equation (2.39), scales formally as N 3 (due to the need to maintain the orthogonality of N orbitals), but in current practice is dropping towards N 1 through the exploitation of the locality of the orbitals. For calculations in which the energy surface is the quantity of primary interest, DFT offers a practical and potential highly accurate alternative to the wave-function methods discussed above. In practice, the utility of the theory rests on the approximation used for E xc [n].

22

2.3

2 Electronic structure methods

Exchange-Correlation Functionals

In the previous section we introduced the Kohn - Sham formalism which allows exact treatment of most of the contributions to the electronic energy of an atomic or molecular system, including the major fraction of the kinetic energy. All remaining - unknown - parts are collectively folded into the exchange-correlation functional Exc [n]. The generation of approximations for Exc [n] has lead to a large and still rapidly expanding field of research. There are now many different flavors of functionals available which are more or less appropriate for any particular study. Ultimately such judgments must be made in terms of results (i.e.: the direct comparison with more accurate theory or experimental data, which will be discussed below), but knowledge of the derivation and structure of functionals is very valuable when selecting which to use in any particular study.

2.3.1

Local Density Approximation (LDA)

The local exchange correlation energy per electron might be approximated as a simple function of the local charge density (say, εxc (n(r))). That is, an approximation of the form: Z LDA Exc [n] = n(r)εxc (n(r))dr. (2.41)

An obvious choice is then to take εxc (n) to be the exchange and correlation energy density of the uniform electron gas of density n(r) - this is the local density approximation (LDA). Within the LDA, εxc (n) is a function of only the local value of the density. It can be separated into exchange and correlation contributions: εxc (n) = εx (n) + εc (n).

(2.42)

The exchange part, εx , which represents the exchange energy of an electron in a uniform electron gas of a particular density is, apart from the pre-factor, equal to the form found by Slater in his approximation of the Hartree-Fock exchange and was originally derived by Bloch and Dirac in the late 1920s: 3 εx (n) = − 4

µ

3n(r) π

¶1/3

,

(2.43)

Inserting equation (2.43) into equation (2.41) retrieves the n4/3 dependence of the exchange energy indicated in equation (2.27). This exchange functional is frequently called Slater exchange and is abbreviated by ”S”. This functional form is much more widely applicable than is implied from its derivation and can be established from scaling arguments [8]. The functional form for the correlation energy density, ε c (n), is unknown. However, highly accurate numerical quantum Monte Carlo simulations of the uniform electron gas are available from the work of Ceperley and Alder [16]. On the basis of these results various authors [17–19] have presented analytical ex-

2.3 Exchange-Correlation Functionals

23

pressions of εc (n) based on sophisticated interpolation schemes.

2.3.1.1

Vosko - Wilk - Nusair (VWN)

It is the most widely used representations of εc (n), developed by Vosko, Wilk, and Nusair in 1980 [19]. VWN usually implies that the correlation energy density of the homogenous electron gas has been obtained in the random phase approximation (RPA), which was first formulated by Bohm and Pines [20]. Later Vosko [19] showed that the previous values are wrong for high densities (rs ≤ 6). The somewhat less frequently used VWN5 variant denotes the use of the parameterization scheme based upon the results of Ceperley and Alder [16].

2.3.2

The Exchange-Correlation Hole

Insight into the behavior of functionals can be obtained by examining how well they approximate P2 . A commonly used device is to convert P2 , the probability of finding an electron at r1 and an electron at r2 , into the conditional probability of finding an electron at r2 given that there is an electron at r1 : Pxc (r1 , r2 ) =

P2 (r1 , r2 ) − n(r2 ). n(r1 )

(2.44)

This quantity is the exchange correlation hole. We may think of Pxc as the hole an electron placed at r1 digs for itself in the surrounding density. There are a number of exact properties of the hole which one would hope to reproduce in approximations. For instance, for any point r1 the reduction in the surrounding density should be one electron, that is: Z Pxc (r1 , r2 )dr2 = −1. (2.45) This result follows immediately, if P2 of equation (2.44) is inserted in equation (2.17). The exact form of the exchange hole for a neon atom is plotted as a function of r2 for r1 = 0.09a0 and 0.4a0 in Fig. 2.3 and compared to that computed within the LDA (see Ref. [21]). It is clear that the LDA is a very poor approximation to P 2 .

24

2 Electronic structure methods

Figure 2.3: The exchange hole, nx (r, r0 ), for a neon atom, comparing the exact result (continuous line) to that within the LD-Approximation (dashed line). The top panel is for r = 0.09a0 and the lower for r = 0.4a0 . See [21] for reference.

We are faced with the question: how can the LDA produce such reasonable energetics if the pair correlation function is so poorly described? The answer is based on the structure of the Coulomb operator. We remember from equation 2.18 that the electron-electron interaction can be written in terms of P2 as: Z 1 1 Vee = P2 (r1 , r2 )dr1 dr2 . (2.46) 2 |r1 − r2 | From this, it seems readily apparent that a poor approximation to P2 leads directly to a poor estimate of the electron-electron interaction. However, the Coulomb operator depends only on the magnitude of the separation of r1 and r2 . Substitution of

2.3 Exchange-Correlation Functionals

25

u = r2 − r1 yields: ¸ Z ·Z 1 1 P2 (r1 , r1 + u)dr1 du = Vee = 2 u ·Z ¸ Z dΩu 1 ∞ 4πu2 du P2 (r1 , r1 + u)dr1 2 0 u 4π

(2.47)

Thus the electron-electron interaction depends only on the spherical average of the pair density - P (u): Z dΩu P (u) = P2 (r1 , r1 + u)dr1 . (2.48) 4π The exact P (u) for the neon atom is compared to that resulting from the LDA in Fig. 2.4 for the same positions of r used in Fig. 2.3.

Figure 2.4: The spherical average of the exchange hole in a neon atom, comparing the exact result (continuous line) with that computed within the LD-Approximation (dashed line). The top panel is for r = 0.09a0 and the lower for r = 0.4a0 . See [21] for reference.

26

2 Electronic structure methods

The LDA makes a reasonable approximation to the spherically averaged hole and preserves the normalization of the hole to 1 (equation (2.45)). This observation explains in part the success of the LDA. We can conclude that the remarkable performance of the LDA is a consequence of its reasonable description of the spherically averaged exchange correlation hole coupled with the tendency for errors in the exchange energy density to be cancelled by errors in the correlation energy density. An understanding of these features is an important pre-requisite to developing functionals that seek to improve on the LDA.

2.3.3

Local Spin-Density Approximation (LSDA)

The initial formulation of DFT dealt only with nonmagnetic systems, i.e. wavefunctions, densities, and potentials with only one spin-component. In this case, the approximate functionals are usually expressed in an unrestricted version, where not the electron density n(r), but the two spin-densities, n↑ (r) and n↓ (r), with n↑ (r) + n↑ (r) = n(r) are employed as the central input. In particular for open shell situations with an unequal number of spin-up (↑) and spin-down (↓) electrons, functionals of the two spin densities consistently lead to more accurate results. But also for certain situations with even number of electrons, such as the H2 molecule at larger separation, the unrestricted functionals perform significantly better because the allow symmetry breaking. Up to this point the local density approximation was introduced to a functional depending solely on n(r). If we extend the LDA to the unrestricted case, we arrive at the local spin-density approximation (LSDA). Formally the two approximations differ only in that instead of equation (2.41) we now write: Z LSDA Exc [n↑ , n↓ ] = n(r)εxc (n↑ (r), n↓ (r))dr (2.49) Just for the simple, spin compensated situation where n↑ (r)=n↓ (r)= 12 n(r), there are related expressions for the exchange and correlation energies per particle of the uniform electron gas characterized by n↑ (r) 6= n↓ (r), the so-called spin polarized case. The degree of spin-polarization is often measured through the spin-polarization parameter n↑ (r) − n↓ (r) . (2.50) ξ= n(r) ξ attains values from 0 (spin compensated) to 1 (filly spin polarized, i.e. all electrons have only one kind of spin). The exchange acts only between electrons having the same spin while the correlation describes interactions between all electrons. The exchange term is given by εx (rs , ξ) = εPx (rs ) + [εFx (rs ) − εPx (rs )]f (ξ), (2.51) where rs , the radius of a sphere whose volume is the effective volume of an electron,

2.3 Exchange-Correlation Functionals

27

is given by: rs =

µ

3 4πn

¶1/3

.

(2.52)

εFx (rs ) and εPx (rs ) in equation (2.51) denote the completely spin-polarized ferromagnetic (n↑ = n and n↓ = 0) and paramagnetic (n↑ = n↓ ) exchange energies with εPx (rs )

=

2−1/3 εFx (rs )

= −3

µ

9 32π 2

¶1/3

rs−1 .

(2.53)

The function f (ξ) from equation (2.51) is given by f (ξ) =

(1 + ξ)4/3 + (1 − ξ)4/3 − 2 . 2(21/3 − 1)

The formula for the corresponding correlation energy is: Z εc [n↑ , n↓ ] = nεc (n, ξ)dr,

(2.54)

(2.55)

but there is no simple like formula like equation (2.51) relating to εc to the limiting spin-compensated and spin-polarized correlation energies per particle, εPc , and εFc . Even εPc is a very difficult problem, only limiting cases are known in analytical form, for example: the high-density limit described by Gell-Mann et al. [22] the low-density limit described by Carr [23] and Nozieres [24] Formulas for the correlation energy obtained by the random phase approximation (RPA), done by Hedin et al. and Vosko et al., are summarized in Section 2.3.3.1 and Section 2.3.3.2, respectively. The correlation part can be also obtained from quantum Monte Carlo simulations and then be parametrized. For example Ceperley and Alder [16] had calculated the total energy for the uniform electron gas in spin-compensated and ferromagnetic states for several different values of rs , using the Monte Carlo quantum method.

2.3.3.1

Hedin - von Barth

Hedin and von Barth [18] derived the correlation potential and the correlation energy from random phase approximation (RPA) calculations and give interpolation formulas for their values not considering density-gradients (i.e. only for the LDA case). They proposed for the correlation energy: P F P εBH c (n, ξ) = εc (rs ) + [εc (rs ) − εc (rs )]f (ξ),

(2.56)

28

2 Electronic structure methods

which has precisely the same form as the exchange energy εx in equation (2.51).

2.3.3.2

Vosko, Wilk and Nusair

The formulas of VWN (see Section 2.3.1.1) usually enhance the magnetic moments and the magnetic energies compared to the values of Barth [18]. The made a careful analysis of the random phase approximation and proposed the more accurate description: ¸ · f (ξ) V WN P [1 + β(rs )ξ 4 ], (2.57) εc (rs , ξ) = εc (rs ) + α(rs ) 00 f (0) where α(rs ) is the spin stiffness, which is related to spin susceptibilities, and β(rs ) is chosen to satisfy εc (rs , 1) = εFc (rs ), namely, 1 + β(rs ) = f 00 (0)

εFc (rs ) − εPc (rs ) . α(rs )

(2.58)

Tables of α(rs ) and β(rs ) can be found in the original paper [19]. Another advance in their work is that Pad´e-approximant interpolations were made of accurate numerical calculations of uniform electron gas εPc and εFc that had been made by Ceperley and Alder [16].

2.3.4

Interpretation of the L(S)DA

For the interpretation of the LDA for the exchange correlation functional let us consider the general case of an open-shell atom or molecule. At a certain position r in this system we have the corresponding spin-densities n↑ (r) and n↓ (r). In the local spin-density approximation we now take these densities and insert them into equation (2.49) obtaining Exc (r). Thus we associate with the densities n↑ (r) and n↓ (r) the exchange and correlation energies and potentials, that a homogenous electron gas of equal, but constant density and the same spin polarizations ξ would have. This is now repeated for each point in space and the individual contributions are summed up (integrated) as schematically indicated in Figure, Fig. 2.5. Obviously, this approximation hinges on the assumption that the exchange-correlation potentials depend only on the local values of n↑ (r) and n↓ (r). This is a very drastic approximation since, after all, the density in our actual system is certainly anything but constant and does not even come close to the situation characteristic of the uniform electron gas. As a consequence one might wonder whether results obtained with such a crude model will be of any value at all. Somewhat surprisingly then, experience tells us that the local (spin) density approximation is actually not bad, but rather delivers results that are comparable to or even better Hartree-Fock approximations.

2.3 Exchange-Correlation Functionals

29

Figure 2.5: The local density approximation. From Ref. [1]

2.3.5

Gradient Expansion Approximation (GEA)

The local density approximation can be considered to be the zeroth order approximation to the semi-classical expansion of the density matrix in terms of the exact Hartree-Fock LDA density and its derivatives [25]. Hence, for many years in which the LDA was the only approximation available for Exc , density functional theory was mostly employed by solid-state physicists and hardly had an impact on computational chemistry. The situation changed significantly in the early 80’s when the first successful extensions to the purely local approximation were developed. The first logical step in that direction was the suggestion of using not only the information of the density n(r) at a particular point r, but to supplement the density with information about the gradient of the charge density, ∇n(r) in order to account for the non-homogeneity of the true electron density. The exchange-correlation energy then becomes: Z GEA Exc [n↑ , n↓ ] = nεxc (n↑ (r), n↓ (r)) + XZ ∇nσ ∇nσ0 σ,σ 0 (2.59) + Cxc (n↑ (r), n↓ (r)) 2/3 2/3 . n n 0 0 σ σ σ,σ This is called the gradient expansion approximation (GEA) in which first order gradient terms in the expansion are included. This approximation has been proposed already by Kohn and Sham in 1965 [11]. Unfortunately, if utilized to solve real molecular problems the GEA does not lead to the desired improved accuracy,

30

2 Electronic structure methods

but frequently performs even worse than the simply local density approximation. The reason for this failure is, that the exchange-correlation hole associated with a functional such as in equation (2.59) has lost many of the properties which made the LDA hole physically meaningful.

2.3.6

Generalized Gradient Approximation (GGA)

In a very straightforward elegant way this problem was solved by enforcing the restrictions valid for the true holes also for the hole of the beyond-LDA functionals. If there are parts in the GEA exchange holes which violate the requirement of being negative everywhere, where just set them to zero. Functionals that include the gradients of the charge density are collectively known as generalized gradient approximations (GGA). These functionals are the workhorse of the current density functional theory and can be generically written as: Z GGA Exc [n↑ , n↓ ] = f (n↑ (r), n↓ (r), ∇n↑ (r), ∇n↓ (r))dr. (2.60) As one will see, several suggestions for the explicit dependence of this integrand f on the densities and their gradients exist, including semi-empirical functionals which contain parameters that are calibrated against reference values rather than being GGA is usually split into its exchange and derived from first principles. In practice, Exc correlation contributions: GGA Exc = ExGGA + EcGGA (2.61) and approximations for the two terms are sought individually. The exchange part can be written as: XZ LDA GGA (2.62) − F (sσ )n4/3 = Ex Ex σ (r)dr. σ

The argument of the function is the reduced gradient for spin σ: sσ (r) =

|∇nσ (r)| 4/3

nσ (r)

,

(2.63)

where sσ is to be understood as a local inhomogeneity parameter. It assumes large values not only for large gradients, but also in regions of small densities, such as exponential tails far from the nuclei. Likewise, small values of sσ occur for small gradients, typical for bonding regions, but also regions for large density. For the function F two main classes of realizations have been put forward. The first one is based on a GGA exchange functional developed by Becke, 1988 [26]. This functional is abbreviated simply by ”B”. Functional which are related to this approach include among others, the following functionals of:

2.3 Exchange-Correlation Functionals

31

• Filatov and Thiel [FT97], 1991 [27] • Perdew [PW91], 1992 [28], see Section 2.3.6.1 for further information • Perdew and Wang [PW86], 1986 [29] • Laming, Termath, and Handy [CAM(A) and CAM(B)], 1993 [30, 31] The second class of GGA exchange functionals uses for F a rational function of the reduced density gradient. Prominent representatives are: • early functionals of Becke [B86], 1986 [32] • early functionals of Perdew [P], 1986 [29] • functional by Lacks and Gordon [LG], 1993 [33] • recent implementations of Perdew, Burke, and Ernzerhofer [PBE], 1996 [34] • implementations of PBE by Zhang and Yang [revPBE], 1998 [35] • implementations of PBE by Hammer, Hansen, and Nørskov [RPBE], 1999 [36] The corresponding gradient-corrected correlation functionals have even more complicated analytical forms and cannot be understood by simple physically motivated reasonings. Among widely used are: • counterpart of the 1986 Perdew exchange functional [P or P86] [28] • Perdew and Wang 1991 [PW91] refined the correction functional [37] • Lee, Yang, and Parr [LYP], 1988 [38]

2.3.6.1

Perdew - Wang (PW91)

The exchange-correlation functional by Perdew and Wang (PW91) [37, 39] is constructed using only quantum Monte Carlo data for the uniform electron gas and exact sum rules and limiting conditions. PW91 includes a real space cut-off also for the correlation functional and takes the Becke exchange with only small refinements. It fulfills almost all of the scaling relations known, including such that were only found after the functional’s formulation. In principle, each exchange functional could be combined with any of the correlation functionals, but only a few combinations are currently in use. The exchange part is almost exclusively chosen by Becke’s functional, which is either combined with Perdew’s 1986 correlation functional, namely, the BP86 functional or the Lee,Yang, Parr one, leading to the BLYP functional, respectively. Sometimes also the PW91 correlation functional is employed, corresponding to BPW91.

32

2.3.7

2 Electronic structure methods

Meta-GGA

Recently functionals that depend explicitly on the semi-local information in the Laplacian of the spin density or of the local kinetic energy density have been developed [40–42]. Such functionals are generally referred to as meta-GGA functionals. The form of the functional is typically: Z M GGA Exc = n(r)εxc (n, |∇n|, ∇2 n, τ )dr, (2.64) where the kinetic energy density τ is: τ=

1X |∇ϕi |2 . 2 i

(2.65)

Meta-GGA can achieve high accuracy, though many of the constructed functionals are semi-empirical by depending on many parameters fitted to chemical data. An example is the functional by van Voorhis (VS98) [43], including 21 fit variables. Such functionals are not only objectionable from an aesthetic point of view but typically fail for the uniform electron gas (as is the case also for VS98) and more generally for solids.

2.3.7.1

(Meta-)GGA in the spinpolarized case

In the case of generalized gradient approximations the expressions for the spinpolarized correlation-potential and correlation-energy have been given individually by the authors of the respective functionals, usually simply extending the LDA formulas. The same is true for meta-GGA functionals, e.g. in the PKZB meta-GGA functional the correlation-energy of the PBE GGA is extended and is self-interaction free. It has to be stressed in this context that in the case of non-selfconsistent implementations of exchange-correlation functionals they do not change the magnetic moments obtained from the preceding self-consistent calculation.

2.3.8

Hybrid Functionals

There is an exact connection between the non-interacting density functional system and the fully interacting many body system via the integration of the work done in gradually turning on the electron-electron interactions. This adiabatic connection approach [44–46] allows the exact functional to be formally written as: Z 1 Z λe2 1 0 dλ Exc [n] = drdr [hn(r)n(r0 )in,λ − n(r)δ(r − r0 )] , (2.66) 2 |r − r| λ=0

2.3 Exchange-Correlation Functionals

33

where the expectation value hn(r)n(r0 )in,λ is the density-density correlation function and is computed at density n(r) for a system described by the effective potential: Vef f = VN e +

1 X λe2 . 2 i6=j |ri − rj |

(2.67)

Thus, the exact energy could be computed if one knew the variation of the densitydensity correlation function with the coupling constant, l. The LDA is recovered by replacing the pair correlation function with that for the homogeneous electron gas. The adiabatic integration approach suggests a different approximation for the exchange-correlation functional. At λ = 0 the non-interacting system corresponds identically to the Hartree-Fock ansatz, while the LDA and GGA functionals are constructed to be excellent approximations for the fully interacting homogeneous electron gas - that is, a system with λ = 1. It is therefore not unreasonable to approximate the integral over the coupling constant as a weighted sum of the end points - that is: λ=1 GGA Exc ≈ aExλ=0 + bExc = aExF ock + bExc (2.68) with the coefficients are to be determined by reference to a system for which the exact result is known. Becke adopted this approach [47] in the definition of a new functional with coefficients determined by a fit to the observed atomization energies, ionization potentials, proton affinities and total atomic energies for a number of small molecules [47]. The resultant (three parameter) energy functional is: B3 Exc = (1 − 0.2)ExLSDA + EcLSDA + 0.2Exλ=0 + 0.72δExB88 + 0.81δEcP W 91

(2.69)

This is the so-called B3PW91 functional, where δExB88 and δEcP W 91 are widely used GGA corrections [37, 48] to the LDA exchange and correlation energies respectively. Hybrid functionals of this type are now very widely used in chemical applications with the B3LYP functional, where the exchange-correlation functional is given by: B3LY P Exc = (1 − 0.2)ExLSDA + 0.2Exλ=0 + 0.72δExB88 +0.81EcLY P + (1 − 0.81)EcV W N ,

(2.70)

where the energy terms are the Slater exchange, the Hartree-Fock exchange, Becke’s 1988 exchange functional correction, the gradient-corrected correlation functional of Lee, Yang and Parr (LYP) [38], and the local correlation functional of Vosko, Wilk and Nusair (VWN) [19], respectively. Computed binding energies, geometries and frequencies are systematically more reliable than the best GGA functionals.

34

2.4

2 Electronic structure methods

Plane Waves and Pseudopotentials

Density functional methods aims to compute properties of interest without recourse to experimental data. Doing this requires finding the wave-function. As this is generally unknown, it is usual to expand it in terms of a set of known functions.

2.4.1

Plane Waves

A single electron wave-function can be written as: ψi (r) =

∞ X

cj φj (r),

(2.71)

j=1

where φj are members of a complete set of functions. Obviously it is impossible to use an infinite number of basis functions so the sum in equation (2.71) is taken over a finite number of functions. This introduces another source of error into the calculations as it is not then possible to describe components of along the missing functions. Any family of functions could, in principle, be used as basis functions. Ideally the basis functions should have the same limiting behavior as the real wavefunction, for an isolated atom or molecules they should decay to zero, and they should be computationally inexpensive. For periodic systems the potential has the property: V (r + ma) = V (r), (2.72) where a is a lattice vector and m is an integer. Using Bloch’s theorem [49] the wave-function can be written as a product of a cell periodic part and a wavelike part: ψi (r) = eikr fi (r). Due to its periodicity f (r) can be expanded as a set of plane waves: X fi (r) = ci,G eiGr ,

(2.73)

(2.74)

G

where G are reciprocal lattice vectors. Thus the electronic wave-functions can be written as: X ψi (r) = ci,G ei(k+G)r . (2.75) G

As in the localized case an infinite number of basis functions would be needed to exactly recreate the real wave-function. The number of wave-functions used is controlled by the largest wave-vector in the expansion in equation (2.74). This is equivalent to imposing a cut-off on the kinetic energy, as the kinetic energy of an

2.4 Plane Waves and Pseudopotentials

35

electron with wave-vector k is given by: Ek =

~2 |k|2 . 2m

(2.76)

Plane-wave basis sets can also be applied to non-periodic systems such as isolated molecules or clusters. To accomplish this, the molecule is placed at the center of a periodic supercell. If the supercell is large enough the interactions between the molecules in neighboring cells becomes negligible. This is illustrated in Fig. 2.6.

Figure 2.6: Supercell geometry for an isolated molecule (icosahedron). The dashed line encloses the periodic supercell.

As wave-functions in isolated molecules bear little resemblance to plane waves many more plane waves are needed compared to a set of STOs (Slater Type Orbitals) or GTOs (Gaussian Type Orbitals). However, the Kohn-Sham equations take on a very simple form using plane waves [50]: ( X ~2 |k + G|2 δGG’ + VN e (G − G’) 2m G0 ) +Vee (G − G’) + Vxc (G − G’) ci,k+G’ = ci,k+G εi ,

(2.77)

where VN e (G − G’), Vee (G − G’), and Vxc (G − G’) are the Fourier transforms of the electron-nuclei, electron-electron Coulomb, and exchange-correlation potentials.

36

2 Electronic structure methods

Plane-wave basis sets also have a few other advantages over Slater or Gaussian functions. These include: • A plane-wave basis set is unbiased, it does not assume any preconceptions of the form of the problem. • Due to Bloch’s theorem plane waves are the natural choice for the representation of electron orbitals in a periodic system. • The kinetic energy operator is diagonal in a plane-wave representation. Similarly the potential is diagonal in real space. The use of Fast Fourier Transforms (FFT) in changing between these representations provides a large saving in computational cost. • As a plane-wave basis set is non-local, meaning plane waves do not depend on nuclear positions, so, unlike localized basis sets, correction terms are not needed for the calculation of forces. The principle disadvantage of a plane-wave basis set is its inefficiency. The number of basis functions needed to describe atomic wave-functions accurately near to a nucleus would be prohibitive.

2.4.2

Pseudopotentials

When using a plane-wave basis set, the region close to an atomic nucleus requires special attention. This is due to two main factors. The first is that the electronnucleus potential varies as 1r , so it diverges as r → 0. Secondly, to ensure the valence electron wave-functions are orthogonal to the core electron wave-functions (as required by the exclusion principle), the valence wave-functions must oscillate rapidly within the core region. These factors lead to large kinetic energies hence the need for large numbers of plane waves. Also a large number of plane waves are needed to describe the tightly bound core states. These problems can be circumvented by the use of the pseudopotential approximation. If all of the electrons in a system were explicitly included when performing a calculation and Vext constructed from the full Columbic potential of the nuclei, the computational cost would still be prohibitive using a plane-wave basis set. The rapid oscillations of the wave-functions near to the nucleus, due to the very strong potential in the region and the orthogonality condition between different states, mean that a very large cut-off energy, and hence basis set, would be necessary. Fortunately, the study of physics and chemistry shows that the core electrons on different atoms are almost independent of the environment surrounding the atom and that only the valence electrons participate strongly in interactions between atoms. Thus, the core electron states may be assumed to be fixed and a pseudopotential may be constructed for each atomic species which takes into account the effects of the nucleus and core electrons [51–53]. This is the so-called frozen-core approximation.

2.4 Plane Waves and Pseudopotentials

37

The core electrons are calculated for a (in general spherical) reference configuration and are kept fixed thereafter. The wave-functions for the valence electrons are ”pseudisized” to give the same energy levels as the all-electron wave-functions. The pseudo wave-functions differ from the all-electron wave-functions only inside a region around the nucleus and are constructed to be node-less (see Fig. 2.7). If the pseudowavefunction still contained nodes they would not describe the lowest valence-state. It can simply be seen that the removal of nodes from the all-electron wave-functions has no effects on the scattering properties of the atom for valence electrons, since the number of nodes of the valence wave-functions in the core region only add multiples of π to the phase shift δ. The nodes in the valence wave-functions are necessary to make valence states orthogonal to the core-states. Node-less pseudo-wave-functions reduces the number of required plane waves considerably.

The pseudo-wave-functions corresponding to this modified potential do not exhibit the rapid oscillations of the true wave-functions, dramatically reducing the number of plane waves needed for their representation. The calculations then need only explicitly consider the valence electrons, offering a further saving in effort.

A pseudopotential is constructed such that it matches the true potential outside a given radius, designated the core radius. Similarly, each pseudo-wavefunction must match the corresponding true wave-function beyond this distance. In detail the procedure therefore is as follows: • Solve the spherical radial Schr¨odinger equation for an atomic or ionic reference configuration −

1 d2 l(l + 1) (rψnl (r)) + ψnl (r) + Vsc ψnl (r) = ²nl ψnl (r) 2 r dr r2

(2.78)

within the framework of density functional theory. This yields the self-consistent potential Vsc as well as the all-electron energy eigenvalues and wave-functions. • Inside some core radius rc the valence wave-functions ψnl (, φv in Fig. 2.7) are replaced by node-less pseudo-wavefunctions ψpseudo (, φpseudo in Fig. 2.7). Outside the radius rc the all-electron and the pseudo-wavefunction are identical. • The corresponding pseudopotential is found by the ”inversion” of the spherical Schr¨odinger equation for the pseudo-wavefunctions. Obviously one gets different pseudopotentials for each spherical quantum number l. To combine them to a single potential, projection operators are used that subsequently also enter the energy functional, see equation (2.37).

38

2 Electronic structure methods

φpseudo φv rC

r

Vpseudo -Z/r

Figure 2.7: A schematic illustration of all-electron (dashed lines) and pseudo- (solid lines) potentials and their corresponding wave-functions. The radius at which all-electron and pseudopotential values match is rc . From Ref. [50], redrawn by S. Dennler [54].

The pseudo-wavefunction inside the core region is to be chosen as to reproduce the scattering properties of the all-electron wave-function in an energy window around the atomic reference energy. A central role in this context plays the logarithmic derivative xl² of the wave-function and its energy dependence, xl² (r) =

d 1 d ln (ψl² (r)) = ψl² (r), dr ψl² (r) dr

(2.79)

since it essentially describes the scattering properties. Topp and Hopfield [55] have found out that the requirement of an identical energy derivative of xl² for all-electron and pseudo wave-function at rc is equivalent to the two wave-functions having the same norm inside the core. This idea was extended to the general concept of the norm-conserving pseudopotentials (NCPP) by Hamann et al. [56]. The atomic properties of the element must be preserved, including phase shifts on scattering across the core. Thus the integral of the squared amplitudes of the real and the pseudo-wavefunctions over the core region must be identical. The norm-conserving pseudopotentials work well for all elements except for the first period and the 3d transition metals. In those elements the requirement of norm-conservation prohibits the use of large cutoff radii and soft pseudopotentials, therefore the energy cutoff for the plane waves still lies at 30-70 Rydberg.

2.4 Plane Waves and Pseudopotentials

39

The reason for this is that for those elements the all-electron valence wavefunctions are already node-less, consequently the construction of the pseudo-potential only shifts the maximum of the valence wave-function and does not improve it considerably. To work around this problem Vanderbilt introduced a ”ultra-soft” pseudopotential, see Section 2.4.2.1. The use of plane-wave basis sets with pseudopotentials is often referred to the Pseudopotential Plane Wave Method [57]. While it originally arose for the study of crystalline systems, it has also been applied to non-periodic systems such as molecules [58] and polymers [59]. The pseudopotential plane-wave method is also commonly used in ab-initio molecular dynamics simulation schemes such as the Car-Parrinello method [60].

2.4.2.1

Ultra-soft Pseudopotentials

As mentioned above for transition metals and elements of the first period energy cut offs of 30-70 Rydberg are necessary, to obtain good results. The reason therefore lies in the concept of the norm-conservation, because this restriction forces the pseudowavefunction to vary rapidly (i.e. a larger basis set must be used). The only compromise one can make to improve the smoothness of the pseudo wave-function is to increase the matching radius, but this always decreases the quality of the PP. In general the minimum basis set size for transition metals is around 400 plane waves per atom making calculations very expensive. To solve these problems, Vanderbilt [61] has developed a new concept for socalled Ultra-soft (US) pseudopotentials, improving previous pseudopotentials in several ways. The two most important ones are the removal of the constraint of norm-conservation as to optimize smoothness and the matching of the logarithmic derivatives at several energies to improve transferability. The drawbacks are twofold: On one hand the hermiticity of the nonlocal pseudopotential operator can only be preserved by the introduction of an overlap operator in the Hamiltonian. On the other hand the non-conservation of the charge density leads to the need for augmentation charge density operators for the valence wave-functions. However the reduction of the number of plane waves by a factor of 3–4 largely outweighs the small computational overhead. Ultra-soft (US) pseudopotentials give results which are very close to or indistinguishable from results obtained with the best all electron first principles methods currently available. For details of the construction of the pseudopotentials used in VASP, see Kresse [62].

40

2 Electronic structure methods

2.4.2.2

The PAW Method

The projector augmented wave method1 is an all-electron electronic structure method, which allows accurate electronic structure calculations and ab-initio molecular dynamics simulations on the basis of density functional theory. It has been developed by Peter Bl¨ochl in the year 1994 [63] in response to the invention of the ab-initio molecular dynamics approach. Whereas the Car-Parrinello method [60], as mentioned above, was based on the plane-wave pseudopotential approach, a new method was needed to enhance the accuracy and computational efficiency of the approach and to provide the correct wave-functions, rather than the fictitious wave-functions provided by the pseudopotential approach. The PAW method describes the wave-function by a superposition of different terms: There is a plane-wave part, the so-called pseudo wave-function, and expansions into atomic and pseudo atomic orbitals at each atom. On one hand, the plane-wave part has the flexibility to describe the bonding and tail region of the wave-functions, but used alone it would require prohibitive large basis sets to describe correctly all the oscillations of the wave-function near the nuclei. On the other hand, the expansions into atomic orbitals are well suited to describe correctly the nodal structure of the wave-function near the nucleus, but lack the variational degrees of freedom for the bonding and tail regions. The PAW method combines the virtues of both numerical representations in one well-defined basis set. In detail the PAW method starts from a simple linear transformation that con˜ which nects the exact valence wave-function |ψi to a pseudo (PS) wave-function |ψi is the one expanded into plane waves, X X ˜ − ˜ + ˜ |ψi = |ψi |φ˜N,i ih˜ pN,i |ψi |φN,i ih˜ pN,i |ψi. (2.80) N,i

N,i

All quantities related to the (PS) representation of the wave-function are indicated by a tilde. The φ are local wave-functions, the index N goes over all sites, i over the quantum numbers n, l, and m. The p˜i are localized projector functions that have to fulfill the condition X ˜ pi | = 1. |φih˜ (2.81) i

Defining the character cN,i of an arbitrary wave-function ψ˜ at the atomic site N as ˜ cN,i = h˜ pN,i |ψi,

(2.82)

the pseudo- and all-electron wave-functions at an atomic site N can be easily recon-

1

Homepage of P. Bl¨ochl: http://www.pt.tu-clausthal.de/atp/

2.4 Plane Waves and Pseudopotentials

41

structed from the plane-wave expanded pseudo wave-functions as X |ψ˜N i = |φ˜N,i icN,i and

(2.83)

i

|ψN i =

X i

|φN,i icN,i .

(2.84)

The decomposition of the all-electron wave-function in the PAW method is schematically displayed in Fig. 2.8.












 























 

 

 

 















 


 


 


  





 
 
  

 

 

  
 
 
   = 


 


 


  

 

 

  

 

 

  


 


 


  

 

 

  AE

− pseudo



























  
























   


 






 


  + 









   




 

 

  



 



 



 

 





 


 


  


 

 

  pseudo−onsite

AE−onsite

FigureP2.8: Decomposition of the all-electron wave-function in the PAW method, |ψi = ˜ − |φ˜N,i icN,i + P |φN,i icN,i , (after [64]). |ψi

The variational quantities that need to be determined during the ground state ˜ However, contrary to the ”simple” pseucalculation are the PS wave-functions ψ. dopotential methods the operators A representing physical quantities need to be consistently extended to their all-electron forms, ³ ´ X A˜ = A + |˜ pN,i i hφN,i |A|φN,j i − hφ˜N,i |A|φ˜N,j i h˜ pN,j |. (2.85) N,i,j

Equation (2.85) is valid for local and quasi-local operators such as the kinetic energy. Truly nonlocal operators need a special treatment but are also tractable. Of course, one does not want to make two electronic structure calculations - one using plane waves and one with atomic orbitals - and thus double the computational effort. Therefore, the PAW method does not determine the coefficients of the ”atomic orbitals” variationally. Instead, they are unique functions of the plane-wave coefficients. It is possible to break up the total energy, and most other observable quantities, into three almost independent contributions: one from the plane-wave part and a pair of expansions into atomic orbitals on each atom. The contributions from the atomic orbitals can be broken down furthermore into contributions from each atom, so that strictly no overlap between atomic orbitals on different sites need to be computed. The PAW method is in principle able to recover rigorously the density func-

42

2 Electronic structure methods

tional total energy, if plane-wave and atomic orbital expansions are complete. This provides us with a systematic way to improve the basis set errors. The present implementation uses the frozen core approximation, even though the general formalism allows extensions in this respect. It provides the correct densities and wavefunctions, and thus allows us to calculate hyperfine parameters etc. Limitations of plane-wave basis sets to periodic systems (crystals) can easily be overcome by making the unit cell sufficiently large and decoupling the long-range interactions [65]. Thus the present method can be used to study molecules, surfaces, and solids within the same approach.

2.5

The Vienna Ab-Initio Simulation Package (VASP)

In the following Section, I will outline the key features of the Vienna Ab-initio Simulation Package (VASP), as it is the tool of our group, namely the Computational Materials Science group at the Materials Physics Institute of the University of Vienna, solving chemical and/or physical problems. VASP is a density functional code developed by Georg Kresse and J¨ urgen Furthm¨ uller [66–69], and maintained and extended under the direction of Georg Kresse by many members from the above mentioned group. The first parts of the code have been written in the early 90’s of the last century, currently the code consists of more than 120000 lines written in Fortran90. VASP solves the Kohn-Sham equations of local density or spin-density functional theory, equation (2.40), iteratively within a plane-wave basis-set. The electronic ground-state is determined either by a conjugate gradient algorithm as optimized by Teter [70], by a blocked Davidson scheme as first proposed by Davidson [71], or via an unconstrained band-by-band matrix-diagonalisation scheme based on a residual minimization method (RMM) [68, 72]. The interaction between ions and electrons is described by ultra-soft Vanderbilt pseudo-potentials (US-PP) or by the projector-augmented wave (PAW) method. US-PP (and the PAW method) allow for a considerable reduction of the number of plane waves per atom for transition metals and first row elements. The approach implemented in VASP is based on the (finite-temperature) local-density approximation with the free energy as variational quantity and an exact evaluation of the instantaneous electronic ground state at each MD time step. VASP uses efficient matrix diagonalisation schemes and an efficient Pulay/Broyden charge density mixing. These techniques avoid all problems possibly occurring in the original Car-Parrinello method, which is based on the simultaneous integration of electronic and ionic equations of motion. Forces and the full stress tensor can be calculated with VASP and used to relax atoms into their instantaneous ground-state.

2.5 The Vienna Ab-Initio Simulation Package (VASP)

43

Besides the pure local density approximation (LDA) for the exchange-correlation functional, the following gradient corrected functionals as presented in Section 2.3 are implemented in VASP to account for the non-locality in exchange and correlation: • LM • BP • PW91 • PBE • RPBE The PKZB meta-GGA functional is included in a non-selfconsistent way based on orbitals obtained by a selfconsistent PBE calculation. Implementation of hybrid functionals is in its initial stages. Another important issue is the energy band dispersion in the Brillouin-zone. Sampling in the reciprocal space is done on points of Monkhorst-Pack special grids [73]. For the integration over the Brillouin zone the tetrahedron method with Bl¨ochl corrections [74] and a generalized Gaussian smearing [75] are available among other less involved methods. Fig. 2.9 shows a brief flow chart of a single self-consistency cycle in VASP. After the determination of the electronic ground-state for a given ionic configuration the forces on the atoms can be evaluated, leading to the absolute energetic minimum of a system. Thereby it is possible to constrain each cartesian direction of every single ion individually. Further information about VASP can be found at http://cms.mpi.univie.ac.at/vasp/.

44

2 Electronic structure methods

¦

¦ ¦

trial-charge ρin and trial-wave-vectors φn

¦

¦ ¦

?

Hartree- and XC-potential and d.c. set up Hamiltonian ?

subspace-diagonalisation φn0 ⇐ Un0 n φn iterative diagonalisation, optimize φn subspace-diagonalisation φn0 ⇐ Un0 n φn ?

new partial occupancies fn new (free) energy E =

P

n ²n f n −

d.c. −σS

?

new charge density ρout (r) =

P

n

fn |φn (r)|2

mixing of charge density ρout , ρin ⇒ new ρin

? (((hhh

hhh ( no h hhh (((( ∆E < E (h hhhh break (((( ( ( hh ( ( h(

Figure 2.9: Typical flow-chart of VASP mination of the Kohn-Sham ground-state. http://cms.mpi.univie.ac.at/vasp/vasp/vasp.html) d.c. . . . double-counting corrections

for

the (from

selfconsistent VASP the

deterguide,

3

Chapter Cluster and Nanostructures

C

lusters are finite aggregates of atoms or molecules that are bound by forces, which may be metallic, covalent, ionic, hydrogen-bonded or van-der-Waals in character and can contain from a few to tens of thousands of atoms. In Section 3.1 we will give a brief introduction, followed by an overview over the structures, Section 3.2. Section 3.3 provides an insight into the production of clusters and Section 3.4 deals with the question of experimental measurement of clusters, a.o. photoelectron spectroscopy, as spectroscopic studies of clusters provide invaluable information related to their geometric and often electronic structure. The last Section outlines the magnetic properties of systems of small size, as this is a topic of considerable current interest.

3.1

Introduction

The field of cluster science has become a very active field of research during the last decade as experimental and theoretical techniques have advanced and computational power has increased. The interest of studying clusters is inspired by both the large number of basic problems, to which studies of clusters have provided new insights, as well as the fast growing field of areas, clusters can be applied to. Especially the fact that aggregates of very small dimensions posses properties that often differ significantly from those of the bulk material fascinates researchers to deal with microscopic particles. Clusters typically display behavior that is not fully characterized as being a solid, liquid or gas. Sometimes they are referred to be a new state of aggregation. Furthermore, their study provides insights into the possibility of using clusters as building blocks for assembling new nanoscale materials in the future. In order to bring these prospects to fruition, detailed investigations of their unique physical and chemical behavior are being actively pursued.

3.1.1

Clusters between Atom and Bulk

As explained above, cluster are, per definition, aggregates of atoms or molecules with regular and arbitrarily scalable repetition of basic building blocks. They are 45

46

3 Cluster and Nanostructures

closing the intermediate gap between atoms and bulk. Cluster can be characterized for example with the formula: Xn (3 . n . 105−7 ),

(3.1)

where the upper limit is hard to define. In contradiction to clusters, molecules usually have a well defined composition and structure, like C6 H6 . Such systems have only a small number of isomers. For even a small cluster, on the other hand, one can find dozens of isomers, like Davis et al. [76] found hundreds of isomers for Ar13 , where the actual number is slightly depending on the detail of the interatomic potential used. The same applies for metal clusters. The reason therefore is the softness of the binding. It is obvious, that within such a number of isomers it’s hard to find the energetically most stable structure. We have already mentioned that clusters have different properties compared to their own bulk, which can easily be seen from the level structure. Finite systems are basically characterized by discrete levels, at least in the low energy part of their spectrum. As known from solid state calculations, single-electron spectra in solids come along in bands each containing a continuum of levels. When more atoms are brought together, this splitting process continues and produces more and more levels which turn out to be closer and closer to each other, the larger the number of involved atoms. There is a thus a continuous path between a fully discrete molecular level scheme and a continuous level. Passing from a discrete to a continuous set of levels is, in fact, also a continuous process (see Fig. 3.1).

Even tough, in a first order approximation, small clusters can be studied with chemistry methods (like atoms) and large ones with techniques coming from solid state physics, more specific techniques have to be developed for accessing clusters. As one can see, size is the key parameter in cluster physics. Considering the fraction of surface atoms to the volume, clusters can be classified as follows: observable number of atoms N diameter d surface fraction f

very small clusters 2 ≤ N ≤ 20 d ≤ 1 nm undefined

small clusters 20 ≤ N ≤ 500 1 nm ≤ 3 nm 0.9 & f & 0.5

large clusters 500 ≤ N ≤ 107 3 nm ≤ 100 nm f ≤ 0.5

Table 3.1: Schematic classification of clusters according to the number N of atoms. As a complement the diameter d for Na clusters is given (second row), together with an estimate of the ratio of surface to volume atoms f (surface fraction, in third row). After [77].

3.1 Introduction

47

majority spin

States [1/eV]

Pd 2

minority spin

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

E-EFermi

States [1/eV]

majority spin

Pd 13

minority spin

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

E-EFermi 6.0 Pd(c)-s Pd(c)-p Pd(c)-d Pd(s)-s Pd(s)-p Pd(s)-d

States [1/eV]

5.0 4.0 3.0 2.0 1.0 0.0 -6.0

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

E-EFermi

Figure 3.1: Total spin-polarized differential density of states for Pd2 (top panel) and for Pd13 (mid panel). Local density of states of Pd (100)(bottom panel). Solid blue line represent surface Pd and black one subsubsurface Pd, which is estimated to be bulk-like in this context.

48

3 Cluster and Nanostructures

As already mentioned above, convergence towards the bulk value with increasing cluster size essentially depends on the nature of the observable and on the resolution with which one looks at it. The experimental resolution of photo-electron spectra puts the transition from well separated discrete electron levels to quasi-continuous bands at about N ≈ 100. Electronic shell effects, as e.g. magic HOMO-LUMO gaps, shrink ∝ N 1/3 . Their importance has triggered large efforts to resolve these up to the range of N ≈ 3000. Atomic shell effects have been resolved up to N ≈ 10000. The peak frequency of optical response converges also towards its bulk value with a term linear in N −1/3 . This means that colors keep drifting with size up to very large clusters in the range > 10000.

3.2

Structures of Clusters

There are, of course, various types of clusters in terms of bonding, depending on the nature of the atoms entering the cluster. Simple molecules as well as bulk can be grouped into the same four classes of bonds: metallic, covalent, ionic, and van-der-Waals bonding. It is natural then to classify bonding in clusters according to these four classes. It reflects deeper physical processes, even if these processes may sometimes differ from the ones observed either in molecules or in the bulk. The famous C60 fullerene provides an example of covalent bonding in clusters. In that case the cluster exhibits a well defined structure with electrons localized along the various links between the atoms. Van-der-Waals bonding prevails in rare-gas clusters, as e.g. for Ar. This cluster corresponds to a closed atomic shell. The material keeps the electrons tightly bound to each mother atom. Clusters with ionic bonding have been in the focus of many studies over a few years, in particular in view of potential applications in photography (e.g. AgBr clusters). The last class of clusters is made out of atoms which exhibit metallic bonding. The simplest example is here the case of alkaline atoms for which the metallic behavior is well assessed and simple to work out. The following figure, Fig. 3.2, shows the N a4 cluster, which has a planar geometry. Both electron density and ionic positions are explicitly represented. As expected in a metallic system, the electronic density extends more or less smoothly over the whole system. The small N a4 cluster exhibits a clearly oblate shape. Correlatively, the electron cloud is strongly oblate, with a shape very similar to the ionic shape. Larger metal clusters tend to favor spherical shapes due to the surface tension of the electron cloud. In any case, the electrons have a large mobility and behave almost like an electron gas in a (spherical) container. Large metal clusters thus provide an ideal basis for the realization of the old Mie idea on the optical response of metallic spheres. The following table, Table 3.2, gives a brief summary over the four bond types mentioned above.

3.2 Structures of Clusters

49

Table 3.2: Classification of binding in clusters. For each one of the four types of bonding, examples of clusters (second column), the nature (third column) and typical binding energies (last column) are given. See [78] for Ref. [77]. Type Ionic clusters

Examples (N aCl)n , N an Fn−1 ...

Covalent clusters

C60 , Sn ...

Clusters of simple and noble metals Transition metal clusters van-der-Waals

N an , Aln , Agn ... P dn , P tn , Rhn ... rare gas clusters Arn , Xen , ...

Nature of binding ionic bonds; strong binding covalent bonding; strong binding metallic bond; moderate to strong binding metallo-covalent bond; moderate to strong binding polarization effects; weak binding

Binding energy 2 - 4 eV 1 - 4 eV 0.5 - 3 eV 0.5 - 3 eV . 0.3 eV

Whatever cluster type, one can see that binding energies lie in the eV range. Elementary bond lengths typically take values of a few a0 . Both these quantities thus fix the range of energies and distances characteristic of cluster physics.

Figure 3.2: Equidensity plot of the electronic density of N a4 computed with LDA. Plotted is the density in the x-y plane integrated over all z. The ionic positions are marked by small diamonds. The x and y scales are in units of a0 . See [79] for Ref.

50

3.2.1

3 Cluster and Nanostructures

Shell Structure of Simple Metal Clusters

The case of Na, see Fig. 3.3, shows an abundance for neutral Na clusters, which had been measured first by Knight et al. [80].

Figure 3.3: Experimental abundances of Na clusters, after [81].

One can see in Fig. 3.3 the pronounced peaks at mass number N = 8, 20, 40, and 58. These can be associated to fully occupied electronic shell in many-electron system enclosed in a spherically symmetric confining potential containing the same number of valence electrons (2, 8, 20,..), in this case of sodium in which Na atoms each provide 1 valence electron in the cluster. A calculation for mid-size cluster confirms the magic N = 20 as well as 40. The binding is strongest if the number of electrons matches a full electronic shell. For clusters, the strong binding makes the magic systems particulary stable in an evaporative ensemble and this yields then the high peaks in the abundance spectra. A similar reasoning applies for the dissociation energy, D(N ) = E(N ) − E(N − 1), the energy required to separate one monomer from a cluster of N atoms. Knights et al. discovered magic shells up to N = 92. This has triggered a long search for larger clusters with possible magic shells, see [82] for further information. Electronic shell effects can be found in simple metal clusters, but the actual sequence of magic electron numbers depends on the material. For the special case of transition metal clusters, see Chapter 4.

3.2.2

Shapes

The shapes of clusters depend, of course, on the details of binding. The binding, on the other hand, emerges from a subtle interplay between ionic and electronic effects. Thus we have the typical situation of molecular physics: little can be said in general, most depends on the actual situation. As examples two limiting cases, namely the

3.2 Structures of Clusters

51

shapes emerging from atomic arrangement and shapes dictated by electronic shell effects will be briefly discussed below.

3.2.2.1

Ionic Shapes

The cleanest cases for atom-atom potentials are rare gases. The interaction is purely van-der-Waals like and spherically symmetric. A sequence of Ar clusters is shown in the following figure, Fig. 3.4.

Figure 3.4: Structure of Ar clusters for various sizes. The larger clusters in the lower part represent atomic shell closures for icosahedral symmetry. See [78] for reference.

For clusters with more than 7 atoms one obtains a pentagonal (five fold) symmetry, actually a bipyramid with a ring of 5 atoms, capped by additional 2 atoms. Pentagonal symmetry is not allowed in standard infinite lattices, as it cannot lead to complete space filling without distortions. But in cluster physics this symmetry becomes possible. And it shows up again, to some extent for clusters with 13 atoms, the first one with icosahedral symmetry. The cluster consists indeed of an interior atom with two pentagonal caps. Larger clusters grow by addition of atoms on their sides, until the next larger icosahedron has been built. This yield the sequence of 55, 147, 309, 561,..., as shown in the bottom of Fig. 3.4. The number of the nearest neighbors in an icosahedral structure turns out to be larger than in a piece if crystalline structure, and this is the reason for the special stability of this structure. Surface energy is minimized by the flat triangular surfaces which are indicated by guiding lines for the largest sample, Ar561 . Clusters’ structures as ”created” from Fig. 3.4 are typically based on the method of surface-minimization. Another example of the determination of clusters’ structures in theoretical studies will be given in Section 4.

52

3.2.2.2

3 Cluster and Nanostructures

Electronically Induced Shapes

Electronic shell effects play a crucial role for the shape of metal clusters. They constitute a typical realization of the Jahn-Teller effect. For deeper details, see [83]. The guiding idea here is that the electronic ground state does not like to be degenerate. Atoms escape a degenerate ground state configuration by spin alignment leading to Hund’s rule [84]. Molecules (and clusters) can change their shape and that turns out to be an even more efficient mechanism to reach an unambiguous ground state. This is demonstrated in the following figure, Fig. 3.5 for small metal clusters, where a deformed harmonic oscillator provides a pertinent shell model.

Figure 3.5: Single electron energies vs. deformation α20 for the axially symmetric deformed harmonic oscillator. The numbers indicate (deformed) shell closures, up to N = 20. The shaded areas indicate bands of deformed magic shells. Typical shapes associated with given deformation α20 are indicated an top. See [79] for reference.

The figure shows the single electron levels as a function of deformation. Each level can carry two electrons (spin up and spin down). The configuration for a given electron number is found by filling the levels from below. The labels in the plot indicate the electron number reached at a given energy and deformation. At spherical shape (α20 =0), the magic electron numbers 2, 8, and 20 can be found.

3.3 Production of Clusters

53

See [79] for reference. It is obviously impossible to obtain another electron number at α20 without running into degeneracies. But different gaps open up at different deformations and the cluster moves to that deformation where it may find a gap at given Nel . From Fig. 3.5 one can obtain that Nel = 4 and 10 are prolate with α20 = 0.3 and Nel = 6, 14 and 18 are oblate with α20 ∼ −0.4. One misses the cases Nel = 12 and 16. These do not fit into the present scheme (the point marked 16’ turns out to be only an isomeric state). Axial symmetry produces a degeneracy of the states with angular momentum ±m for m = 0. One needs to break that degeneracy and to deal with fully triaxial shapes to create space for a gap at Nel = 12 and 16. However, this symmetry breaking is not as efficient as the first step from spherical to axial symmetry. As a consequence, there is a competition between Hund’s rules (spontaneous spin alignment) and the Jahn-Teller effect for clusters with Nel = 12 and 16 [85]. Fig. 3.5 illustrates the principle mechanism generating deformed shapes. The actual determination of the optimal configuration in microscopic calculations uses simulated annealing for detailed ionic models (see Section 4) or dedicated search in the landscape of deformations arm for the Jellium model.

3.3

Production of Clusters

The study of clusters requires a proper understanding and management of their production and handling. As for many other physical systems, production conditions strongly limit the level of details accessible in experiments. It is thus of importance to discuss production mechanisms in order to better understand what is measurable and how. The study of free clusters started with the increasing availability in the 1970’s. Without proper and controlled means of production, researching properties of clusters was bound to the study of embedded or adsorbed clusters. In this case the phenomena mix in a complex manner the effects of clusters and substrate or matrix. As we already know, clusters are finite objects, finite pieces of material. Cluster sources hence basically rely either on condenstaion/aggregation or on break up, and sometimes on both. One can thus sort cluster production sources into three main classes. In supersonic jet, gas aggregation and surface sources.

3.3.1

Supersonic Jets

The widely supersonic jets are the best understood cluster sources. The basic idea behind is the formation of clusters by condensation of an expanding gas of atoms. A highly compressed gas (typically around 10bar) with atoms of the material to be aggregated is allowed to expand through a small nozzle. The ensuring adiabatic expansion slows down the atoms up to a point at which binding between neighboring atoms becomes energetically favorable. This leads to a successive aggregation of the atoms in clusters. Supersonic sources are often used for producing clusters of low

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3 Cluster and Nanostructures

melting metals (usually alkali metals). A furnace contains melted metal which is heated to produce a metal vapor of pressure around 10-100 mbar. This vapor is mixed with a rare gas introduced into the source at a pressure of several bar. The hot mixture of metal vapor and rare gas is driven through the nozzle and expanded after the nozzle. The resulting clusters can then be mass-selected and subjected to many types of high-resolution spectroscopies. However, although this technique allows very detailed and sensitive studies to be performed, it is not so suitable for producing large quantities of clusters and it is hard to make direct measurements of structure.

3.3.2

Gas aggregation sources

In this type of cluster production, atoms are injected into a stationary or streaming gas and cluster formation again proceeds via condensation due to cooling of a gas of atoms. It is even efficient to produce large clusters. The basic production mechanism is simple and well known [86]. A liquid or solid is evaporated into a colder gas, which cools down the evaporated atoms or molecules until condensation starts. Condensation then roughly proceeds as in supersonic jets. The major difference to supersonic jets lies in the kinematics of the condensation process which is directional in a jet and not so in an aggregated source. As a consequence, there is ”no” automatic stopping of growth in an aggregation source compared to the case of supersonic jets, in which the expansion stops clusterization at some stage. But the intensities of the cluster beam extracted from the gas aggregation is nevertheless much lower than from supersonic jets.

3.3.3

Surface Sources

The principle of surface sources is primarily different from both supersonic and aggregation sources in the sense that initial cluster formation mostly results from break up. Nevertheless the ablation phase is usually complemented by further clustering phase which again proceeds via condensation. The first ”erosion” phase can be achieved by heavy particle impact at a few keV kinetic energy, in the so called ”Surface Erosion Source” or sputtering sources. By hitting the surface, the projectile ejects atoms, molecules and clusters. When the charge projectile is replaced by photons, typically from a laser with intensity I ∼ 108 W/cm2 , one speaks of a Laser Evaporation Source (LES). Extraction can also be achieved by a high-current pulsed arc discharge in the so called Pulse Arc Cluster Ion Source (PACIS). Thus the formation process is violent, clusters are usually formed at high temperatures. In order to overcome this temperature problem, such sources have often been coupled to supersonic jets and aggregation sources to cool down the formed clusters.

3.3 Production of Clusters

55

Figure 3.6: Basic layout of typical cluster sources. A seeded supersonic nozzle source is represented in (a), a gas aggregation source in (b) and surface sources, namely laser evaporation source in (c) and pulsed arc cluster ion source in (d). See [87] for reference.

3.3.4

Pick-up Sources

These sources are typically used to produce mixed clusters composed of various materials. The idea is to mix clusters as formed in supersonic jets with a jet of other molecules. This for example allows the attachment of the latter molecules on clusters formed in the jet.

3.3.5

Embedded and deposit clusters

Sources, discussed above, explain how to produce free clusters. However the field of embedded and deposit clusters is of a great practical importance and it allows

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several studies which are not easily possible with free clusters. The substrate fixes the cluster and gives therefore much more time for growth. This allows to collect much larger clusters and clusters with a higher density. The densities of scatters thus enable measurements with peak signals, e.g. second-harmonic generation (SHG) or Raman spectroscopy . There are two main different methods to produce embedded clusters or clusters deposit on surfaces. The first method is the growth from supply of atoms or ions. An example of deposited clusters is the growth of Na clusters on an isolating substrate by exposing the substrate to a Na vapor. The average cluster size can be controlled to some extent by the growth conditions, such as temperature, pressure and time. The method is applicable only if the combination of materials tends to clustering of the vapor atoms or ion respectively. The alternative method relies on producing free clusters first and depositing them in a second phase onto the substrate or into the matrix. Their structure can then be probed by techniques such as High Resolution Electron Microscopy [88], and Scanning Tunnelling Microscopy (STM) [89, 90]. But even in this method problems arise, as possible effects, surfaces may have on their adsorbed clusters, have to be taken into account. The oldest way to prepare clusters dates back until 1856. The time, when Faraday made his famous investigations of the optical properties of gold colloids1 [91]. Typically, clusters produced by this method are stabilized by the addition of a passivating layer, as compared to the naked clusters produced in molecular beams. One of the main advantages of this method is that large quantities of clusters can be produced. Furthermore, significant advances have now been made in controlling the size, shape and structure of these particles.

3.4 3.4.1

Experimental Measurements Mass Spectrometers

Mass spectrometers constitute a key device in cluster physics for cluster identification before, or after, measurements. There are two main classes of machines involving either time dependent or time independent electromagnetic fields. Wien filters and instruments with magnetic and electric sections are typical examples of devices built with time independent electromagnetic fields (see Fig. 3.7(b)). In this case of device, crossed homogeneous static electric E and magnetic B fields, acting perpendicular to the cluster beam, deviate charged particles and sort them according to their charge over mass ratio, q/m. Such devices require initially accelerated particles (via a potential V) and are tuned to ”one” given q/m ratio for a given combination of electric and magnetic fields. Wien filters’ advantage is to preserve a straight line trajectory for selected charged particles, but they have on 1

The word ’colloid’ was coined by Thomas Graham in 1861 to describe systems, which exhibited slow rates of diffusion through a porous membrane, of which glue solutions were a typical example.

3.4 Experimental Measurements

57

the other side a limited resolution due to the dispersion of cluster velocities on entrance. Thus the practical attainable mass range is typically 1 - 5000 atomic mass units (amu). Devices with time dependent fields are quadrupols, Ion Cyclotron Resonance systems and the widely used Time Of Flight (TOF) set-up. These quadrupol filters exploit a proper combination of static (ac) and time dependent (dc) electric fields to select clusters with a given q/m ratio (see Fig. 3.7(a)). It is composed of four rods constituting two pairs of opposite poles. One pair is held at the potential U + V cos(ωt), the other at opposite potential. Charged particles acquire an oscillatory motion along the poles. Mass resolution attained this way is of order 10−3 and therefore better than in Wien filters’ and masses up to 8000 amu can be analyzed.

Figure 3.7: Schematic representation of basic mass spectrometers: (a) Radio-Frequency (RF) quadrupole and (b) Wien filter. See [87] for reference.

3.4.2

Optical Response

Optical photons have energies between 2 and 3 eV, which is typically in the range of excitations of clusters. Optical spectroscopy thus constitutes an important tool of investigation in cluster physics. Optical measurements can be divided into two main classes, in photo-absorption, where cluster under consideration may be preserved (dissociation or fragmentation of the system is not necessary) and in photodepletion methods, where the system is promoted to a dissociative state and thus eventually destroyed after the measurement. See the following figure, Fig. 3.8, for details.

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Figure 3.8: Schematic view of absorption and depletion spectroscopy. The left panel describes the various situations in the spectral representation. The right panel shows the corresponding observed strengths. Several situations are considered: typical ionization potential (IP) measurement (top), access to excited (non-dissociative) level, and access to excited (dissociative) level (bottom). From [79].

3.4.3

Vibrational Spectra

Vibrational spectroscopy provides the most definitive means of identifying the surface species generated upon molecular adsorption and the species generated by surface reactions. Vibrational bands have thus been widely studied in molecular physics in many kinds of molecules [84]. In principle, any technique that can be used to obtain vibrational data from solid state or gas phase samples (IR, Raman etc.) can be applied to the study of surfaces - in addition there are a number of techniques which have been specifically developed to study the vibrations of molecules at interfaces (EELS, SFG etc.). Vibrational spectra have energies in the range of several tens of meV, thus in deep infrared region. There are, however, only two techniques that are routinely used for vibrational studies of molecules on surfaces - these are: • IR Spectroscopy (of various forms, e.g. RAIRS, MIR) • Electron Energy Loss Spectroscopy (EELS)

3.4 Experimental Measurements

3.4.3.1

59

IR Spectroscopy

There are a number of ways in which the IR technique may be implemented for the study of adsorbates on surfaces. For solid samples possessing a high surface area : Transmission IR Spectroscopy employing the same basic experimental geometry used for liquid samples and mulls. This is often used for studies on supported metal catalysts where the large metallic surface area permits a high concentration of adsorbed species to be sampled. The solid sample must, of course, be IR transparent over an appreciable wavelength range. Diffuse Reflectance IR Spectroscopy (DRIFTS) in which the diffusely scattered IR radiation from a sample is collected, refocused and analyzed. This modification of the IR technique can be employed with high surface area catalytic samples that are not sufficiently transparent to be studied in transmission. For studies on low surface area samples (e.g. single crystals) : Reflection-Absorption IR Spectroscopy (RAIRS) where the IR beam is specularly reflected from the front face of a highly-reflective sample, such as a metal single crystal surface. Multiple Internal Reflection Spectroscopy (MIR) in which the IR beam is passed through a thin, IR transmitting sample in a manner such that it alternately undergoes total internal reflection from the front and rear faces of the sample. At each reflection, some of the IR radiation may be absorbed by species adsorbed on the solid surface - hence the alternative name of Attenuated Total Reflection (ATR). Within the study of adsorbates on metallic surfaces by reflection IR spectroscopy (RAIRS) it can be shown theoretically that the best sensitivity for IR measurements on metallic surfaces is obtained using a grazing-incidence reflection of the IR light. Furthermore, since it is an optical (photon in/photon out) technique, it is not necessary for such studies to be carried out in vacuum. The technique is not inherently surface-specific, but there is no bulk signal to worry about and the surface signal is readily distinguishable from gas-phase absorptions using polarization effects. One major problem, is that of sensitivity (i.e. the signal is usually very weak owing to the small number of adsorbing molecules). Typically, the sampled area is ca. 1 cm2 with less than 1015 adsorbed molecules (i.e. about 1 nanomole). With modern FTIR spectrometers, however, such small signals (0.01% - 2% absorption) can still be recorded at relatively high resolution (ca. 1 cm−1 ). For a number of practical reasons, low frequency modes (< 600 cm−1 ) are not generally observable this means that it is not usually possible to see the vibration of the metal-adsorbate bond and attention is instead concentrated on the intrinsic vibrations of the adsorbate species in the range 600 − 3600 cm−1 . The observation of vibrational modes

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3 Cluster and Nanostructures

of adsorbates on metallic substrates is subject to the surface dipole selection rule. This states that only those vibrational modes which give rise to an oscillating dipole perpendicular (normal) to the surface are IR active and give rise to an observable absorption band.

IR Source

IR Detector

J

J J

Figure 3.9: Schematic view of a RAIRS spectrometer.

3.4.3.2

Electron Energy Loss Spectroscopy (EELS)

This is a technique utilizing the inelastic scattering of low energy electrons in order to measure vibrational spectra of surface species: superficially, it can be considered as the electron-analogue of Raman spectroscopy. To avoid confusion with other electron energy loss techniques it is sometimes referred to as High Resolution EELS (HREELS) or Vibrational EELS (VEELS). Since the technique employs low energy electrons, it is necessarily restricted to use in high vacuum (HV) and UHV environments - however, the use of such low energy electrons ensures that it is a surface specific technique and, arguably, it is the vibrational technique of choice for the study of most adsorbates on single crystal substrates. The basic experimental geometry is simple as illustrated schematically below, see Fig. 3.10 - it involves using an electron monochromator to give a well-defined beam of electrons of a fixed incident energy, and then analyzing the scattered electrons using an appropriate electron energy analyzer.

Monochromator

Analyser E0

Electron Source

E 1. (3.3) Then the system can lower its energy by bringing enough majority spin electrons down in energy by opening up the ferromagnetic exchange splitting.

3.6 Magnetism of Free Clusters

71

Figure 3.15: Stoner criterion for ferromagnetism. The two important factors are (a) the density of states at the Fermi level, D(EF ), and (b) an atomic exchange integral, I. (c) Their product has to be larger than unity to drive the transition to ferromagnetism . See [115] for Ref.

As with the opening of a gap in superconductivity, the density of states at the Fermi level and a coupling parameter - in this case the exchange integral I - are the critical quantities. The Stoner criterion explains why Fe, Co, and Ni are singled out for ferromagnetism. Several other elements are close to fulfill the criterion, like Pd. In some thin-film structures or clusters, these elements are transformed into ferromagnets, for example V, Cr, Mn, Mo, Ru, Rh, Pd, and Pt. In the case of thinfilms, magnetism can either be induced by exchange coupling to a ferromagnetic substrate or occur spontaneously owing to a higher density of states in a monolayer.

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The magnetic moment is dominated by the electrons with higher angular momentum, the 3d electrons in Cr, Mn, Fe, Co, Ni and the 4f electrons in the rare earths. With increasing atomic volume, one approaches the free-atom limit where Hund’s first rule postulates maximum spin, that is all the individual spins of the electrons in a shell are aligned parallel. Electrons with parallel spin have different spatial wave functions, owing to Pauli’s exclusion principle which is reflected in the exchange interaction. That reduces the Coulomb repulsion. When the atoms are squeezed into a solid, some of the electrons are forced into common spatial wave functions which forces their spins antiparallel and reduces the overall magnetic moment. For example the moment of 5 µB in the free Cr atom is reduced by an order of magnitude in the solid. In a somewhat oversimplified atomic orbital picture, the moment is created by interaction between electrons on the same atom and the coupling between electrons on different atoms. Ferromagnetic instabilities result from the interplay between the electronic Coulomb interaction and the Pauli principle: the minimum spin state minimizes the kinetic energy while the maximum spin reduces the effect of the Coulomb repulsion (a familiar example of this is Hund’s rule for atoms). This leads to the Stoner instability, which gives a spontaneous magnetization when the typical interaction exchange energy between two particles close to the Fermi level is of the order of the single particle level spacing. The knowledge that exchange forces are responsible for ferromagnetism has led to many semi-quantitative conclusions of great value. For example, it allows one to rationalize the appearance of ferromagnetism in some metals and not in others. Variations of the exchange integral with interatomic distance can be predicted. When the atoms are far apart, the 3d orbitals of transition metals only overlap slightly and the exchange integral J is small and positive. When the atoms come closer together and the 3d electrons approach one another more closely, the positive exchange interaction, favoring spins, becomes stronger and then decreases to zero. A further decrease in the interatomic distance brings 3d electrons so close together, that their spins must become antiparallel, leading to negative J. This condition is called antiferromagnetism. These remarks are particularly relevant to clusters where interatomic distances are modified due to the presence of the surface. It is commonly admitted today that bulk ground state properties of transition metals are well described by an itinerant electron picture, developed by Stoner and implemented in connection with modern band theory. In contrast rare earth metals are better described by the localized electron picture. Contrary to the case of 3d wave functions in transition metals, 4f wave functions of rare earths only overlap weakly. The standard model of rare earth magnetism is based upon the approximation that the 4f states are essentially the same in the solid as in free atoms. The electronic configuration is defined by Hund’s rules. Since the 4f electrons lie well within the ion core, the interaction with the environment is adequately represented by local

3.6 Magnetism of Free Clusters

73

exchange-interactions between 4f and (5d, 6s) conduction electrons. As a result the exchange-interaction between the localized 4f spins S is of indirect nature and is mediated by conduction electrons in the RKKY fashion [116]. Due to the strong spin orbit coupling in the 4f shell, J rather than S is a constant of the motion. The exchange is then determined by the projection of S on J, which is (g − 1)J and the exchange energy is proportional to (g − 1)2 J(J + 1). Therefore the exchange energy is highest for elements in the middle of the rare earth series. Characteristic exchange energies correspond to temperatures ranging from tens to hundreds of Kelvin. The large magnetic anisotropy of rare earths are due indirectly to unquenched orbital momentum and spin orbit coupling in the incomplete f-shells of atoms. As a result, most properties of rare earths can be explained in terms of a strong competition between the exchange energy and the magnetic anisotropy energy [117].

3.6.2

Implications for Clusters

Cluster magnetism will be particularly sensitive to the local environment of the atoms at the surface. In particular, it is well known that the reduced coordination of surface atoms leads to a band narrowing. Surface sites become more atomic like and increase their local moments. For transition metals, the electronic d band, which is responsible for magnetism can be adjusted to the second moment of the local density of states on atom a. The band width can then be expressed as: Wi = Wbulk (Zi /Zbulk )1/2

(3.4)

Assuming equal band width for up and down spins, and assuming that the exchange splitting is the same as in the bulk, the magnetic moment on atom i can be written: µi = (Zi /Zbulk )1/2 µbulk

(3.5)

This relationship is only valid for not too small value of Zi . It shows that for small clusters, the average magnetic moment is largely dominated by surface atoms since small Zi implies large µi . Atoms in the center of the clusters, have all their near neighbors and therefore Zi = Zbulk . On the other hand, the contraction of interatomic distances that are often observed at surfaces may produce a reduction of the total moment. At first glance, these effects appear to balance one another. Since surface relaxed ions are hard to evaluate in clusters, most calculations only account for the first effect. On the basis of these calculations, speculations arose to the possibility of magnetic ordering in VN , RuN , PdN and CrN clusters. Recent measurements have ruled out significant magnetic ordering in RuN and PdN clusters [105, 118]. Because of their different sensitivity to local environment, the 3d and 4f magnetism will manifest themselves in quite different ways in clusters. Moreover, the symmetry breaking at the surface may result in large surface-induced magnetic anisotropies.

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While magnetic ground-state properties of clusters have been calculated with various principle degrees of accuracy for transition metals (TMs), no first principle calculations have been performed for excited states. In particular, serious problems arise, if one wishes to extend the itinerant electron model to finite temperatures and the approach breaks down entirely for the paramagnetic regime above the critical temperature. Therefore at finite temperature, only the localized spin picture is available to explain magnetic properties. For magnetically order materials, there is a size, typically 20 nm, below which the giant magnetic moments of the particles begin to fluctuate in direction at temperatures well below the bulk Curie temperature. The particle looses its spontaneous magnetization in the laboratory reference frame but responds strongly to an applied field This type of rotation fluctuation of the magnetic was first pointed out by Neel [119]. This so called supernaturalistic behavior. The switching rate of the magnetization is thermally activated and therefore depends exponentially on the barrier U and the temperature T: Γ = Γ0 exp(−U/kT ),

(3.6)

where Γ0 is related to the precession frequency and is estimated to be 109 to 1013 s−1 . While shape anisotropy accounts for the energy barrier of non-spherical particles, crystal anisotropy is responsible for the variation of energy due to orientation of the magnetization with respect to the crystal axes. When the crystal contribution to the anisotropy is the dominant term, U = KV , where K is the anisotropy energy (for typical ferromagnets K ≈ 0.01 − 1 J/cm3 ) and V is the volume of the single domain particle. At the temperature, Tbl =

KV , k ln(τexp Γ0 )

(3.7)

below which thermal energy fluctuations are too slow to be observable on the time scale of the experiment τexp , particles are said to have reached their blocking temperature Tbl .

3.6.3 3.6.3.1

Stern-Gerlach Experiments Experimental Principles

All facilities in use are of the type described by Cox et al. [120]. This technique couples the conventional Stern-Gerlach deflection scheme with laser vaporization cluster source (LVCS) technology and time-of-flight mass spectrometry TOFMS. The metal clusters are grown in a helium filled source from atoms produced by the pulsed laser vaporization of a sample material. In recent designs, the source retains the clusters in a cavity for 1 to 2 ms (for experimental details and source operation mode see for example [121]). During this time, the clusters stop growing and thermalize with the

3.6 Magnetism of Free Clusters

75

source cavity. The helium-cluster mixture then undergoes a free jet expansion into the vacuum, producing a supersonic cluster beam. While the free jet expansion that forms the beam cools the translational and rotational temperatures of the clusters, it has little or no effect on their vibrational temperatures [121]. By restricting the studies to clusters that have reached thermal equilibrium with the source cavity, the clusters vibrational temperature is established to within a few K. The source chamber can be attached to a closed cycle helium refrigerator and electric heater, so that its temperature can be controlled between 60 and 350K. In the design of the University of Virginia [121], see Fig. 3.16, a mechanical chopper near the source allows only a brief pulse of these equilibrated clusters to pass; it also serves as a reference event for calculating the clusters beam velocity. The beam is collimated and passes through gradient magnet, where the clusters are accelerate perpendicular to their trajectory according to the projection of their magnetic moment onto the field axis. They are deflected in a drift tube before entering a ionization region of a TOFMS. These clusters are ionized by a narrow ultraviolet beam from an ArF excimer laser (193 nm), directed antiparallel to the cluster beam. This laser beam scans back and forth, mapping out the profile of deflected clusters. Once ionized, the clusters are dispersed perpendicular to the beam trajectory and detected in a mass spectrometer. By analyzing the spectra of clusters obtained at many positions of the excimer laser beam, it is possible to determine cluster deflections as functions of cluster size, vibrational temperature, and magnetic field. In this way deflection of individual clusters can be measured up to several hundred atoms.

Figure 3.16: Perspective view of the quadrupol sector pole faces used at the University of Virginia. This configuration offers optimal field gradient homogeneity. Observable parameters correspond to equation 3.8. See [98] for reference.

Initial results on transition metals, i.e. Fe and Co [100, 101] showed that these clusters deflect exclusively towards increasing magnetic field. They produce a relatively narrow, well defined deflection profile. Therefore the following expression relating the deflection d to the average magnetic moment per atom projected onto

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3 Cluster and Nanostructures

the field axis hµz i can be used: d = L2

1 ∗ 2D/L ∂B/∂zhµz (N, B, T )i 2mvx2

(3.8)

where ∂B/∂z is the magnetic field gradient in the z direction, L is the length of the gradient field, D is the distance from the exit of the magnet to the TOFMS, m is the atomic mass, vx is the velocity of the clusters. hµz i must be considered as a time average value of the projection of the magnetic moment onto the field axis and depends implicitly on B, T and the number N of atoms in the cluster. De Heer and coworkers already recognized that the cluster magnetic moments determined in this way were far below the bulk value [100]. The reduced magnetization was contrary to theoretical predictions that lowering the coordination enhances local moments and hence clusters should be more magnetic than bulk. The comparison between theory and experiment seemed quite controversial, specially concerning the temperature dependence of d. It was until the work of Bucher, Douglass and Bloomfield [101] that the relation between the measured deflection d and hµz i became clear.

3.6.3.2

Magnetic Moment Measurements of free Clusters

Over the temperature range experimentally accessible, the clusters of transition metals behave superparamagnetically. Although each cluster exhibits ferromagnetic order and has a giant magnetic moment, the orientation of that moment fluctuates on the nanosecond time scale. The moment is only weakly attached to the cluster’s atomic lattice, and thermal agitation from the vibrational modes causes the moment to explore all possible orientations during the several 100 microseconds the clusters spend in the gradient field magnet. This fluctuating giant moment responds paramagnetically to an applied magnetic field, yielding a smaller time averaged magnetic moment aligned with that field (in the laboratory reference frame). The applied field modifies the Boltzmann factors weighting the moment’s possible orientations, so that the cluster exhibits a magnetic moment per atom, hµz i, that is reduced from its full internal moment per atom µ, by the Langevin function L, · ¸ N µB N µB kB T hµz i = µL = µ coth − (3.9) kB T kB T N µB where N is the number of atoms in the cluster, B is the magnitude of the applied field, kB is Boltzmann’s constant and T is the cluster’s vibrational temperature. Since the deflection measurement yields a value of hµz i, the cluster’s internal moment per atom, µ, is obtained by inverting this relationship. The superparamagnetic behavior - due to its clear signature - allows a systematic study of internal magnetic moments

3.6 Magnetism of Free Clusters

77

of clusters. It becomes manifested by the scaling behavior of magnetization as a function of N, B and T. The magnetization per atom of a cobalt clusters increases linearly as a function of magnetic field. A linear increase of the magnetization per atom is also observed as a function of clusters sizes. For reference see [101, 103]. Finally, the magnetization is observed to vary as the inverse of cluster temperature. This scaling behavior of the magnetization as a function of N, B, and T was demonstrated for the first time by Bucher et al. [121] thanks to a new cluster source conceived for equilibrium measurements. A strait line passing through the origin is obtained for a given size N, when hµz i is reported as a function of N B/T . The proportionality factor must be the square of a magnetic moment its value is always in excess of the bulk magnetic moment per atoms. These findings are essential and appear almost like a signature since hµz i/µ = N µb/3kB T is the first term of the development of a Langevin function of argument N µB/kB T and hints strongly at the statistical nature of excitations in these clusters. It was Khanna, who first emphasized that a superparamagnetic interpretation could fit the experimental [122]. For comparison concerning the thermal behavior of certain cluster of theoretical data with Stern-Gerlach measurements, the following technique can be applied. For reference see [123]. Neglecting the vibrational and entropic differences between the isomers, the probability to find a cluster with spin S, irrespective of its atomic isomeric structure, is given by: ´ ³ P N EB (N,I,S) exp I kB T PN,T (S) = (3.10) ZN,T where kB denotes Boltzmann’s constant, T the ensemble temperature, I the structural isomer index, and ZN,T the (normalizing) partition function. The thermally averaged magnetic moment per atom hµiN,T can be derived via: P 2SµB PN,T hµiN,T = S (3.11) N This method has been applied to P dN and RhN clusters, see Chapter 4, Section 4.5 for further information.

Part II Applications

4

Chapter

Structure, electronic and magnetic properties of Pd and Rh clusters: An ab-initio density-functional study

I

n this chapter we present a comprehensive investigation of the structural, electronic, and magnetic properties of PdN and RhN clusters with up to N = 13 atoms, based on ab-initio density-functional calculations. The novel aspects of our investigation are the following: (i) The structural optimization of the cluster by a symmetry-unconstrained static total-energy minimization have been supplemented for larger clusters (N ≥ 7) by a search of the ground-state-structure by dynamical simulated annealing. The dynamical structural optimization has led to the discovery of highly unexpected ground-state configurations. (ii) The spin-polarized calculations have been performed in a fixed-moment mode. This allows to study coexisting magnetic isomers and leads to a deeper insight into the importance of magneto-structural effects. For both Pd and Rh the larger clusters adopt groundstate structures that can be considered as fragments of the face-centred cubic crystal structure of the bulk phase. For Pd clusters, the magnetic ground-state is a spintriplet (S = 1) for N ≤ 9, a spin-quintuplet (S = 2) for N = 10, and a spin-septet (S = 3) for the largest clusters. Large magnetic moments with up to S = 8 have been found for Rh clusters. Magnetic energy differences have been found to be small, such that there is an appreciable probability to find excited magnetic isomers even at ambient temperatures. In several cases, also structural energy differences are sufficiently small to permit the coexistence of polytypes.

4.1

Introduction

The investigation of the physico-chemical properties of small metallic clusters is a very active field of research, for many reasons. For example, metallic clusters are known to possess unique catalytic properties. Ultrafine dispersed Pd clusters sup81

82

4 Pd and Rh clusters

ported on alumina were found to be superior CO oxidation catalysts [124] than single crystals of Pd, and also show an enhanced activity for the reduction of nitric oxide by carbon monoxide [125], as well as a higher activity and better selectivity in hydrogenation processes [126]. Another reason for the current interest in metallic clusters are their fascinating magnetic properties. It is now well known that in addition to the common ferromagnetic metals Fe, Co, and Ni some transition metals that are non-magnetic in the bulk may become magnetic when the dimensionality is reduced, as in ultrathin films [127–129], in nanowires [130–132], and in clusters [105, 118, 133, 134]. An important factor distinguishing clusters from other low-dimensional structures is the possibility to assume non-crystallographic (icosahedral, dodecagonal or other) arrangements, while in ultrathin films and in nanowires, the structure of the adsorbate is determined by the epitaxial relation with the support. Non-crystallographic symmetry, together with variations in bond lengths and coordination could lead to important changes in the electronic properties, with significant consequences for the magnetic and chemical properties. Experimentally, 4d-metals from the end of the transition series have been investigated most extensively. While for Rh clusters large magnetic moments have been reported [105, 118], the magnetism of small Pd clusters remains a controversial issue. Early Stern-Gerlach experiments showed the absence of magnetic moments in Pd clusters [102, 105, 118]. Photoemission experiments [133] suggested a Ni-like spin-distribution in Pdn clusters with n ≤ 6, and non-magnetic behavior for n ≥ 15. In contrast, dc susceptibility measurements [135] found magnetic moments of 0.23 ± 0.19 µB per atom in Pd clusters with diameters in the range of 50 to 70 ˚ A. A large number of theoretical studies [123, 136–146] has been devoted to the investigation of Pd clusters. These studies are based on a wide variety of different approaches, ranging from multi-configuration selfconsistent field calculations for the smallest clusters [136, 138, 143] over density functional calculations for small and medium sized aggregates [123, 137, 139, 141, 142, 146, 147] to extended H¨ uckel [140] and tight-binding methods [144, 145] applied to large clusters. Rh clusters have also been studied using various techniques [134, 148–157]. The central problem of theoretical cluster studies is the determination of the ground state geometry - with increasing cluster-size the number of conceivable configurations increases tremendously. Experimental information is only indirect and in most cases not sufficient to determine the structure precisely. Ab-initio density-functional calculations mostly follow a strategy of optimizing the structure of different isomers under the constraint of conserving point-group symmetry. While this is an acceptable procedure for those numbers of atoms allowing for the formation of compact, close-packed structures that can be expected to be favored by a substantial structural energy difference, it is not very likely that the ground state will be found in clusters where one or two atoms have been added to or subtracted from a compact cluster. In addition, there are indications for the coexistence of structural isomers. For this reason attempts have been made to combine ab-initio electronic structure calculations with force-field molecular dynamics simulations for the cluster-structure. However, this

4.2 Computational Method

83

strategy is limited by the fact that force fields do not account for the correlation between structure and interatomic forces that in clusters not only structural, but also magnetic isomers can exist. It has been demonstrated [123, 154] that different magnetizations can lead to different bond lengths and different geometries. The present work has been devoted to extensive investigations of structural and magnetic isomers in Pd and Rh clusters with up to 13 atoms. Our investigations are based on ab-initio molecular-dynamics using spin-densityfunctional theory with gradient-corrected exchange correlation functionals, supplementing the static optimization of the cluster-structure by dynamical simulated annealing calculations to verify the dynamical stability of the optimized structures. The calculations are performed in a fixed-moment mode. This allows to perform an independent structural optimization of spin-isomers and to explore the correlation between magnetism and geometric structure. An important result of our study is that, especially for the larger clusters, not only energetically almost degenerate magnetic isomers exist, but that in addition the structural energy differences between certain isomers are small enough to allow two or more structural isomers to occur at ambient temperatures. This coexistence of structural and magnetic isomers must be taken into account when theoretical predictions and experimental observations are confronted. Our paper is organized as follows: In Sec. 4.2 we review very briefly our methodology for ab-initio density-functional calculations, in Sec. 4.3 we present an overview on the stable cluster structures, the binding energies, magnetic moments and HOMO-LUMO gaps as a function of cluster-size. In Sec. 4.4 we discuss the results for individual clusters, in particular their atomic and magnetic structures and the correlation between structure and magnetism. Because of the large amount of information contained in our data, only the most important results can be reproduced in this paper - a additional material can be found on our web-site.[158] An analysis of the possible coextistence of magnetic and structural isomers at finite temperatures, together with concluding remarks, is presented in Sec 4.5

4.2

Computational Method

All calculations were performed using the Vienna Ab-initio Simulation Package (VASP) [66–69]. VASP is based on density-functional theory (DFT) theory and works in a plane wave basis set. The electronic ground state is determined by solving the Kohn-Sham equations using an iterative unconstrained band-by-band matrix digitalization scheme based on a residual minimization method [68, 72]. Exchange and correlation were treated in the generalized gradient approximation (GGA), based on the parameterization by Perdew and Zunger [17] of the local-density functional of Ceperley and Alder [16], with the gradient corrections following Perdew and Wang [37]. Spin polarization was taken into account according to Von Barth and Hedin’s [18] local-spin-density theory, using the spin-interpolation proposed by

84

4 Pd and Rh clusters

Vosko, Wilk, and Nusair [19]. The electron-ion interaction was described by the full-potential all-electron projector augmented wave (PAW) method, introduced by Bl¨ochl [63], as implemented in VASP by Kresse and Joubert [159]. To explore coexisting magnetic isomers, all spin-polarized calculations have been performed in a fixed-moment mode. The plane-wave basis set included plane waves up to a kinetic energy cutoff of 250 eV. For clusters consisting of 2 to 10 atoms a 10×10×10 ˚ A3 cubic supercell was used (this was found to be large enough to ensure that the periodically repeated cluster images do not interact with each other). For clusters consisting of more A3 cubic supercell was used. Electronic than 10 atoms per supercell a 15×15×15 ˚ eigenstates have been calculated at the center of the Brillouin zone of the supercell only. To improve convergence, a modest Gaussian smearing (σ = 0.02 eV) has been used for the calculation of the electronic density of states. The geometry of the clusters has been determined by static relaxation, using a conjugate-gradient minimization and the exact Hellmann-Feynman forces. For the smallest clusters (up to six atoms) it was found to be sufficient to optimize the geometries of a few structural isomers, allowing several spin-isomers for each structure. For larger clusters we performed in addition a dynamical simulated annealing of the cluster structure. Each simulated annealing runs starts with a molecular dynamic simulation at a high temperature of 1500 K, i.e. far above the melting temperature of the cluster. The system was then gradually cooled down to room temperature before the final structural refinement using a static conjugated-gradient approach.

4.3

Trends in binding energies, geometries, magnetic moments and electronic properties as a function of cluster sizes

Figure 4.1 shows the cluster structures explored in this work. For the smallest clusters, the evident structural isomers extensively discussed in the literature were considered: dumbbell and triangle for dimers and trimers, square, rhombus and tetrahedron for the tetramer, square pyramid and trigonal bipyramid for the pentamer, pentagonal pyramid and octahedron for the hexamer. These structures have been chosen as the starting configurations for the structural optimization. The optimizations have been performed without any symmetry constraint, i.e. distortions of the idealized geometries induced by a Jahn-Teller effect [83] or by a magnetostructural effect were not excluded during the structural optimization. For larger clusters, the choice of possible structural isomers is not so evident and for the structures discussed previously in the literature, rather small binding energies (in comparison with slightly smaller or larger clusters) were calculated. In these cases we attempted to find energetically more favorable geometries via the dynamical simulated annealing (DSA) method described above. Indeed, for clusters with N ≥ 7

4.3 Trends in binding energies, geometries, magnetic moments ...

85

DSA runs produced some surprising structures to be discussed in detail below. For all starting geometries - those assumed a priori as well as the DSA-optimized structures, fixed spin-moment calculations for possible magnetic isomers have been performed. For PdN clusters with N ≤ 7, only non-magnetic and M = 2 µB solutions have been considered (exploratory calculations show that high-spin isomers are unstable), for larger clusters magnetic moments of up to M = 8 µB have been admitted. A much wider range of magnetic moments must be considered for RhN clusters, with M up to 18 µB . Tables 4.1 and 4.2 list the binding energies, the average nearest-neighbor bondlengths, the magnetic moments, and the energy gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) for all structural and magnetic isomers of PdN and RhN clusters with N = 2 to 13. For each cluster size we have listed the starting structure used to initialize the static optimization, or the structure generated by a DSA run (these configurations are marked by an asterisk). As no symmetry constraint is imposed during the relaxation, the initial symmetry may be broken - in some cases symmetry breaking depends on the magnetic state of the cluster. The final relaxed configuration is characterized by its point group symmetry. For Pd we have systematically explored all possible magnetic isomers. For clusters with up to seven atoms these are the non-magnetic (S = 0) and S = 1 solutions, no solutions with higher moments exist. For larger Pd clusters, magnetic isomers with spin up to S = 3 are at least metastable. RhN clusters can carry even larger magnetic moments, and spin-isomers with spins up to S = 9 (M = 18 µB ) have been considered .

86

4 Pd and Rh clusters 2

3

4a

4b

4c

5a

5b

5c

6a

6b

6c

7b*

7a

8a

8b*

9a

9b*

10a

11b*

11a

12a

13a

10c

10b

12b

12c*

13b

13c*

Figure 4.1: Final symmetric structures and structures produced by dynamical simulated annealing (labelled by an asterisk) for clusters with 2 to 13 atoms: XN with X=Pd or Rh and N = 2 − 13. Structures produced by dynamical simulated annealing are labelled with an asterisk. N = 2: dumbbell; N = 3: triangle; N = 4: square (4a), rhombus (4b), and tetrahedron (4c); N = 5: square pyramid (5a), trigonal bipyramid (TBP 5b), and flat trigonal bipyramid (5c); N = 6: pentagonal pyramid (6a), octahedron (6b), and incomplete pentagonal bipyramid (6c); N = 7: centred hexagon (7a) and pentagonal bipyramid (PBP - 7b? ); N = 8: bicapped octahedron I (8a) and bicapped octahedron II (8b? ); N = 9: capped PBP (9a) and double trigonal antiprism (9b? ); N = 10: tetragonal antiprism (TAP) with capped square faces (10a), edge sharing double octahedra (10b) and trigonal pyramid (10c); N = 11: polytetrahedral cluster (11a) and edge sharing octahedra plus adatom (11b? ); N = 12: incomplete cubo-octahedron cube (12a), incomplete icosahedron (12b), and cluster of octahedra (12c? ); N = 13: capped cube with central atom (13a), centred icosahedron (13b), and cluster of octahedra (13c ? ). For details, see text.

4.3 Trends in binding energies, geometries, magnetic moments ...

87

Table 4.1: Final structure (notation according to Fig. 4.1), point group symmetry PGS, total magnetic moment M (in µB ), average coordination number NC , average nearestneighbor distances d (in ˚ A), HOMO-LUMO gap Eg (in eV), and binding energies (in eV/atom) for structural and magnetic isomers of PdN clusters with N = 2 to 13. We use the Schoenflies notation for the point group symmetry. An asterisk indicates that the antiferromagnetic or ferrimagnetic configuration breaks the PGS of the cluster geometry. Present results are compared with those of Barreteau et al. (Ref. [144]) using a spd tightbinding model and with ab-initio DFT calculations of Kumar and Kawazoe (Ref. [146]). The last two columns list the magnetic energy differences 4Emag for each structural isomer, and the structural energy difference 4Estruct calculated for the respective magnetic ground state (in eV/atom). N

structure

2

Dimer

3

Triangle

4a

Square

4b

Rhombus

4c

Tetrahedron

5a

Square Pyramid

5b

Trigonal Bipyramid

5c Flat Trigonal Bipyramid 6a

Pentagonal Pyramid

6b

Octahedron

6c

Incomplete PBP

7a

Centered Hexagon

7b?

Pentagonal Bipyramid

8a

Bicapped Octahedron I

8b? Bicapped Octahedron II

9a

Capped PBP

9b?

Double Trigonal Antiprism

PGS M NC D∞h D∞h C2v D3h D4h D4h D2h D2h S4 * Td C4v C4v C2v * D3h D3h D3h C1h C1h D4h Oh C2v C2v D2h D2h C2v C2v C2v C2v C2v C2v C2v C2v C2v C2v C1h C1h C1h C1h * C1h D3h

0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 0 2 4 6 0 2 4 6 0 2 4 0 2 4

1 1 2 2 2 2 2.5 2.5 3 3 3.2 3.2 3.6 3.6 2.4 2.4 3.3 3.3 4 4 4 4 3.4 3.4 4.6 4.6 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.5 4.9 5.1 5.1 4.7 4.7 4.7

d

Eg

2.57 2.48 2.49 2.52 2.60 2.49 2.54 2.56 2.61 2.61 2.61 2.61 2.64 2.64 2.55 2.55 2.63 2.63 2.66 2.66 2.67 2.67 2.66 2.64 2.70 2.70 2.69 2.69 2.68 2.67 2.68 2.67 2.67 2.67 2.68 2.71 2.71 2.66 2.66 2.67

0.18 0.32 0.27 0.00 1.04 0.03 0.16 0.02 0.12 0.00 0.00 0.19 0.00 0.08 0.00 0.48 0.10 0.00 0.10 0.00 0.24 0.19 0.18 0.03 0.08 0.07 0.16 0.13 0.00 0.04 0.17 0.21 0.10 0.07 0.05 0.07 0.10 0.13 0.00 0.05

Binding energy 4Emag 4Estruct this work Ref. [144] Ref. [146] 0.473 0.646 1.250 1.250 1.234 1.485 1.466 1.465 1.654 1.675 1.748 1.798 1.760 1.805 1.671 1.729 1.756 1.777 1.940 1.949 1.897 1.907 1.700 1.688 1.975 1.985 1.995 1.994 1.998 1.968 2.058 2.076 2.065 1.995 2.108 2.118 2.129 2.135 2.139 2.134

0.611 1.456 1.203 1.463

1.781 1.857

1.628

1.766

2.451 2.465

2.457 2.490

1.922 1.919

1.917 1.953 2.036

2.094

173 251 1 21 50 45 58 21 9 10 12 10 3 4 28 18 11 81 21 11 4 5

190 209

7 76 172 42 285

-

78

-

10 -

88

4 Pd and Rh clusters

Table I (continued) N

structure

10a

TAP with Capped Square Faces

10b

Edge Sharing Double Octahedra

10c

Trigonal Pyramid

11a Polytetrahedral Cluster 11b? Edge Sharing Octahedra Plus Adatom 12a

Capped Cube

12b Incomplete Icosahedron

12c? Edge Sharing Octahedra Plus Two Adatoms

13a

13b

13c?

bulk

Capped Cube with Central Atom

Centred Icosahedron

Cluster of Octahedra

PGS M NC C4v C4v C4v D2h D2h D2h T C2v T T C2v * C2v C2v C1 C1 C1 D4h D4h D4h D4h C1h C1h C5v C1h C1h C1h C1h C1h C2h C4v C2h C2h C2h C2h C2h Ih Ih C1h C1h C1h C1h fcc

0 2 4 2 4 6 0 2 4 6 2 4 6 2 4 6 2 4 6 8 2 4 6 8 2 4 6 8 2 4 6 8 0 2 4 6 8 2 4 6 8 0

4.8 4.8 4.8 5 5 5 4.8 4.8 4.8 4.8 5.5 5.5 5.5 5.1 5.1 5.1 4.7 4.7 4.7 4.7 6 6 6 6 5.3 5.3 5.3 5.3 5.5 4.9 4.9 4.9 6.5 6.5 6.5 6.5 6.5 5.5 5.5 5.5 5.5 12

d

Eg

2.68 2.68 2.68 2.68 2.68 2.68 2.67 2.67 2.67 2.66 2.71 2.71 2.68 2.68 2.68 2.68 2.64 2.64 2.64 2.65 2.75 2.74 2.74 2.74 2.69 2.69 2.69 2.69 2.73 2.69 2.68 2.70 2.75 2.75 2.75 2.75 2.75 2.69 2.69 2.69 2.69 2.80

0.13 0.10 0.09 0.07 0.01 0.08 0.00 0.22 0.00 0.00 0.25 0.08 0.18 0.09 0.07 0.08 0.07 0.17 0.00 0.17 0.09 0.03 0.15 0.13 0.19 0.07 0.02 0.02 0.14 0.09 0.00 0.11 0.05 0.09 0.00 0.01 0.06 0.00 0.04 0.04 0.03

Binding energy 4Emag 4Estruct this work Ref. [144] Ref. [146] 2.131 2.128 2.140 2.175 2.185 2.166 2.103 2.116 2.109 2.110 2.203 2.214 2.218 2.228 2.226 2.230 2.222 2.233 2.207 2.184 2.242 2.250 2.257 2.253 2.254 2.259 2.274 2.238 2.278 2.285 2.280 2.266 2.273 2.283 2.287 2.285 2.304 2.323 2.321 2.324 2.302 3.704

2.529 2.543 2.119

2.511 2.231

2.834

2.290

3.718

9 12 10 19 13 7 6 15 4 2 4 11 26 49 15 7 4 20 15 36 7 5 19 31 21 17 19 1 3 22

45 -

69

12

41

17

-

39

20

-

4.3 Trends in binding energies, geometries, magnetic moments ...

89

Table 4.2: Final structure (notation according to Fig. 4.1), point group symmetry PGS, total magnetic moment M (in µB ), average coordination number NC , average nearestneighbor distances d (in ˚ A), HOMO-LUMO gap Eg (in eV), and binding energies (in eV/atom) for structural and magnetic isomers of RhN clusters with N = 2 to 13. We use the Schoenflies notation for the point group symmetry. An asterisk indicates that the antiferromagnetic or ferrimagnetic configuration breaks the PGS of the cluster geometry. Present results are compared with those of Barreteau et al. (Ref. [144], using an spd tightbinding model), Reddy et al.(Ref. [155], using an ab-initio LCAO approach), Jinlong et al. (Ref. [134]) and Wang and Ge (Ref. [156]) using an ab-initio DFT approach. The last two colums list the magnetic energy differences 4Emag for each structural isomer, and the structural energy difference 4Estruct calculated for the respective magnetic ground state (in eV/atom). N

structure

2

Dimer

3

4a

4b

4c

5a

5b

5c

6a

6b

6c

PGS M NC

D∞h * D∞h D∞h Triangle D3h D3h D3h D3h Square D4h D4h D4h D4h Rhombus D2h D2h D2h D2h Tetrahedron Td C1h S4 Td Square Pyramid C4v C4v C4v Trigonal D3h Bipyramid D3h D3h Flat Trigonal D3h Bipyramid D3h D3h Pentagonal C1h Pyramid C1h C5v * C5v * Octahedron Oh D4h D4h Oh D4h Incomplete PBP C2v C2v C2v

0 2 4 1 3 5 7 0 2 4 6 0 2 4 6 0 2 4 6 3 5 7 3 5 7 3 5 7 2 4 6 8 0 2 4 6 8 4 6 8

1 1 1 2 2 2 2 2 2 2 2 2.5 2.5 2.5 2.5 3 3 3 3 3.2 3.2 3.2 3.6 3.6 3.6 2.8 2.8 2.8 3.3 3.3 3.3 3.3 4 4 4 4 4 3.7 3.7 3.7

d

Eg

2.24 2.18 2.21 2.38 2.38 2.43 2.43 2.31 2.32 2.35 2.38 2.42 2.43 2.43 2.48 2.45 2.48 2.51 2.52 2.47 2.49 2.51 2.50 2.52 2.55 2.46 2.42 2.44 2.49 2.51 2.50 2.51 2.51 2.52 2.53 2.54 2.56 2.51 2.51 2.54

0.52 0.10 0.57 0.35 0.38 0.08 0.09 0.00 0.11 0.30 0.00 0.24 0.01 0.10 0.15 0.60 0.15 0.13 0.08 0.07 0.43 0.26 0.27 0.13 0.29 0.00 0.00 0.13 0.38 0.31 0.02 0.38 0.35 0.04 0.15 0.20 0.00 0.17 0.13 0.04

Binding energy 4Emag 4Estruct this work Ref. [134] Ref. [144] Ref. [155] Ref. [156] 1.288 1.387 1.686 2.234 2.308 2.280 1.890 2.611 2.662 2.724 2.639 2.399 2.445 2.508 2.555 2.750 2.677 2.693 2.646 3.006 3.028 3.017 2.959 2.948 2.971 2.832 2.807 2.843 2.944 2.985 2.995 3.007 3.190 3.153 3.169 3.204 3.174 3.168 3.123 3.028

1.477 1.488 1.520

1.88

1.74

2.35

2.36

2.253

2.034 1.955

2.640

2.180

2.728

2.118

2.79

2.77

2.370

2.91

2.81

3.11 3.13

3.08

2.655 2.733 2.950

2.369 2.293

3.062

3.457

3.129

3.28

3.390

3.244 3.251

3.27

3.26

398 299 74 28 418 113 62 85 156 111 47 73 57 104 22 11 12 23 11 36 63 22 12 14 51 35 30 45 140

26

195 -

-

57

185

197

36

90

4 Pd and Rh clusters

Table II (continued) N

structure

PGS M NC

7a Centered hexagon D2h 5 3.4 D2h 7 3.4 D2h 9 3.4 D2h 11 3.4 D2h 13 3.4 D2h 15 3.4 7b? Pentagonal C5h * 5 4.6 Bipyramid C2v 7 4.6 C2v 9 4.6 C2v 11 4.6 C5h * 13 4.6 8a Bicapped C2v 0 4.5 Octahedron I C2v 2 4.5 C2v 4 4.5 C2v 6 4.5 C2v 8 4.5 8b? Bicapped C2v 4 4.5 Octahedron II C2v 6 4.5 C2v 8 4.5 C2v 10 4.5 9a Capped PBP C1h 9 5.1 C1h 11 5.1 C1h 13 5.1 9b? Double Trigonal C1 9 4.7 Antiprism D3h 11 4.7 C1 13 4.7 10a TAP with Capped C4v 0 4.8 Square Faces C4v 6 4.8 C4v 8 4.8 C4v 10 4.8 C4v 12 4.8 C4v 14 4.8 C4v 16 4.8 10b Edge Sharing D2h 10 5 Double Octahedra D2h 12 5 D2h 14 5 D2h 16 5 10c Trigonal Pyramid C1h 0 4.8 C1h 2 4.8 C1h 4 4.8 C1h 6 4.8 11a Polytetrahedral C2v 3 4.9 Cluster C2v 5 4.9 C2v 7 4.9 C2v 9 4.9 C2v 11 4.9 C2v 13 4.9 11b? Edge Sharing C1 5 5.1 Octahedra C1 7 5.1 Plus Adatom C1 9 5.1 C1 11 5.1 C1 13 5.1

d

Eg

2.47 2.49 2.51 2.52 2.52 2.54 2.57 2.58 2.58 2.59 2.58 2.53 2.54 2.55 2.54 2.55 2.55 2.55 2.55 2.56 2.59 2.60 2.60 2.56 2.56 2.56 2.55 2.55 2.55 2.55 2.55 2.56 2.57 2.56 2.57 2.58 2.59 2.53 2.54 2.54 2.55 2.54 2.55 2.55 2.55 2.56 2.57 2.56 2.56 2.57 2.57 2.58

0.35 0.08 0.02 0.12 0.03 0.00 0.17 0.00 0.00 0.26 0.52 0.00 0.04 0.15 0.31 0.10 0.10 0.27 0.00 0.00 0.17 0.17 0.20 0.05 0.23 0.20 0.00 0.19 0.00 0.00 0.00 0.00 0.00 0.18 0.19 0.04 0.13 0.09 0.21 0.28 0.00 0.11 0.14 0.16 0.15 0.10 0.07 0.20 0.18 0.12 0.11 0.20

Binding energy 4Emag 4Estruct this workRef. [134]Ref. [144]Ref. [155]Ref. [156] 2.888 2.889 2.896 2.922 2.927 2.795 3.180 3.213 3.259 3.285 3.304 3.317 3.327 3.323 3.325 3.329 3.388 3.401 3.426 3.430 3.461 3.473 3.471 3.479 3.498 3.488 3.574 3.572 3.553 3.573 3.606 3.634 3.618 3.600 3.607 3.597 3.604 3.551 3.566 3.545 3.518 3.658 3.667 3.666 3.659 3.653 3.654 3.648 3.654 3.662 3.667 3.674

2.96

3.433

3.762 3.773

3.246

3.409

3.442 3.438 3.755

3.33

39 38 31 5 31 124 91 45 19 12 2 6 4 48 29 4 12 2 19 11 141 101 81 61 28 16 7 10 3 15 21 48 9 1 8 14 13 26 20 12 7 -

377

-

11

25

-

-

27

68

7

-

4.3 Trends in binding energies, geometries, magnetic moments ...

91

Table II (continued) N

structure

12a

Capped Cube

PGS MNC d Eg

C2h 12 C4v 14 C4v 16 12b Incomplete Icosahedron C1h 8 C1h 10 C1h 12 C1h 14 C1h 16 C1h 18 12c? Edge Sharing C1h 12 Octahedra Plus C1h 14 Two Adatoms C1h 16 C1h 18 13a Capped Cube C4v * 5 with Central Atom C2h 7 C4v 9 C2h 11 13b Centred Icosahedron C2h 7 Ih 15 Ih 17 C2h 21 13c? Cluster of Octahedra C1h 13 C1h 15 C1h 17 bulk

4.7 4.7 4.7 6 6 6 6 6 6 5.3 5.3 5.3 5.3 4.3 4.9 5.5 5.2 7 7 7 7 5.5 5.5 5.5

fcc 0 12

2.55 2.54 2.55 2.64 2.63 2.63 2.63 2.64 2.64 2.59 2.59 2.59 2.60 2.52 2.58 2.63 2.58 2.64 2.65 2.65 2.67 2.59 2.59 2.60

Binding energy 4Emag 4Estruct this workRef. [134]Ref. [144]Ref. [155]Ref. [156] 0.06 3.629 22 0.23 3.654 60 0.25 3.651 3 0.18 3.646 35 0.04 3.641 40 0.16 3.654 27 0.15 3.649 32 0.15 3.662 19 0.16 3.681 33 0.00 3.705 9 0.13 3.710 4 0.16 3.714 0.09 3.713 1 0.12 3.691 99 0.05 3.685 6 0.12 3.690 1 0.17 3.685 6 0.18 3.692 3.955 53 0.31 3.739 4.012 3.847 6 0.26 3.742 3.79 3 0.18 3.745 3.985 45 0.01 3.788 2 0.14 3.790 0.06 3.779 11

2.715 5.744

92

4.3.1

4 Pd and Rh clusters

Binding energy

Fig. 4.2 summarizes for the stable structural and magnetic isomers the variation of the binding energy, the average coordination number NC , the nearest-neighbor bond-length d, of the magnetic moment M, and of the HOMO/LUMO gap with increasing cluster size. For N ≥ 3 the binding energy increases essentially proportional √ to NC , but this trend is clearly recognizable only on the basis of the geometries optimized using the dynamical simulated annealing approach. The smooth variation of the binding energy with the cluster size contrasts the shell-structure of simple-metal cluster [80, 81, 160]. For the cluster-sizes considered here, the shell-model based on a jellium approximation would predict an enhanced stability of the N = 8 atom cluster, which we find to be one of the difficult cases for structural optimization. For transition-metal clusters, a tight-binding model seems to be a more appropriate zeroth-order approximation. Assuming that the hopping integral h is the same between all neighboring atoms, tight-binding theory predicts [161] the binding energy in a condensed system to be proportional to the hopping √ integral and to the square-root of the average coordination number Nc , Ebind = Nc | h |. We find that the optimized binding energies follow this simple relation rather well (see Fig. 4.2). Extrapolation of the trend derived from the small clusters to the face-centred cubic crystal with Nc = 12 gives for Pd a binding energy of 3.43 eV/atom, in reasonable agreement with a calculated cohesive energy of 3.70 eV/atom. Only for the smallest Pd clusters, an excess stability of the tetrahedron and the octahedron (as measured by the difference of its binding energy and the average binding energies with one atom more or less) is recognizable, and the trigonal and pentagonal bipyramids of the N = 5 and N = 7 clusters have a binding energy that is lower than the average binding energies of the tetrahedron and the octahedron, and of the octahedron and the double trigonal antiprism (for the N = 9 cluster), respectively. For larger N no clusters with particular stability can be identified. For Rh clusters the binding energy is a convex function of the cluster-size even for the smallest clusters, neither the tetrahedron nor the octahedron excel by an enhanced stability compared to their neighbors. For the tetrahedron this is eventually related to a frustration of magnetism (for details, see below). Tables 4.1 and 4.2 report the same information (plus the magnetic and structural energy differences) for all structural and magnetic isomers considered in our study. For comparison we have also listed binding energies from calculations published in the literature. For PdN clusters, we refer to the work of Barreteau et al. [144] based on tight-binding (TB) techniques and to the calculations of Kumar and Kawazoe [146], also using VASP. Due to a less stringent setting of the computational parameters, the binding energies given by Kumar and Kawazoe are lower by about 20 to 30 meV than our values, but structural energy differences are in better agreement. The important difference between our work and that of Kumar and Kawazoe is that with their restricted set of starting structures and symmetry-constrained relaxations, the stable configurations for larger clusters could not be identified.

4.3 Trends in binding energies, geometries, magnetic moments ... 3.8

2 1.8 1.6 1.4 1.2 1

relaxed simulated annealed ÖNC

0.8

average coordination number

binding energy/atom [eV]

Pd

2.2

3 2.8 2.6 2.4 2.2

relaxed simulated annealed ÖNC

2 1.6

7

6 5 4 3

relaxed simulated annealed

6 5 4 3 2

relaxed simulated annealed

1

1

2

relaxed simulated annealed assumed mag. moment of 2µB/cluster

0.8

magnetic moment/atom [µB]

magnetic moment/atom [µB]

3.2

7

1

0.6 0.4 0.2 0

relaxed simulated annealed

1.75 1.5 1.25 1 0.75 0.5 0.25 0

2.8

average bond length [Å]

2.7

2.7

2.6

2.5

relaxed simulated annealed

2.4

2.5 2.4 2.3

relaxed simulated annealed

0.6

relaxed simulated annealed

0.4

2.6

2.2

HOMO/LUMO gap [eV]

average bond length [Å]

3.4

0.6

2

Rh

3.6

1.8

average coordination number

binding energy/atom [eV]

2.4

HOMO/LUMO gap [eV]

93

0.3

0.2

0.1

0

relaxed simulated annealed

0.5 0.4 0.3 0.2 0.1 0.0

1

2

3

4

5

6

7

8

9

number of atoms

10

11

12

13

1

2

3

4

5

6

7

8

9

10

11

12

13

number of atoms

Figure 4.2: Binding energy, average coordination number, magnetic moment/atom, average bond length and HOMO/LUMO gap of energetically preferred Pd N and RhN clusters with N=2-13. The filled rhombi represent the stable structural and magnetic isomers obtained by static relaxation calculations, whereas the open squares show the results of simulated annealing. The dashed lines in the uppermost panels display the variation of the binding energy with the square root of the average coordination number, as expected from simple tight-binding experiment, cf. text.

94

4 Pd and Rh clusters

Much larger differences are found relative to the TB work of Barreteau et al.. Their binding energies are much larger and - more importantly - the TB calculations tend to grossly overestimate the magnetic energy differences, while underestimating the structural energy difference. Take for example the Pd4 cluster: for the structural energy difference between the S = 0 isomers of the square and the tetrahedron we find a value of 0.419 eV/atom, compared to the TB-result of 0.318 eV/atom; for the magnetic energy differences between the S = 0 and S = 1 isomers of the tetrahedron, the values are 22 meV/atom (present work) and 76 meV/atom (TB result). For RhN clusters we have chosen a larger data-base for comparison, including again the TB work of Barreteau et al. [144], the work of Reddy et al. [150] based on an ab-initio LCAO technique, and the work of Jinlong et al. [134] and of Wang and Ge [156]. Wang and Ge also used our VASP software, their binding energies differ from ours by a constant increment of 0.06 eV/atom to be attributed to a different choice of reference state for the isolated Rh atom. Interestingly, the TB calculations [144] now produce significantly smaller binding energies than the ab-initio DFT calculations, emphasizing the uncertainties in determining the TB parameters. The older work of Jinlong et al. [134] is based on the local spin-density approximation and does not include gradient corrections, but this is one of the few papers looking into the relative stability of different spin-isomers. For the smaller clusters agreement with our predictions for the most stable isomers is good, but for the octahedron where we predict a stable S = 3 state, the LSDA calculations find a non-magnetic ground state. Such differences are not unexpected, as the gradient corrections tend to stabilize magnetism in solids as well as in clusters. A similar preference for a lowspin S = 3 solution instead of our S = 7 state is found for the Rh10 cluster and for the Rh13 icosahedron. For the Rh12 cluster in an icosahedral structure with a vacant center Jinlong et al. also report a low-spin (S = 4) state, whereas we find the empty icosahedron to be structurally unstable. A vacant site is more easily accommodated at an outer vortex of the icosahedron at its center. This configuration breaks the icosahedral symmetry. This result demonstrates the importance of an unconstrained structural optimization.

4.3.2

Cluster geometry

Optimization of the cluster-geometry has been performed without any symmetry constraints. Structural distortions may be driven by different mechanism: a genuine instability of the assumed structure under the action of the interatomic forces, a Jahn-Teller mechanism if partially occupied eigenstates exist, or via magnetostructural effects. To monitor structure changes during the relaxation, we list in Tables 4.1 and 4.2 the chosen starting structure together with the point-group symmetry of the final relaxed configuration. A striking example of a magnetically induced distortion is observed, e.g, for a Pd4 tetrahedron: The magnetic S = 0 isomer adopts an antiferromagnetic configuration with local moments of ±0.26 µB , the symmetry is lowered to S4 (we use the

4.3 Trends in binding energies, geometries, magnetic moments ...

95

Schoenflies notation for the point group symmetry). The ground state is the S = 1 isomer which conserves the full tetrahedral Td symmetry. An even more complex situation is found for the Rh4 clusters where the non-magnetic S = 0 (ground state) and the high-spin S = 6 isomers conserve the full tetrahedral symmetry, whereas at intermediate magnetic moments (S = 2 and S = 4) the symmetry is reduced. For the S = 1 state the geometric and magnetic symmetry is C1h . Three Rh atoms carry the magnetic moments of ∼ 0.64 µB , whereas the fourth atom has a very low moment of only 0.09 µB . The magnetic moments appear to be compatible with a threefold axis through the low-spin state, but the trigonal symmetry is broken by the distortion of the triangular face opposite to this site, with edge-lengths of 2.71 ˚ A and two times 2.43 ˚ A (for details see below and supporting material). The symmetry of the S = 2 isomer is reduced to S4 (as for the nonmagnetic Pd4 tetrahedron) with two long edges of 2.71 ˚ A and four short edges of 2.40˚ A. Similar magneto-structural effects are detected also in larger clusters. Geometry-optimization of the larger clusters offers some surprises: the stable structures of the clusters with N = 11, 12, and 13 are not based on icosahedral or cubo-octahedral motifs as frequently assumed in the literature, but can be described as clusters of octahedra with adatoms (for details see below). These results contradict simulations [162] based on semi-empirical many-atom potentials of the embedded-atom type which predict a stabilization of near-crystalline motifs only at much larger cluster sizes. The results of the structure optimizations also serve as a warning against over-stretching of simple tight-binding arguments: for N = 13 the stable structure is neither a centred icosahedron (Nc = 6.46), nor a centred cube with capping atoms (Nc = 5.54) or the cubo-octahedron (NC = 4.61), but a polyoctahedral cluster with NC = 5.54 representing the optimal compromise between maximizing the number of nearest-neighbor bonds (as in the icosahedron) and the building of symmetry-adapted hybrid-orbitals based on s- and d-states. The cubooctahedron is even found to be completely unstable and relaxed for both Pd and Rh to a an eventually distorted centered cube with four capped square faces. Again magneto-structural effects are important. For Pd13 , the cluster geometry is compatible with C4v symmetry for all spin isomers, but the low-spin S = 1 isomer has a completely asymmetric magnetic configuration. For the Rh13 cluster, only the S = 9/2 isomer has full C4v symmetry in its geometric and magnetic structure. The S = 5/2 isomer has a geometric structure compatible with C4v , but slight differences int the magnetic moments lower the symmetry of the magnetic structure. For the S = 7/2 and the S = 11/2 isomers, the symmetry is reduced to C2h - the driving factor in the symmetry breaking are significant differences in the magnetic moments between two pairs of capping atoms (see supporting information for details). The average bond-lengths in the smaller clusters are considerably shorter than the equilibrium nearest-neighbor distance in the bulk metals (Pd: d = 2.80 ˚ A, Rh: ˚ d = 2.715 A) and increase slowly with cluster size. For our largest clusters, the average nearest neighbor distance reaches about 95 pct. of the bulk value. In general, the geometries optimized by dynamical simulated annealing are more compact

96

4 Pd and Rh clusters

(shorter bond length) than the highly symmetric geometries generally assumed in the literature.

4.3.3

Magnetic moment

For the magnetic moment, we find a clear trend with increasing cluster size for the Pd clusters: The isolated Pd atom is non-magnetic, for all clusters up to N = 9 the stable magnetic isomer is S = 1, for N = 10 the magnetic ground state is S = 2, and for N ≥ 11 the stable isomer is S = 3. The low-spin isomers are metastable for the larger clusters. The magnetic energy differences become very small, of the order of a few meV/atom. Hence room temperature experiments will always average over all possible spin-isomers. Magneto-structural effects are evident - different geometries also lead in most cases to different magnetic ground states (details to be discussed below). For Rh clusters, the situation is more complex. Due to its open-shell configuration an isolated Rh atom has a magnetic moment of 3 µB in its DFT ground state. For the three smallest clusters, the magnetic moment per atom of the most stable structural and magnetic isomers (dimer, triangle and tetrahedron) decreases linearly from 2 µB /atom in the dimer to 1 µB /atom in the trimer to zero for the tetramer. The Rh4 tetrahedron is hence the smallest non-magnetic Rh clusters. For N = 5 to N = 13, the magnetic moment of the most stable isomer fluctuates around 1 µB /atom. However, magnetic energy differences are rather small so that the measured magnetic moment must be compared with a weighted average over the thermodynamically accessible spin isomers. If this is done (for details see below), we find reasonable agreement with the experimental estimates of Cox et al. [105, 118].

4.3.4

Electronic properties

The HOMO/LUMO gap generally decreases with increasing cluster size, but due to a smaller exchange splitting, this decrease is much faster for PdN clusters where the gap is smaller than 0.1 eV for all N ≥ 8 clusters and almost vanishes for the stable N = 9, 10, and 12 isomers. A partially occupied HOMO is found for the magnetic ground state (S = 1) of triangular Pd3 and tetrahedral Pd4 , whereas for the antiferromagnetic (S = 0) isomers, where magneto-structural effects break the symmetry, a small HOMO/LUMO gap exists (see also Fig. 4.3). For a partially occupied HOMO, a Jahn Teller distortion is expected, but this cannot be produced by DFT calculations (see below for more details). Similarly, the octahedral Pd 6 cluster has a partially occupied HOMO for the fully symmetric (Oh ) S = 1 isomers, whereas for the nonmagnetic isomers the symmetry is reduced to D4h and a small gap is created. A partially occupied HOMO exists also for the metastable structural isomer of Pd8 , but not for the structure created by DSA - although both have C2v symmetry. Vice versa, for Pd9 the double trigonal antiprism created by DSA has a partially occupied HOMO in the stable S = 1 isomer - in spite of a reduced

4.3 Trends in binding energies, geometries, magnetic moments ...

97

symmetry (C1h ), whereas the only slightly less stable S = 2 isomer (4Emag = 5 meV) with full D3h symmetry has a small gap. For Rh clusters partially occupied HOMO’s are found for the high-spin isomers of Rh8 (bicapped octahedron) and of Rh10 (tetragonal antiprism with capped square faces).

4 Pd and Rh clusters

3.0

12.0 10.0 8.0 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0

majority spin

Pd 2

States [1/eV]

2.0 1.0 0.0 1.0

M=2µB

2.0

10.0

minority spin

3.0 -4.0

Number of Electrons

98

12.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

Pd 3

1.0 0.0 1.0 2.0

M=0µB minority spin

-4.0

-3.0

-2.0

-1.0

0.0

1.0

majority spin

Pd 3

2.0 1.0 0.0 1.0 2.0

M=2µB minority spin

3.0 -4.0

2.0

-3.0

-2.0

22.0

States [1/eV]

2.0

14.0 10.0

1.0

6.0 2.0

0.0

2.0 6.0

1.0

10.0

2.0

M=0µB

minority spin

3.0

-2.0

-1.0

2.0

0.0

1.0

22.0

14.0

Pd 4 2.0

18.0 14.0 10.0

1.0

6.0 2.0

0.0

2.0 6.0

1.0

10.0

2.0

18.0

14.0

M=2µB

minority spin

3.0 -4.0

22.0

-3.0

1.0

majority spin

18.0

States [1/eV]

Pd 4

Number of Electrons

majority spin

3.0 -4.0

0.0

E-EFermi

E-EFermi 3.0

-1.0

16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0

2.0

Number of Electrons

3.0

3.0

18.0 22.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

E-EFermi

E-EFermi 3.0

26.0 22.0 18.0 14.0 10.0 6.0 2.0 2.0 6.0 10.0 14.0 18.0 22.0 26.0

majority spin

Pd 5

States [1/eV]

2.0 1.0 0.0 1.0 2.0

M=2µB

minority spin

3.0 -4.0

-3.0

-2.0

-1.0

0.0

1.0

Number of Electrons

States [1/eV]

2.0

16.0 14.0 12.0 10.0 8.0 6.0 4.0 2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0

States [1/eV]

majority spin

Number of Electrons

3.0

Number of Electrons

E-EFermi

2.0

Pd 6

States [1/eV]

2.0 1.0 0.0 1.0 2.0

M=0µB

minority spin

3.0 -4.0

-3.0

-2.0

-1.0

E-EFermi

0.0

1.0

2.0

32.0 28.0 24.0 20.0 16.0 12.0 8.0 4.0 0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0

3.0

majority spin

Pd 6 2.0

States [1/eV]

majority spin

Number of Electrons

3.0

1.0 0.0 1.0 2.0

M=2µB

minority spin

3.0 -4.0

-3.0

-2.0

-1.0

0.0

1.0

32.0 28.0 24.0 20.0 16.0 12.0 8.0 4.0 0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0

Number of Electrons

E-EFermi

2.0

E-EFermi

Figure 4.3: Total spin-polarized differential (left scale) and integrated (right scale) density of states for Pd2 , Pd3 , tetrahedral Pd4 , trigonal-bipyramidal Pd5 and octahedral Pd6 , calculated for different spin isomers.

4.4 Structure, magnetic and electronic properties ...

4.4 4.4.1

99

Structure, magnetic and electronic properties of PdN and RhN clusters Pd2 and Rh2

For the Pd dimer, the binding energy amounts to 0.646 eV/atom and the total magnetic moment equals 2 µB (i.e. 1 µB /atom). In comparison with bulk Pd (d = 2.80 ˚ A), the bond length of the dimer is contracted to 2.48 ˚ A. As can be seen from the DOS represented in Fig. 4.3, the HOMO/LUMO gap has a value of Eg = 0.3174 eV. In agreement with the findings of Kumar et al. [146], we find sp-d hybridization to lead to a full occupation of the 4d majority spin states while the 4d minority spin states are depleted. The bond length and magnetic moment are in good agreement with other theoretical studies, e.g. Kumar et al. [146], Efremenko et al. [140], Moseler et al. [123] and Lee et al. [137]. In disagreement with our work, Barreteau et al. [144], using a spd tight-binding model, found a non-magnetic solution to be the ground state of the Pd dimer. The experimental binding energies reported in the literature [163, 164] for the Pd2 dimer scatter between 0.37 eV/atom and 0.57 eV/atom. High-level quantum chemistry calculations [136] also predict a triplet ground state with the identical bond-length of 2.48 ˚ A, but a lower binding energy of 0.43 eV/Atom in better agreement with experiment. Calculations [139] based on hybrid functionals mixing DFT and exact (Hartree-Fock) exchange predict a substantially larger bond-length of 2.53 ˚ A and a binding energy of 0.48 eV. The magnetic ground state of the Rh dimer we found to be the M = 4 µB (2 µB /atom) spin isomer with a binding energy of 1.686 eV/atom. Its bond length amounts to 2.21 ˚ A (strongly contracted compared to the Rh-Rh nearest neighbor distance of 2.715 ˚ A in bulk Rh). The magnetic moment of 2 µB /atom is in good agreement with other theoretical studies [134, 153–156] and with experiment [165]. The low spin isomers have a much higher energy, the S = 0 configuration is weakly antiferromagnetic with local moments of ± 0.14 µB and a bond length increased by 0.03 ˚ A. In contrast to the good agreement for the spin multiplicity of the ground state, the values for the binding energy of the Rh dimer found in literature scatter widely. Gingerich and Cocke [165] report an experimental binding energy of 1.46 eV/atom for a Rh2 dimer. The comparison between theory and experiment for both dimers shows that even in the GGA the over-binding tendency characteristic for density-functional techniques is not completely removed.

4.4.2

Pd3 and Rh3

For the N = 3 clusters, the triangular configuration has been shown to have a much lower energy than the linear chain. However, the HOMO of the triangular Pd 3 cluster in its M = 2 µB ground state is degenerate and partially occupied so that a Jahn-Teller distortion is expected. The degeneracy of the HOMO is a consequence of the exchange-splitting. In the S = 0 isomer of a triangular Pd3 cluster the symmetry

100

4 Pd and Rh clusters

˚) and is lowered to C2v (although the difference in the bond length is only 0.01 A the magnetization densities (see Fig. 4.4) show a non-vanishing spin-polarization of the occupied eigenstates (although integration over atomic spheres leads to almost vanishing local moments). Due to the mirror symmetry, occupation by spin-up and spin-down electrons is interchanged on sites 1 and 2, and a similar symmetry is also found on site 3.

Figure 4.4: Isosurfaces of the magnetization densities for M=0µB (left) and M=2µB (right) for the Pd3 triangle. Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

In the S = 1 isomer, with full D3h symmetry, the nondegenerate state is shifted below the Fermi-level, and the now degenerate threefold minority HOMO is occupied by only two electrons (see Fig. 4.3). The theorem of Jahn and Teller [83] states that if the HOMO is degenerate and partially occupied, a structural distortion that breaks the symmetry will remove the degeneracy and lower the total energy. A D3h → C2v distortion of the equilateral triangle would lift the degeneracy of this state, which means the highly symmetric D3h equilateral triangular geometry of the Pd3 cluster should be unstable against a Jahn-Teller distortion. Our calculations, however, did not show this instability and the equilateral triangle was found to be the energetically most stable Pd3 isomer. The stability against Jahn-Teller distortions is an artefact introduced by the way exchange and correlation are commonly approximated in DFT. In the conventional DFT the electron density distribution n(~r) is represented by that of a single-determinant ground state of noninteracting electrons in an external potential v(~r) - such systems are called pure-state v-representable (P-VR). Most physical systems are P-VR. However, there are also densities which can be represented only by a weighted sum of the densities of several degenerate single-determinant ground states. These systems are called ensemble v-representable (E-VR). Prominent examples are the C2 molecule and the H2 +H2 reaction [166], but also the triangle-cluster with a degenerate ground state. Formal Kohn-Sham equations for E-VR systems have been developed by Ullrich and Kohn [167], but so far this generalized Kohn-Sham equation has been used only for very simple systems (e.g. the Be2 dimer) as an ex post correction to a selfconsistent

4.4 Structure, magnetic and electronic properties ...

101

Kohn-Sham calculation with a conventional functional. That the stability of the equilateral Pd3 triangle is a DFT artefact is also made plausible by the work of Valerio and Toulhoat [139] and of Balasubramanian [136], which shows Pd3 clusters to be stable against Jahn-Teller distortions at the GGA level, but to undergo a distortion when a hybrid functional mixing DFT and exact exchange is used or if the calculation is performed at the Hartree-Fock + Configuration Interaction (HF-CI) level. In the case of the Rh3 equilateral triangle (D3h ) we compared four spin isomers with magnetic moments of 1 µB , 3 µB , 5 µB and 7 µB , respectively. All spinisomers are ferromagnetic and conserve the D3h symmetry of the equilateral triangle, all have a non-zero HOMO/LUMO gap (see Table 4.2). In agreement with the results of Nayak et al. [153] and Wang and Ge [156], but in disagreement with Chien et al. [154] and Reddy et al. [155] we found the M = 3 µB isomer with a binding energy of 2.308 eV/atom and a bond length of d = 2.38 ˚ A to be the ground state. The bond length increases slightly for the metastable high-spin isomers. Chien et al. found that within the LDA the ground state has D3h symmetry and a magnetic moment of M = 3 µB , whereas GGA calculations predicted a C2v symmetry for an S = 5/2 isomer. Reddy et al. also found the S = 5/2 isomer in an isosceles triangle to be slightly lower in energy.

4.4.3

Pd4 and Rh4

For the Pd4 cluster we compared two planar structures, the square (D4h ) and the rhombus (C2h ), and the smallest three-dimensional cluster, the tetrahedron (Td ) (see Fig. 4.1). Table4.1 lists the binding energy/atom for the non-magnetic and 2 µB spin isomers of these three Pd4 clusters. In case of the square and tetrahedral geometries the magnetic solution is energetically preferred over the non-magnetic solutions by 0.251 eV/atom and 0.021 eV/atom, respectively. In case of the rhombus the non-magnetic and magnetic spin isomers are almost energetically degenerate (the difference in the free energy is 0.001 eV/atom). Bond lengths and HOMOLUMO gaps for these isomers are listed in Table4.1. The energetically most favorable structural and spin isomer is the tetrahedral Pd4 cluster with an average bond length of 2.61 ˚ A a total magnetic moment of 2 µB (0.5 µB /atom), and a binding energy of 1.675 eV/atom, which is in good agreement with the results of previous work [123, 138, 144]. The HOMO (minority spin) of the tetrahedral Pd 4 is threefold degenerated and partially occupied by only one electron (see Fig. 4.3). In disagreement with Kumar et al. [146] and despite the similarity of our theoretical approach and the one used these authors, our calculations did not show tetrahedral Pd4 to undergo a Jahn-Teller distortion. However, Dai et al. [138], who studied the Pd4 clusters at a considerably higher level of theory, using a complete active space self-consistent field (CASSCF) approach followed by multi-reference singles+doubles configuration interaction (MRSDCI), also consider the ground state of Pd4 to be an undistorted tetrahedron. The tetrahedral Pd4 cluster is the simplest example of a

102

4 Pd and Rh clusters

magneto-structural effect. The antiferromagnetic S = 0 isomer has local magnetic moments of ± 0.26 µB on pairs of Pd atoms occupying opposite edges of a distorted tetrahedron (see Fig. 4.5). The symmetry is lowered from Td to S4 , with interatomic distances of 2.56 ˚ A between ferromagnetically occupied Pd atoms and 2.63 ˚ A between antiferromagnetically coupled pairs. The magnetization densities of both spin isomers are shown in Fig. 4.5.

Figure 4.5: Isosurface plots of the magnetization densities of the S = 0 (left) and S = 1 (right) spin-isomers of Pd4 . Dark surfaces surround regions of negative, light surfaces of positive magnetization. For the S = 0 isomer the symmetry is lowered from T d to C2v .

It is interesting to observe that in the S = 0 isomer, the magnetization densities around atoms with like moments extend along the connecting line, while the overlap is minimized between atoms with opposite moments. An analysis of the local DOS projected onto a pair of atoms with like moments (see Fig. 4.6) demonstrates the absence of exchange-splitting on most eigenstates which show an equal occupation of spin-up and spin-down states, while for certain strong-coupling eigenstates a splitting of up to 0.56 eV is calculated. These states are occupied preferentially by one spin-component on one pair of atoms, while on the pair with opposite magnetic moments the preferred spin-orientation is reversed so that the total DOS does not show any spin-polarization. The HOMO is a non-spinpolarized eigenstate, while the LUMO is a spin-split state whose counterpart lies about 0.4 eV below the Fermi energy (the exchange-splitting of these states is 0.45 eV). The magnetization density of this antiferromagnetic S=0 configuration shown in Fig. 4.5 is determined by this spin-polarized eigenstate. It should be noted that the structural distortion of the non-magnetic isomer is driven not by a Jahn-Teller effect, but by the imposed constraint S = 0 which cannot be met while tetrahedral symmetry is conserved. We also note that due to a small magnetic energy difference, the distorted S = 0 isomer will be observed with a non-negligible probability at finite temperatures. In the case of Rh4 we studied the same structural isomers as mentioned above for Pd4 . As can be seen in Table4.2 we found the non-magnetic (undistorted) tetrahedron, with a binding energy of 2.750 eV/atom, to be the ground state of Rh4 . It is energetically more favorable than its own 2 µB spin isomer (EB = 2.677 eV/atom),

4.4 Structure, magnetic and electronic properties ...

1.5

103

spin up

Pd 4

States [1/eV]

1.0 0.5 0.0 0.5 1.0 spin down

1.5 -4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

E-EFermi Figure 4.6: Local density of states at sites 1 and 4 of the antiferromagnetic S = 0 P d 4 tetrahedron, carrying negative moments of - 0.26 µB . Solid (black) lines correspond to states, which are hybridized over all atoms. The dashed (red) lines represent states localized at atoms No. 2 and 3, carrying a magnetic moment of 0.26 µB . For states localized on atoms 1 and 4, spin-up and spin-down occupation is reversed. Cf. text.

and the most stable spin isomers of the square (EB = 2.724 eV/atom, 4 µB ) and the rhombus (EB = 2.555 eV/atom, 6 µB ). The analysis of the magnetization densities and of the local DOS demonstrates that in contrast to Pd4 , the S = 0 isomer of a Rh4 is genuinely paramagnetic. This result is in good agreement with those published in the literature [134, 144, 153–155, 168]. For the square and the rhombus, all spin isomers conserve the full symmetry (D4h and D2h , resp.). While for the square, the S = 0 isomer is nonmagnetic and all magnetic states ferromagnetic, the low-spin isomers of the rhombus (S = 0 and S = 1) are antiferromagnetic with pairs of parallel moments coupled along the short and long diagonals. The energetically more favourable high-spin isomers (S = 2 and S = 3) are ferromagnetic, with almost equal local moments on all four sites. Similar to the Pd4 tetrahedron, we note a very interesting magneto-structural effect: while for the paramagnetic and for the high-spin S = 3 isomers, both the geometric and the magnetic structures show full tetrahedral (Td ) symmetry, for the S = 1 and S = 3 isomers the symmetry is reduced to C1h and S4 , respectively, with differences in the interatomic distances of up to 0.33 ˚ A. For the S = 1 isomer the magnetic symmetry is almost that of a trigonal pyramid, with an almost nonmag-

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4 Pd and Rh clusters

netic atom (M1 = 0.09 µB ) at the vertex and three almost equal (0.64, 0.64 and 0.63 µB ) moments at the corners of the base (the magnetization densities for the S = 1 to S = 3 isomers are shown in Fig. 4.7), but the geometric distortion of the base (two edges measures 2.43 ˚ A one 2.71 ˚ A) lowers the symmetry to C1h . For the S = 2 isomer, all local magnetic moments are almost equal (M = 1.00 ± 0.01 µB ), but the symmetry of the cluster geometry is only S4 , with two long (2.73 ˚ A) and ˚ four short edges (2.40 A). The structural distortion also influences the magnetic energy differences (see Table4.2) which increase first with increasing magnetic moment, but decrease again for the S = 2 isomer - evidently for the low-symmetry S = 1 isomer, the energy for the structural distortion adds to the magnetic energy difference. The magnetization induced changes in the cluster geometry are also reflect in the electronic spectrum, as demonstarted in Fig. 4.8. While for the S = 0 and S = 3 isomers with full Td symmetry we observe a number of three- and twofold degenerate eigenstates, the reduced symmetry of the S = 1 and S = 2 isomers lifts the degeneracy and leads to a much denser spectrum. For the highest spin isomer, the state-dependent exchange-splitting varies between 0.34 eV and 1.09 eV for the occupied eigenstates.

Figure 4.7: Isosurface plots of the magnetization densities of the S = 1 (left), S = 2 (center) and S = 3 (right) spin-isomers of Rh4 . Dark surfaces surround regions of negative, light surfaces of positive magnetization. For the S = 1 and S = 2 isomers the symmetry is lowered from Td to C1h and S=4 , respectively. Cf. text.

majority spin

20.0

2.0

Rh 4

16.0

4.0

0.0

4.0 8.0 12.0

M=0µB

3.0 3.0

20.0

majority spin

20.0

Rh 4

16.0

2.0

12.0 8.0

1.0

4.0 0.0

0.0

4.0

1.0

3.0 3.0 2.0

8.0 12.0

M=2µB

16.0

minority spin

20.0

majority spin

21.0 18.0 15.0 12.0 9.0 6.0 3.0 0.0 3.0 6.0 9.0 12.0 15.0 18.0 21.0

Rh 4

1.0 0.0 1.0 2.0 3.0 3.0 2.0

M=4µB minority spin majority spin

22.0

Rh 4

18.0 14.0 10.0

1.0

6.0 2.0

0.0

2.0 6.0

1.0 2.0

Number of Electrons

16.0

minority spin

10.0

M=6µB

14.0

Number of Electrons

States [1/eV]

0.0

1.0

2.0

States [1/eV]

8.0

1.0

2.0

States [1/eV]

12.0

Number of Electrons

3.0

105

Number of Electrons

States [1/eV]

4.4 Structure, magnetic and electronic properties ...

18.0

3.0

minority spin

-5.0

-4.0

-3.0

22.0

-2.0

-1.0

0.0

1.0

2.0

E-EFermi Figure 4.8: Total spin-polarized differential and integrated DOS for tetrahedral Rh 4 cluster with S = 0 to S = 3 (top to bottom)

106

4.4.4

4 Pd and Rh clusters

Pd5 and Rh5

For the pentamer Pd5 , we considered a square pyramid (D4h symmetry), an acute and a flat trigonal bipyramid (D3h ) (see Fig. 4.1). The most favorable structural isomer of Pd5 is the trigonal bipyramid (TBP) - in agreement with the results of Moseler et al. [123]- with a binding energy of 1.805 eV/atom and a total magnetic moment of 2 µB (0.41 µB /atom for the atoms making up the triangle and 0.38 µB /atom, for the two atoms forming the caps). The length of the bonds forming the triangle is 2.65 ˚ A, whereas the bonds with the caps are slightly shorter at 2.64 ˚ A. Hence both the geometric and the magnetic structure differ only slightly from that of a double icosahedron. The S = 0 isomer shows an antiferromagnetic distorted (C2v ) structure with two atoms in the central triangle carrying moments of ± 0.37 µB , while the remaining three are almost nonmagnetic. However, as shown by the magnetization densities (Fig. 4.9) this does not mean that there is no spin-polarization at these sites. The C2v point group has two mirror planes (parallel to the central triangle and passing through the three nonmagnetic sites). The magnetization densities of the antiferromagnetically coupled sites have essentially dx2 −y2 character in this last plane, while on the other three sites spin-up and spin-down electrons occupy orthogonal states extending perpendicular to this plane. The most favorable spin isomer of the square pyramid structure has a binding energy of 1.798 eV/atom and a magnetic moment of 2 µB (0.43 µB /atom for the atoms making up the base, and 0.28 µB /atom for the atom forming the cap). The S = 0 isomer shows zero moment on all atomic sites. The structural energy difference between the stable magnetic isomers of the trigonal bipyramid and the square (both S = 1) is only 7 meV/atom, so that both structures should be present in thermodynamic equilibrium. A flat trigonal bipyramid squeezed along the trigonal axis is a metastable configuration with a larger structural energy difference.

Figure 4.9: Isosurface plots of the magnetization densities in trigonal bipyramids of Pd 5 for the S = 0 (left) and S = 1 (right) spin-isomers. Note that for the S = 0 isomer the symmetry is reduced from D3h to C2v . Same symbols as in Fig. 4.5.

4.4 Structure, magnetic and electronic properties ...

107

For the Rh5 cluster we found the S = 5/2 (M = 5 µB ) spin-isomer of the square pyramid to be the ground state with EB =3.028 eV/atom. The atoms building the base have a bond length of 2.39 ˚ A and a magnetic moment/atom of 0.97 µB , the capping atom is located at a distance of 2.58 ˚ A from the atoms making up the base and has a magnetic moment/atom of 1.13 µB . Magnetic energy differences relative to the S = 3/2 and S = 7/2 isomers are very small, only 11 meV/atom and 22 meV/atom, respectively. For the trigonal bipyramid, the structural distortions relative to a double tetrahedron are much more pronounced than for Pd5 , and we calculate also larger differences in the magnetic moments of the atoms at the basis and at the vertices. However, again in contrast to Pd, no antiferromagnetic components are found in any of the isomers of Rh5 . The stable magnetic isomer for both the acute and the flat trigonal bipyramid is S = 7/2. For the acute bipyramid the structural energy difference relative to the square pyramid is only 57 meV, and we note also only very small magnetic energy difference between 12 and 23 meV for the low-spin isomers. Hence for the Rh5 cluster a coexistence of different structural and magnetic isomers has to be expected. Our results for the ground state are in good agreement with those of Reddy et al. [155], Chien et al. [154] and Wang et al. [168], but disagree with those of Aguilera-Granja et al. [157] and Jinlong et al. [134], who found the trigonal bipyramid with a magnetic moment of 1 µB resp. 3 µB to be the ground state of Rh5 .

4.4.5

Pd6 and Rh6

For the six-atom clusters the octahedron (symmetry Oh ), the pentagonal pyramid (C5v symmetry), and a pentagonal bipyramid with a vacant site in the central plane have been considered as possible structural isomers (see Fig. 4.1). For Pd6 the S = 1 isomers are the magnetic ground state for all three configurations, but the magnetic energy differences for the S = 0 isomer are only of the order of ten meV/atom. For the Pd6 octahedron, full Oh symmetry is conserved in both magnetic isomers, although the HOMO (minority spin) is threefold degenerate and only partially occupied (see Fig. 4.3). However, no Jahn-Teller distortion is produced. In the nonmagnetic state, the nondegenerate HOMO of the majority electron is emptied, filling the HOMO of the minority spins. For the pentagonal pyramid, the fivefold symmetry is reduced to C1h (mirror plane through the axis of the pyramid) for both magnetic isomers, the S = 0 isomer is in fact weakly antiferromagnetic. The investigation of the non-magnetic spin isomer led to another metastable structure, a pentagonal bipyramid with a missing atom in the central plane (see Fig. 4.1). This structure has a higher symmetry (C2v with two perpendicular mirror planes through the central axis and perpendicular to it) and a lower structural energy difference relative to the octahedron. Its S = 0 isomer (which is higher in energy by only 10 meV/atom) has an antiferromagnetic configuration with non-magnetic atoms in the central plane and moments of ± 0.36 µB at the vertices. Alternatively this structure may also be considered as consisting of three

108

4 Pd and Rh clusters

face sharing tetrahedra clustered along a central axis, or as a distorted octahedron. This interpretation is also suggested by the close similarity of the local magnetic moments, which all range with ± 0.01 µB around 0.33 µB . The structural energy difference of 42 meV is however, not small enough to suggest a coexistence of both isomers in thermal equilibrium. Our results are in good agreement with those of Barreteau et al. [144] and Moseler et al. [123]. For the Rh6 we found the octahedron with a magnetic moment of 6 µB to be the ground state. In addition to the magnetic ground state we identified four metastable spin-isomers with S = 0 to S = 4, iso-surfaces of their spin-polarization densities are shown in Fig. 4.10.

Figure 4.10: Isosurface plots of the magnetization densities of the S = 1 to S = 3 (from left to right) spin-isomers of a (nearly) octahedral Rh6 cluster. Dark surfaces surround regions of negative, light surfaces of positive magnetization. For the S = 1, 2, and 4 isomers the symmetry is lowered from Oh to D4h . Cf. text.

Only in the S = 0 state with non-magnetic atoms and in the S = 3 ground state with local atomic moments of exactly 1 µB on all atoms, the full octahedral symmetry is conserved, for all other isomers it is lowered to D4h . The magnetic configuration for S = 1 is ferrimagnetic, with small negative moments on the atoms located along the tetragonal axis. In contrast, the S = 2 state shows moments at the vertex sites which are twice as larger as those of the atoms in the basal plane (for exact values of local moments and interatomic distances, see again the supporting material). The structural distortions imposed by the constraint to assume an intermediate spin-state lead to larger magnetic energy differences for the S = 2 and 4 configurations than for the non-magnetic isomer (see Table4.2). For the pentagonal pyramid, the symmetry is reduced to C1h for the low-spin isomers (S=1 and S=2), due to large differences in the magnetic moments of the atoms in the basis of the pyramid. For the S = 3 and S = 4 states, the cluster geometry is compatible with pentagonal (C5v ) symmetry, but the magnetic structure has lower symmetry (C1h with a mirror plane through the axis and one of the basis atoms). Again we find an incomplete pentagonal bipyramid to be lower in energy than the pentagonal pyramid. In both cases, non-magnetic isomers are unstable, for the former the favored

4.4 Structure, magnetic and electronic properties ...

109

magnetic isomer is S = 2, no isomers with a lower magnetic moment are stable. The small structural energy difference is 36 meV. Studies of other researchers give conflicting results. Wang et al. [156], Chien et al. [154] (DFT-GGA) and Barreteau et al. [144], agree with our predictions, while Jinlong et al. [134] and Reddy et al. [155] (DFT-GGA) found the non-magnetic octahedron to be the ground state of Rd6 .

4.4.6

Pd7 and Rh7

For Pd7 the cluster geometries explored in earlier studies [123, 144, 146] were a centred hexagon, a pentagonal bipyramid (PBP), an octahedron with a capped triangular face, and a polytetrahedral cluster. Because of this large number of structural isomers, we decided to search for the optimal geometry using a dynamical simulated annealing run. The DSA run converged both for Pd and Rh clusters - as expected from results of former researchers - to a distorted pentagonal bipyramid (PBP). For Pd7 , the spin isomer with a total magnetic moment of 2 µB (composed by magnetic moments of 0.33 µB (four atoms) and 0.30 µB (one atom) for the atoms forming the pentagon and 0.19 µB for the atoms located along the axis) with a binding energy of 1.985 eV/atom was found to be the ground state. The symmetry around the fivefold axis is broken, the relaxed cluster has C2v symmetry. The same reduced symmetry is found for the S = 0 state with the antiferromagnetic configuration shown in Fig. 4.11. On the sites located in the central plane, states with negative spin extend in the peripheral direction, whereas states extending perpendicular to the plane are occupied preferentially by spin-up electrons.

Figure 4.11: Isosurface plots of the magnetization densities of the S = 0 (left) and S = 1 (right) spin-isomers of a Pd7 cluster forming a slightly distorted pentagonal bipyramid. Dark surfaces surround regions of negative, light surfaces of positive magnetization. Cf. text.

As an example for the electronic eigenstates of the clusters we present in Fig. 4.12 the spin-polarized electronic density of states of the stable Pd7 isomer (pentagonal bipyramid with S = 1), together with the electron-density distributions of the states in the vicinity of the Fermi level. This analysis allows to identify the exchangesplit eigenstates with the same symmetry, all are compatible with an overall C 2v

110

4 Pd and Rh clusters

symmetry of the clusters. The exchange splitting is formed to be strongly-state dependent. Within this interval the splitting varies between ∆Eex ∼ 0.10 eV for the empty eigenstates 37 and 38 (for occupied eigenstates the smallest splitting of ∼ 0.12 eV is found for states 30 and 31) and ∆Eex ∼ 0.27 eV for states 35 and 36 which are occupied for majority and empty for minority electrons. Only these to degenerate states contribute to the spin-density of the S = 1 isomer - a comparison of Fig. 4.11 and Fig. 4.12 demonstrates the magnetization density is just the sum of the electron densities of these two degenerate states. This demonstrates that the magnetic structure is determined by states in the immediate vicinity of the Fermi energy. Table 4.3: Energies of eigenstates for spin-up and spin-down states and exchange splitting (∆Eex ) of the stable Pd7 isomer (pentagonal bipyramid with (S = 1) Band No. 19-20 21-22 23-25 26-28 29 30-31 32-33 34 35-36 37-38 majority -1.1464 -1.0138 -0.8152 -0.5994 -0.4530 -0.3636 -0.3466 -0.2426 -0.2268 0.4400 minority -0.962 -0.8731 -0.6600 -0.4335 -0.3155 -0.2410 -0.0905 -0.0383 0.0383 0.5322 ∆Eex 0.1844 0.1407 0.1552 0.1659 0.1375 0.1226 0.2561 0.2043 0.2651 0.0922

A planar configuration in the form of a centred hexagon is metastable in S = 0 and S = 1 states, albeit with a relatively large structural energy difference of nearly 0.3 eV/atom. For both magnetic isomers the hexagonal symmetry is reduced to D2h , with two atoms at opposite corners of the hexagon having a distance from the center which is 0.45 ˚ A (S = 0), respectively 0.17 ˚ A (S = 1) larger than that of the other four atoms. Our results are in good agreement with those of Barreteau et al. [144], Moseler et al. [123] and Kumar et al. [146]. For the Rh7 cluster we found the pentagonal bipyramid (PBP) with a magnetic moment of 13 µB to be the ground state, but the energy of a 11 µB isomer is only 5 meV/atom higher. For all metastable magnetic isomers the pentagonal symmetry is reduced to C2v , while the magnetic ground state conserves the fivefold symmetry C5h around the axis of the bipyramid. Quite generally, the reduced symmetry is more apparent in the magnetic moments than in the cluster geometry. The strongest structural distortion is found for the S = 11/2 isomer, for which the interatomic distances along the periphery of the central plane differ by as much as 0.2 ˚ A (for details see the supporting material), with decreasing magnetization the structural distortion is gradually reduced, for S = 5/2 (isomers with lower moments are unstable) the fivefold symmetry of the cluster geometry is nearly completely recovered (up to differences of 0.01 ˚ A in the distances), but the magnetic symmetry is distinctly lower with moments of 2 times 0.72 µB , 2 times 0.782 µB and 0.752 µB on the atoms forming the fivefold ring. The binding energy of the distorted centred hexagon (D 2h symmetry) with a magnetic moment of 13 µB amounts to 2.927 eV/atom, the struc-

38

37

38

111

EF

36

35 34

1.0

33

32

33 31

31

30

-1.0 E-EFermi

3.0

2.0

1.0

0.0

1.0

2.0

3.0

-0.5

32 30

0.0

0.5

majority spin

minority spin

36 34

35

EF

37

4.4 Structure, magnetic and electronic properties ...

27

28

26

27

28

24

25

23

24

25

29

26 23

29

States [1/eV]

Figure 4.12: Spin-polarized electronic density of states and iso-surface plots of the electronic density distributions of eigenstates in the vicinity of the Fermi-level, as calculated for the stable (S = 1) spin-isomer of a Pd7 cluster forming a slightly distorted pentagonal bipyramid. Eigenstates for majority electrons are numbered in the sequence of increasing energy, eigenstates for minority electrons carry the number of their symmetry-equivalent counterparts. Cf. text.

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4 Pd and Rh clusters

tural energy difference is nearly 0.4 eV/atom (see Table 4.2). Our results agree with other researchers concerning the equilibrium geometry, but disagree are in the magnetic ground state. Reddy et al. [155] and Barreteau et al. [144] found the 9 µB PBP to be the ground state, whereas according to Aguileira-Granja et al. [157], the non-magnetic PBP has the lowest energy.

4.4.7

Pd8 and Rh8

For the Pd8 cluster we assumed a octahedral geometry with two adatoms capping triangular facets. Initially we considered a configuration in which the two capping atoms were placed in front of opposite triangular faces on one side of the central square (see Fig. 4.1, configuration 8a), DSA produced a slightly different structure enabling the formation of a nearest-neighbor bond between the capping atoms (see Fig. 4.1, configuration 8b? ). As this configuration leads to a breaking of the bond between the two atoms forming the common edge of the capped triangles, the average coordination number remains the same, NC = 4.5. However, in configuration 8a, the local coordination number varies between 3 and 6, whereas in configuration 8b we have only four fourfold and four fivefold coordinated sites. For configuration 8a, the magnetic ground state has a magnetic moment of 4µB (e.g.: 0.52 µB for the atoms forming the base of the octahedron, 0.38 µB and 0.44 µB for the atoms at the vertices of the octahedron and 0.55 µB for the capping atoms). For configuration 8b a lowspin isomer with a magnetic moment of 2µB (one half of atoms having a magnetic moment/atom of 0.28µB and the other a magnetic moment/atom of 0.22µB ) was found to be the ground state. This configuration has the lowest total energy, the structural energy difference is 78 meV/atom compared to the most stable magnetic isomer of structure 8a. For all isomers, the point group symmetry is C2v . Our result is in good agreement with that of Kumar et al. [146], who found the distorted bicapped octahedron with a magnetic moment of 2µB to be the energetically most preferred one. For the Rh8 cluster we compared both bicapped octahedron structures and spinisomers up to S = 5. The ground state is configuration 8b with a total magnetic moment of 10 µB (1.15 µB for the fourfold coordinated sites and 1.35 µB for the fivefold coordinated atoms). However, the magnetic energy difference compared to an isomer with M = 8 µB is only 4 meV/atom. Also the stable magnetic isomer of configuration 8a with M = 8 µB is only slightly higher in energy. We also calculate only minimal magnetic energy differences of a few meV’s for the low-spin isomers (see Table 4.1). Hence for the Rh8 cluster many structural and magnetic isomers will coexist in thermodynamic equilibrium.

4.4 Structure, magnetic and electronic properties ...

4.4.8

113

Pd9 and Rh9

For the Pd9 cluster we assumed a capped pentagonal bipyramid (PBP) as starting configuration for the static relaxations, DSA leads to a structure describable either as a trigonal antiprism with capped square faces or as an elongated trigonal prism with capped rectangular faces (see Fig. 4.1) - incidentally, this structure is one of the canonical coordination polyhedra discussed by Bernal as the building blocks of liquid and amorphous structures [169] - and compared different spin isomers. For the capped PBP (which may be considered as a distorted polytetrahedral arrangement fragment of an icosahedron) the symmetry is reduced to C1h for all spin isomers, the (meta)stable spin isomer with a magnetic moment of 4 µB has a binding energy of 2.129 eV/atom. The magnetic energy differences compared to the low-spin isomers are only 11 meV and 21 meV for S = 1 and S = 0, respectively. The S = 0 isomer has an antiferromagnetic structure with small local moments. The final structure of after simulated annealing is a distorted double trigonal antiprism (see Fig. 4.1), consisting of three layers of nearly equilateral triangles, with the middle triangle rotated by 60◦ . Only for the high-spin isomer with M = 4 µB the point group has full trigonal D3h symmetry, but the magnetic and structural ground state is the S = 1 isomer where the point-group symmetry is reduced to C1h . All symmetry is lost for the S = 0 isomer with an antiferromagnetic spin-density distribution (see supporting material for details). Again magnetic and structural energy differences are very small, only a few meV’s (see Table 4.1), so that all structural and magnetic isomers will be observed in thermal equilibrium. For Rh9 clusters we considered the same structures assumed as for Pd9 . The stable spin isomers of both the capped PBP and the double trigonal antiprism have a magnetic moment of 11 µB , the structural energy difference is 25 meV/atom, magnetic energy differences relative to the states with moments of 9 and 13 µB are even smaller. The tendency to break the trigonal symmetry of the double antiprism is weaker than for Pd9 , both the magnetic ground state and the high-spin isomer have full D3h symmetry, while the state with M = 9 µB is completely asymmetric (C1 ). All spin-isomers with the capped PBP have C2v symmetry. For both Pd9 and Rh9 clusters the lowest spin states of both structural isomers have ferrimagnetic, respectively antiferromagnetic character. Aguilera-Granja et al. [157] found a 14.58 µ B spin isomer to be the ground state of Rh9 (the origin of a non-integer value for the moment is unclear), whereas Reddy et al. [155] (DFT-GGA) concluded the 9 µB spin isomer to be the most stable one. Both calculations consider only the capped PBP structure. Experimental studies [118] predict a magnetic moment/atom of 0.8 µB ± 0.2 µB .

114

4.4.9

4 Pd and Rh clusters

Pd10 and Rh10

For Pd10 clusters we compared three geometries, a tetragonal antiprism with capped square faces (C4v - see configuration 10a in Fig. 4.1), a structure consisting of two edge-sharing octahedra (10b) and a trigonal pyramid (10c) with atoms at the corners and the midedge positions. The 4 µB spin isomer of the edge sharing octahedra with EB =2.185 eV/atom was found to be the ground state, the global symmetry is D2h . The two individual octahedra are only slightly distorted, their common edge is elongated by ∼0.1˚ A. Magnetic energy differences relative to the S = 1 and S = 3 isomers are only 10 mev/atom and 19 meV/atom, respectively. The magnetization of the S = 2 and S = 3 isomers is quite homogeneous, with local moments that are only slightly larger in the central plane. In contrast, the moments on these sites are strongly reduced in the S = 1 isomer. The structural energy difference relative to the stable 4 µB spin isomer of the capped square antiprism is 45 meV/atom. The low-spin isomers of this configuration are only slightly higher in energy, their magnetization densities display an interesting antiferromagnetic structure in the S = 0 state (see Fig. 4.13), whereas in the stable S = 2 state all atoms have almost exactly the same moment of 0.40 µB .

Figure 4.13: Isosurfaces of the magnetization densities for M = 0 µB to M = 4 µB for the Pd10 tetragonal antiprism with capped square faces (from left to right).

A third structural isomer is a tetrahedron with atoms at the vertices and slightly off the center of all edges. The motivation for studying this structure is that all facets correspond to the closely packed (111) surfaces of the fcc crystal which have the lowest surface energy. The tetrahedral symmetry is broken in the low-spin isomer with S = 1 representing the magnetic ground state - in this both the geometric and magnetic structure have C2v symmetry. In the nonmagnetic (S = 0) and in the high-spin isomers which differ in energy only by a few meV, the tetrahedral geometry (point group T) is completely recovered. Local magnetic moments and magnetization densities for S = 1 and S = 2 are shown in Fig. 4.14. This analysis shows that the equilibrium magnetic configuration is ferrimagnetic, with a substantial spin-down

4.4 Structure, magnetic and electronic properties ...

115

component on all low-moment sites. The structural energy difference relative to the bi-octahedral cluster is 69 meV. 10

0.15

9

7

0.17

1

0.28

-0.01

8

0.17

6

4

0.17

2

0.47

0.15

5

0.17

3

0.28

10

0.37

9

7

0.42

1

0.38

8

0.41

0.41

6

4

0.42

2

0.41

3

0.38

5

0.42

0.38

Figure 4.14: Local magnetic moments and isosurfaces of the magnetization densities in tetrahedral Pd10 clusters with S = 1 (distorted C2v symmetry) and S = 2 (T symmetry).

For Rh10 the S = 7 spin isomer of the capped tetragonal antiprism with EB = 3.634 eV/atom represents the ground state of Rh10 . For this structure all spinisomers between S = 0 and S = 6 are metastable, the spin-density distribution (see Fig. 4.15) of the low-spin-isomers shows an antiferromagnetic component at the apices which is, however, weaker than that found for the isostructural Pd10 cluster, compatible with C4v symmetry. The S = 0 isomer is completely nonmagnetic, the magnetization is more homogeneous with increasing spin. The stable magnetic isomer of the double octahedron is S = 6, with a magnetic energy difference of 27 meV/atom, for the large distorted tetrahedron a low-spin isomer with S = 1 represents the ground state, with a larger structural energy difference of 68 meV/atom. For the large Rh10 tetrahedron we observe a reduction of both the geometric and the magnetic symmetry to C1h for all isomers (details see the supporting material). Jinlong et al. [134], using a spd tight binding model, found the 6 µB spin isomer of the twisted double square pyramid to be the ground state. Experimental studies [118] predict a magnetic moment/atom of 0.8 µB ± 0.2 µB .

116

4 Pd and Rh clusters

Figure 4.15: Isosurfaces of the magnetization densities in Rh10 clusters with the structure of a capped tetragonal antiprism, for S = 3 to S = 8 (top left to bottom right). The stable magnetic isomer is S = 8. Cf. text.

4.4.10

Pd11 and Rh11

For Pd11 our starting configuration for the static relaxation calculations consisted of a pentagonal bipyramid with four capping atoms, the structural optimization led to a polytetrahedral cluster with C2v symmetry. The stable magnetic isomer has a moment of 6 µB , the magnetization is distributed quite homogeneously over all atoms, varying only between 0.52 µB and 0.59 µB . The low-spin isomers are only slightly higher in energy, in the S = 1 state the magnetic symmetry is lower than the geometric symmetry (see supporting material for details). DSA leads to a configuration consisting essentially of two distorted edge-sharing octahedra with one additional atom. The position of this atom is such that a third, strongly distorted octahedron is formed. The optimized configuration has no apparent symmetry, the energetically most favorable spin isomer is S = 3, it is lower in energy by 12 meV/atom than the polytetrahedral cluster. Magnetic energy differences to the low-spin configurations are only a few meV/atom. The same configurations have also been explored for the Rh11 cluster, allowing for a larger range of magnetic moments. The ground state is the distorted poly-octahedral structure (point group C1 ) with a magnetic moment of 13 µB and a binding energy of 3.674 eV/atom. For the polytetrahedral cluster the magnetic ground state has a lower moment of 5 µB only, all magnetic isomers have C2v sym-

4.4 Structure, magnetic and electronic properties ...

117

metry. The low-spin isomers are ferromagnetic. The structural energy difference is only 7 meV/atom, the magnetic energy differences are also very small for both structures (see Table4.2). Hence at finite temperatures, many structural and magnetic isomers will occur with comparable probabilities. Experimental results [118] indicate a magnetic moment/atom of 0.8 µB ± 0.2 µB , in reasonable agreement with the weighted average over the calculated magnetizations.

4.4.11

Pd12 and Rh12

For the Pd12 cluster the starting configurations for the static relaxations were an icosahedron and a cubo-octahedron, both with a vacant center. The final configuration after relaxation of the cubo-octahedral structure with a missing central atom consisted of a central cube with four capping atoms building four pyramids with a rectangular basis (see Fig. 4.1 configuration 12a). The magnetic ground state is S = 2 with full D4h symmetry, which is also adopted by the magnetic isomers with higher spin. In a low-spin isomer with S = 1, the magnetic structure breaks the geometric symmetry, this ferrimagnetic configuration is completely asymmetric (see supporting material for details). The icosahedral structure with a vacant center is found to be unstable, it relaxes to a centred icosahedron with a vacant site in the outer shell (see Fig. 4.1 configuration 12b). The energetically most favorable magnetic isomer is S = 3, but with only minimal magnetic energy differences if the spin is reduced or enhanced by one unit. The fivefold rotational symmetry about an axis through the center and the vacant site (point group C5v ) is realized only in the magnetic ground state, in the metastable magnetic isomers the symmetry is reduced to C1h . This structure is lower in energy by 24 meV/atom than the capped cube. DSA leads to a structure which is best described as a poly-octahedral cluster: two edge-sharing octahedra plus two more atoms added such that a further halfoctahedron is formed, the optimized structure is plotted in Fig. 4.1 configuration 12c? , the symmetry is only C1h . The stable magnetic isomers is S = 3, this structure is still lower in energy by 17 meV than the incomplete icosahedron. For Rh12 we compared the same structures as for Pd12 . The poly-octahedral cluster structure (configuration 12c* in Fig. 4.1, symmetry C1h ), which has been determined by dynamical simulated annealing is found to have the lowest energy. The stable spin isomer of this structure has a magnetic moment of 16 µB (resp. ˙ B /atom) and a binding energy of 3.714 eV/atom. The magnetic energy differ1.3µ ence compared to an isomer with a moment of 15 µB is only in the order of a few meV/atom. The stable magnetic isomers of the incomplete icosahedron (S = 9) and of the capped cube (S = 7) are higher in energy by 33 meV/atom and 60 meV/atom, respectively. The symmetry of all isomers of the incomplete icosahedron is reduced to C1h , states with S ≤ 4 have a pronounced ferrimagnetic character. The capped cube has point group symmetry C4v for the stable isomer, the symmetry is reduced to C2v for lower moment. Results of other researchers give conflicting results. Reddy et al. [155] found the 8 µB spin isomer of the incomplete icosahedron to be the ground

118

4 Pd and Rh clusters

state of Pd12 , whereas Aguilera-Granja et al. [157] using a spd tight-binding model, found the 3 µB spin isomer to be the energetically most stable one. Experimental studies by Cox et al. [118] provide a magnetic moment/atom of 0.59 µB ± 0.12 µB .

4.4.12

Pd13 and Rh13

For the Pd13 cluster our starting structures for the static optimizations were the cubo-octahedron (Oh ) and the icosahedron (Ih ) (in contrast to the Pd12 cluster, the central site is now occupied by a Pd atom). In addition, a third structure was generated by a dynamical simulated annealing run. Surprisingly, the cubo-octahedron was found to be unstable also in this case, the static relaxation led to a body-centred tetragonal structure with four capping atoms (see Fig. 4.1 configuration 13a). The stable spin isomer is S = 2, in this case (and for the higher spin isomers) the symmetry is C4v . The magnetic moment at the body-centred site is 0.69 µB , the atoms occupying the corners of the distorted cube have moments of 0.28 µB , those of the capping atoms are only slightly lower with 0.26 µB . For the low-spin isomer with S = 1 both the molecular and the magnetic symmetry are reduced to C2h , the magnetic structure has a weak antiferromagnetic component. Magnetic energy differences relative to the S = 2 state are 7 meV for S = 1 and 5 meV for S = 3. Similar observations apply also to the icosahedron. The stable spin isomer is S = 4, for this high-spin solution the icosahedral Ih symmetry is conserved in the geometric and the magnetic symmetry. The symmetry is reduced to C2h for the low spin isomers (S = 0 to S = 2), their magnetization acquires a strong antiferromagnetic component (see Fig. 4.16). The icosahedron is energetically favored over the tetragonal structure derived from the cubo-octahedron, but DSA has identified an energetically more favorable structural isomer.

Figure 4.16: Isosurface plots of the magnetization densities of the S = 0 and S = 4 spinisomers of a (nearly) icosahedral Pd13 cluster. Dark surfaces surround regions of negative, light surfaces of positive magnetization. Cf. text.

4.4 Structure, magnetic and electronic properties ...

119

The structure resulting from the dynamical simulated annealing run is shown in Fig. 4.1, configuration 13c*. It may be considered as a cluster of three distorted edgesharing octahedra: two octahedra sharing one edge and a third one is added in such a way that it shares edges with both and that one of its edges connects the vertices of the two octahedra. The average bond length is equal to 2.69 ˚ A, individual bonds measure between 2.64 ˚ A and 2.79 ˚ A, and the point group symmetry is only C1h . This structure has the shortest average bond-length of all structural isomers, but a lower average coordination number than the icosahedron. The stable magnetic isomer is S = 3, but the magnetic energy differences to the low-spin isomers are exceedingly small (4Emag ≤ 3meV). The magnetization of this structure is surprisingly uniform, in the stable isomer the largest atomic moment is 0.51 µB , the smallest 0.45 µB . Fig. 4.17 shows a graph of the relaxed cluster structure, featuring also the magnetic moments. 13

13

0.45

0.49

0.45

0.45

0.45

7 0.49

3 2

0.49

10 0.45

0.51

12 0.45

11 0.45

7

4 1

0.45

0.45

0.49

3

0.45

6

0.45

0.51

0.45

1

5

2

0.45

9

8

0.49

4

6

0.45

0.45

12

8

5

11

0.45

0.49

9 0.45

10

0.45

Figure 4.17: Structure of the stable Pd13 cluster with local magnetic moments (left), side-view of the relaxed cluster (right). Cf. text.

A side view of the cluster is particulary instructive as it demonstrates that it consists essentially of two close-packed sheets of atoms, in a close-packing stacking. This applies not only to the N=13 cluster, but also to the stable structural isomers of the N=12 and N=11 clusters. This is a rather surprising result suggesting that surface energies are a decisive structure-determining factor even for these very small clusters. The structural energy differences is 20 meV/atom compared to the icosahedron. Reddy et al. [150] (DFT-GGA) found the icosahedron with a magnetic moment/atom of 0.43 µB for the central atom and 0.12 µB for the surface atoms to be the ground state of Pd13 .

120

4 Pd and Rh clusters

Figure 4.18: Isosurfaces of the magnetization densities for Pd13 clusters with a polyoctahedral structure, for S = 1 to S = 3 (from left to right). Cf text.

For Rh13 we considered the same structures as for Pd13 . The spin isomer of the octahedral cluster produced by DSA, with a magnetic moment of 15 µB was found to represent the ground state (EB = 3.790 eV/atom). The structural distortions of the individual octahedra are slightly larger than for Pd13 , with interatomic distances ranging between 2.53 ˚ A and 2.79 ˚ A. The magnetization is quite homogenous, with the smallest moments (0.45 µB ) at the four-coordinated atoms at the corners and the largest moment (0.61 µB ) at the eight-fold coordinated site shared by all three octahedra. The icosahedron with a magnetic moment of 21 µB has a binding energy of 3.745 eV/atom. The capped body-centred structure (configuration 13a in Fig. 4.1) is most stable in an S = 5/2 state, with a structural energy difference of 99 meV/atom relative to the ground state. Here it is interesting to observed that while in the Pd13 cluster with this structure the largest magnetic moment was found on the body-centred position, in the Rh13 cluster the magnetic moment is almost completely quenched (to 0.10 µB ) at the central atom, while the by far largest moments are found on the capping atoms. The low-spin isomers of the icosahedron display again a strong antiferromagnetic component in the spin-density distribution (see Fig. 4.19) which triggers also a substantial geometric distortion to C 2h symmetry, while in the high-spin isomers with S = 15/2 and S = 17/2 the full Ih symmetry is conserved. These two isomers are higher in energy by only 3 and 6 meV respectively than the ground state with S = 21/2. Interestingly, at the highest magnetization, Ih symmetry is reduced to C2h symmetry, with more pronounced distortions of the geometric than of the magnetic structure. On the octahedral-cluster configuration, the magnetization is quite homogenous, with local magnetic moments between 1.03 µB and 1.27 µB for the stable isomer (see Fig. 4.20). The distribution of the local moments, with equal moments on a hexagon surrounding the central atoms, also suggests a slightly different view of this structure as a fragment of a face-centred cubic structure: a centred hexagon forms the fragment of a close-packed layer, atoms 2, 7 and 8 start the next layer of a close sphere packing, atoms 1, 5 and 13 are positioned such as to terminate dangling

4.5 Summary and Conclusions

121

Figure 4.19: Isosurface plots of the magnetization densities of the S = 7/2 and S = 21/2 isomers of an icosahedral Rh13 cluster. Dark surfaces surround regions of negative, light surfaces of positive magnetization. Cf. text.

bonds at the fringe of the fragment. A side-view of the structure demonstrates the layer-structure of the optimized geometry. 13 1.03

13 11

12

8

1.09

1.03

11

7 1.09

1.27

1.27

7 4

3 2

1.09

10 1.27

1.04

4 1

1.03

12

1.27

8

1.09

6

5

1.27

1.03

1.27

1.27

1

1.27

5

2

1.03

1.27

6

1.04

1.27

9

1.09

3

1.03

1.09

9 1.27

10

1.27

Figure 4.20: Structure of the stable Rh13 cluster with local magnetic moments (left), side-view of the relaxed cluster (right). Cf. text.

4.5

Summary and Conclusions

In this work we have presented a comprehensive investigation of the structural, electronic, and magnetic properties of PdN and RhN clusters with up to N = 13 atoms. The novel aspects of our investigation are the following: (i) The structural optimization of the cluster by a symmetry-unconstrained static total-energy minimization has been supplemented for larger clusters (N ≥ 7) by a search of the ground-statestructure by dynamical simulated annealing. The dynamical structural optimization has led to the discovery of highly unexpected ground-state configurations. (ii) The

122

4 Pd and Rh clusters

spin-polarized calculations have been performed in a fixed-moment mode. This allows to study coexisting magnetic isomers and leads to a deeper insight into the importance of magneto-structural effects. For all clusters with nine or more atoms, the dynamical simulated annealing strategy has identified novel structures with a lower energy than any of the structural variants discussed so far in the literature: for N = 9 a double trigonal antiprism (similar to the canonical Bernal polyhedron for nine atoms), for N = 11 a structure of edge-sharing octahedra, completed by one ad-atom, for N = 12 a similar configuration, but with two ad-atoms completing a half-octahedron, and for N = 13 a cluster of three octahedra. Alternatively, the last three configurations can also be considered as a stacking of fragments of two close-packed planes, slightly distorted at the edges, i.e as fragments of the bulk fcc structures of both elements. That even for such small clusters crystalline fragments should be preferred over non-crystallographic motifs achieving a higher average coordination (such as, e.g., the icosahedron) is certainly quite surprising. It is also somewhat surprising that while an icosahedron represents at least a local minimum on the potential-energy surface for the N = 13 clusters, a cubo-octahedron was found to be unstable and relaxed to a tetragonally distorted capped cube (which represents in turn only a local minimum). With these new ground-state configurations, the binding energy shows a very smooth variation with cluster size - it varies essentially proportional to the squareroot of the average coordination number, emphasizing that in these transition-metal systems, the size-dependence is better understood in terms of even oversimplified tight-binding arguments than in terms of shell-model considerations relevant for jellium-like simple metal clusters. Our calculations demonstrate that structural energy differences can be small enough (a few tens of meV/atom) to lead to a finite probability for the formation of metastable cluster structures at higher temperatures. A large effort has been spent, especially for Rh clusters, to investigate the coexistence of magnetic isomers. Quite generally, magnetic energy differences are much smaller than structural energy differences - for the larger clusters, the magnetic ground-state is favored over other magnetic isomers only by a few meV/atom. As a consequence, excited magnetic isomers exist even at ambient temperatures. As examples, we show in Fig. 4.21(a) the probability to find an octahedral Pd6 cluster in the S = 1 triplet and in the S = 0 singulet states - at T ∼ 500 K, about 25 pct. of the clusters loose their magnetic moment. In Fig. 4.21(b) a similar analysis for a Rh9 cluster is presented. In this case, besides the S = 11/2 magnetic ground-state of double trigonal antiprism, isomers with both higher (S = 13/2) and lower (S = 9/2) magnetic moment coexist at higher temperatures. In this case the diagram also shows that there is a finite probability of a coexisting structural isomer (the capped pentagonal bipyramid).

4.5 Summary and Conclusions

123

1.00 0.90 0.80 oct S=1

PN=6,T(I,S)

0.70 0.60

pbp-1 S=1 pbp-1 S=0 pent.pyr S=1, pent.pyr S=0

0.50 0.40 0.30

oct S=0

0.20 0.10 0.00 100

200

300

400

500 600 700 temperature [K]

800

900

1000

1.00 0.90 0.80

dta S=11/2

PN=9,T(I,S)

0.70 0.60 dta S=9/2 capped pbp S=11/2 capped pbp S=13/2 capped pbp S=9/2

0.50 0.40 0.30

dta S=13/2

0.20 0.10 0.00 100

200

300

400

500 600 700 temperature [K]

800

900

1000

Figure 4.21: Probability PN =6 and PN =9 to find a P d6 and Rh9 cluster with Spin S, respective to its isomeric structure I, with respective to temperature [K], see Section 3.6.3 for theory.

124

4 Pd and Rh clusters

Fig. 4.22 summarizes the results for the expected temperature-dependence of the magnetic moments of PdN and RhN clusters. The average is over both magnetic and structural isomers. The surprising result is that the magnetic moment can increase as well as decrease with temperature, as in some cases low- and high-spin isomers are excited with almost the same probability. Examples for a magnetic moments increasing with temperature are the Rh4 (which is non-magnetic in the groundstate) and the Rh12 cluster with a lower magnetic energy difference for a high- than for a low-spin isomer. For a Pd9 cluster, the magnetic moment shows even a nonmonotonous variation as a function of T - this is due to the fact that for the structural ground-state configuration, magnetic isomers with higher and lower moment occur with comparable probability and that in addition, at higher temperatures there is in addition a finite probability to form a high-spin structural isomer (cf. the magnetic and structural energy differences listed in Table4.1 and 4.2). Our study also presents ample evidence for magneto-structural effects - the simples examples are the Pd3 triangle, the Pd4 tetrahedron, and the Pd5 trigonal bipyramid - in all three cases, the symmetry of the cluster is lower in the non-magnetic S = 0 isomer than in the S = 1 ground state - in all three cases the S = 0 isomer is in fact not paramagnetic, but antiferromagnetic. Antiferromagnetic components of the magnetization densities and magnetically-induced symmetry breaking have also been found in a number of other cases. Striking examples are the Rh6 octahedron, where only the paramagnetic S = 0 and the stable S = 3 isomer adopt the full Oh symmetry, while all other magnetic isomers are tetragonally distorted, and the icosahedral clusters: Pd13 has full icosahedral (Ih ) symmetry only for the high-spin isomers, while for Rh13 icosahedral symmetry is reduced to orthorhombic both in the high- and low-spin limit. Altogether, the magnetism of Pd and Rh clusters shows a rich variety of phenomena which had hardly been explored up to now. The chemical properties of small clusters depend on their HOMO-LUMO gap and on the nature of the frontier orbitals. Our results compiled in Table4.1 and 4.2 demonstrate a strong variation of the gap with the magnetization for a given structural isomer, and also between possible metastable structural variants. The analysis of the electronic spectrum (not reported in detail here) also point to changes in the nature of the frontier orbitals - the consequences on the chemical reactivity of the clusters remain to be explored.

thermally averaged magnetic moment/atom [µB]

4.5 Summary and Conclusions

125

0.55 0.5

Pd 12

0.45

Pd 4 Pd 5

0.4

Pd 10

Pd 11 0.35

Pd 3 Pd 13

0.3

Pd 6

0.25

Pd 8

Pd 7 Pd 9

0.2 100

200

300

400

500

600

700

800

900

1000

thermally averaged magnetic moment/atom [µB]

temperature [K] 2 Rh 7

1.8 1.6 Rh 10

1.4

Rh 12

Rh 9 1.2

Rh 8

1

Rh 11

0.8

Rh 13

Rh 6 Rh 3 Rh 5

0.6 0.4

Rh 4

0.2 0 100

200

300

400

500

600

700

800

900

1000

temperature [K]

Figure 4.22: Thermally averaged magnetic moment/atom [µB ] for Pd and Rh clusters with respective to temperature [K], see Section 3.6.3 for theory.

126

4 Pd and Rh clusters

Part III Appendices

A

Appendix Pd-Cluster Pd2: Dumbbell Total magnetic moment: 0µB Bond lengths [˚ A]

2

-0.01

1- 2: 2.57 Average bond length: 2.57 ˚ A

1

-0.01

4Emag : 173 meV Point symmetry: D∞h

Total magnetic moment: 2µB Bond lengths [˚ A]

2

1.00

1- 2: 2.48 Average bond length: 2.48 ˚ A

1

1.00

4Emag : - meV Point symmetry: D∞h

Pd3: Equilateral Triangle Total magnetic moment: 0µB Bond lengths [˚ A]

3 0.00

1- 2: 2.50 1- 3: 2.49

2- 3: 2.49

Average bond length: 2.49 ˚ A

1

0.05

2

-0.05

4Emag : - meV Point symmetry: C2v - mm2

129

130

Appendix A

Total magnetic moment: 2µB Bond lengths [˚ A]

3 0.67

1- 2: 2.52 1- 3: 2.52

2- 3: 2.52

Average bond length: 2.52 ˚ A

1

0.67

2

0.67

4Emag : - meV Point symmetry: D3h - ¯ 62m

Isosurfaces of the magnetization densities for M=0µB (left) and M=2µB (right) for the Pd3 triangle. Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

Pd4a: Square Total magnetic moment: 0µB 4

0.00

Bond lengths [˚ A]

3

0.00

1- 2: 2.60 1- 4: 2.60

2- 3: 2.60

3- 4: 2.60

Average bond length: 2.60 ˚ A

1

0.00

2

0.00

4Emag : 251 meV Point symmetry: D4h - 4/mmm

Pd-Cluster

131

Total magnetic moment: 2µB Bond lengths [˚ A]

4

3

0.50

0.50

1- 2: 2.49 1- 4: 2.49

2- 3: 2.49

3- 4: 2.49

Average bond length: 2.49 ˚ A

1

4Emag : - meV 4Estruct : 190 meV

2

0.50

0.50

Point symmetry: D4h - 4/mmm

Pd4b: Rhombus Total magnetic moment: 0µB Bond lengths [˚ A]

4

3

0.00

0.00

1- 2: 2.56 1- 4: 2.56

2- 4: 2.47

3- 4: 2.56

2- 3: 2.56

Average bond length: 2.54 ˚ A

1

2

0.00

4Emag : - meV 4Estruct : 209 meV

0.00

Point symmetry: D2h - mmm

Total magnetic moment: 2µB Bond lengths [˚ A]

4

3

0.54

0.46

1- 2: 2.56 1- 4: 2.56

2- 4: 2.57 2- 3: 2.56

Average bond length: 2.56 ˚ A

1

2

0.46

4Emag : 1 meV

0.54

Point symmetry: D2h - mmm

Pd4c: Tetrahedron Total magnetic moment: 0µB 4

Bond lengths [˚ A]

-0.26 1- 2: 2.63 1- 3: 2.63 1- 4: 2.56

1

3

0.26

-0.26

2 0.26

2- 3: 2.56 2- 4: 2.63

Average bond length: 2.61 ˚ A 4Emag : 23 meV Point symmetry: S4 * - ¯ 4

3- 4: 2.63

3- 4: 2.56

132

Appendix A

Total magnetic moment: 2µB Bond lengths [˚ A]

4

0.50

1- 2: 2.61 1- 3: 2.61 1- 4: 2.61

1

3

0.50

0.50

2- 3: 2.61 2- 4: 2.61

3- 4: 2.61

Average bond length: 2.61 ˚ A 4Emag : - meV 4Estruct : - meV

2 0.50

Point symmetry: Td - ¯ 43m

Isosurfaces of the magnetization densities for M=0µB (left) and M=2µB (right) for the Pd4 tetrahedron. Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

Pd5a: Square Pyramid Total magnetic moment: 0µB Bond lengths [˚ A]

5

0.00

4 1

0.00

0.00

1- 2: 2.57 1- 4: 2.57 1- 5: 2.65

3 2 0.00

0.00

2- 3: 2.57 2- 5: 2.65

Average bond length: 2.61 ˚ A 4Emag : 50 meV Point symmetry: C4v - 4mm

3- 4: 2.57 3- 5: 2.65

4- 5: 2.65

Pd-Cluster

133

Total magnetic moment: 2µB Bond lengths [˚ A]

5

0.28

1- 2: 2.56 1- 4: 2.56 1- 5: 2.66

4

0.43

1

0.43

3

0.43

2 0.43

2- 3: 2.56 2- 5: 2.66

3- 4: 2.56 3- 5: 2.66

Average bond length: 2.61 ˚ A 4Emag : - meV 4Estruct : 7 meV Point symmetry: C4v - 4mm

Pd5b: Trigonal Bipyramid Total magnetic moment: 0µB Bond lengths [˚ A]

5 0.00 1- 2: 2.63 1- 3: 2.63 1- 4: 2.64

2 -0.36

3

2- 3: 2.74 2- 4: 2.61 2- 5: 2.64

4- 5: 2.64

0.38 A Average bond length: 2.64 ˚

4 -0.02

1

3- 4: 2.62 3- 5: 2.63

0.00

4Emag : 45 meV Point symmetry: C2v * - mm2

Total magnetic moment: 2µB Bond lengths [˚ A]

5 0.38

2 0.41

3

1- 2: 2.64 1- 3: 2.64 1- 4: 2.64

2- 3: 2.65 2- 4: 2.65 2- 5: 2.64

4- 5: 2.64

0.41

4 1 0.38

3- 4: 2.65 3- 5: 2.64

0.41

Average bond length: 2.64 ˚ A 4Emag : - meV 4Estruct : - meV Point symmetry: D3h - ¯ 62m

4- 5: 2.66

134

Appendix A

Isosurfaces of the magnetization densities in trigonal bipyramids of Pd5 for M=0µB (left) and M=2µB (right). Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

Pd5c: Flat Trigonal Bipyramid Total magnetic moment: 0µB 3

Bond lengths [˚ A]

0.00 1- 2: 2.55 1- 3: 2.55 1- 4: 2.55

1

0.00

5 4

0.00

3- 5: 2.55 2- 5: 2.55 4- 5: 2.55

Average bond length: 2.55 ˚ A

0.00

4Emag : 58 meV

2 0.00

Point symmetry: D3h - ¯ 62m

Total magnetic moment: 2µB 3

Bond lengths [˚ A]

0.36 1- 2: 2.55 1- 3: 2.55 1- 4: 2.55

1 5 4

0.36

0.46

3- 5: 2.55 2- 5: 2.55

Average bond length: 2.55 ˚ A

0.46

2 0.36

4Emag : - meV 4Estruct : 76 meV Point symmetry: D3h - ¯ 62m

4- 5: 2.55

Pd-Cluster

135

Pd6a: Pentagonal Pyramid Total magnetic moment: 0µB 5

Bond lengths [˚ A]

0.06

4

6 0.01

-0.14

1

1- 2: 2.56 1- 5: 2.55 1- 6: 2.71

2- 3: 2.55 2- 6: 2.71 3- 4: 2.55

0.01

3 0.06

2

0.01

3- 6: 2.70 5- 6: 2.70 4- 5: 2.55 4- 6: 2.68

Average bond length: 2.63 ˚ A 4Emag : 21 meV Point symmetry: C1h - m

Total magnetic moment: 2µB Bond lengths [˚ A]

5

0.36

4

6 0.19

0.35

1

1- 2: 2.56 1- 5: 2.57 1- 6: 2.68

2- 3: 2.57 2- 6: 2.68 3- 4: 2.57

0.38

3- 6: 2.69 5- 6: 2.69 4- 5: 2.57 4- 6: 2.69

Average bond length: 2.63 ˚ A

3 0.36

2

0.38

4Emag : - meV 4Estruct : 172 meV Point symmetry: C1h - m

Pd6b: Octahedron Total magnetic moment: 0µB 5

Bond lengths [˚ A]

0.00

2 1

3

0.00

0.00

0.00

4 0.00

1111-

2: 4: 5: 6:

2.65 2.65 2.66 2.66

2- 3: 2.65 2- 5: 2.66 2- 6: 2.66

3- 4: 2.65 3- 5: 2.66 3- 6: 2.66

Average bond length: 2.66 ˚ A 4Emag : 9 meV

6 0.00

Point symmetry: D4h - 4/mmm

4- 5: 2.66 4- 6: 2.66

136

Appendix A

Total magnetic moment: 2µB Bond lengths [˚ A]

5 0.34

2 3

0.33

1 0.33

0.33

4

1111-

2: 4: 5: 6:

2.66 2.66 2.66 2.66

2- 3: 2.66 2- 5: 2.66 2- 6: 2.66

3- 4: 2.66 3- 5: 2.66 3- 6: 2.66

4- 5: 2.66 4- 6: 2.66

Average bond length: 2.66 ˚ A

0.33

4Emag : - meV 4Estruct : - meV

6 0.34

Point symmetry: Oh - m3m

Isosurfaces of the magnetization densities in octahedron of P6 for M=2µB . Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

Pd6c: Incomplete Pentagonal Bipyramid Total magnetic moment: 0µB Bond lengths [˚ A]

6 -0.36

5

3

0.00

1

0.00

2

0.00

0.00

4

0.36

1111-

2: 4: 5: 6:

2.71 2.65 2.65 2.65

2- 3: 2.65 2- 4: 2.65 2- 6: 2.65 3- 4: 2.62

Average bond length: 2.67 ˚ A 4Emag : 10 meV Point symmetry: C2v - mm2

3- 6: 2.62 4- 5: 2.62 4- 6: 2.95

5- 6: 2.62

Pd-Cluster

137

Total magnetic moment: 2µB Bond lengths [˚ A]

6 0.32

5

3

0.35

1

0.35

2

0.33

0.33

4

0.32

1111-

2: 4: 5: 6:

2.70 2.66 2.67 2.66

2- 3: 2.67 2- 4: 2.66 2- 6: 2.66 3- 4: 2.61

Average bond length: 2.67 ˚ A 4Emag : - meV 4Estruct : 42 meV Point symmetry: C2v - mm2

3- 6: 2.61 4- 5: 2.61 4- 6: 2.93

5- 6: 2.61

138

Appendix A

Pd7a: Centred Hexagon Total magnetic moment: 0µB 6

7

Bond lengths [˚ A]

5

0.00

0.00

4

1

0.00

0.00

0.00

11111-

2: 3: 4: 5: 6:

2.50 2.50 2.95 2.50 2.50

1- 7: 2.95

3- 4: 2.63

2- 3: 2.73 2- 7: 2.63

4- 5: 2.63

6- 7: 2.63

5- 6: 2.73

Average bond length: 2.66 ˚ A 4Emag : - meV 4Estruct : 285 meV

2

3

0.00

0.00

Point symmetry: D2h - mmm

Total magnetic moment: 2µB 6

5

0.29

7

4

1

0.21

Bond lengths [˚ A]

0.29

0.43

0.21

11111-

2: 3: 4: 5: 6:

2.58 2.58 2.75 2.58 2.58

1- 7: 2.75

3- 4: 2.64

2- 3: 2.64 2- 7: 2.64

4- 5: 2.64

6- 7: 2.64

5- 6: 2.64

Average bond length: 2.64 ˚ A 4Emag : 12 meV

2

3

0.29

0.29

Point symmetry: D2h - mmm

Pd7b∗: Pentagonal Bipyramid Total magnetic moment: 0µB Bond lengths [˚ A]

6 0.00

5

1

-0.26

4

0.10

2

3

0.10

-0.26

7 0.00

0.33

1111-

2: 5: 6: 7:

2.64 2.67 2.69 2.69

2- 3: 2.67

2- 6: 2.69 2- 7: 2.69 3- 4: 2.65 3- 6: 2.72 3- 7: 2.72

Average bond length: 2.70 ˚ A 4Emag : 10 meV Point symmetry: C2v - mm2

5- 7: 2.72 4- 5: 2.65 4- 6: 2.67 4- 7: 2.67 5- 6: 2.72

6- 7: 2.96

Pd-Cluster

139

Total magnetic moment: 2µB Bond lengths [˚ A]

6

1111-

0.19

5

1

0.33

4

0.33

2

0.30

3

0.33

0.33

2: 5: 6: 7:

2.65 2.63 2.70 2.70

2- 3: 2.63

2- 6: 2.70 2- 7: 2.70 3- 4: 2.65 3- 6: 2.70 3- 7: 2.70

5- 7: 2.70 4- 5: 2.65 4- 6: 2.70 4- 7: 2.70

6- 7: 3.00

5- 6: 2.70

A Average bond length: 2.70 ˚

7

4Emag : - meV 4Estruct : - meV

0.19

Point symmetry: C2v - mm2

Isosurface plots of the magnetization densities in pentagonal bipyramids of Pd 7 for M=0µB (left) and M=2µB (right). Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

Pd8a: Bicapped Octahedron I Total magnetic moment: 0µB Bond lengths [˚ A]

5

7

-0.21

-0.10

-0.10

2 1

8

0.03

0.03

3 4 0.03

6 0.28

0.03

11111-

2: 4: 5: 6: 8:

2.97 2.59 2.73 2.66 2.56

2222-

3: 5: 6: 8:

2.59 2.73 2.66 2.56

3- 4: 2.97 Average bond length: 2.69 ˚ A 4Emag : 3 meV Point symmetry: C2v - mm2

3- 5: 2.73 3- 6: 2.66 3- 7: 2.56 4- 5: 2.73 4- 6: 2.66

4- 7: 2.56 5- 7: 2.76 5- 8: 2.76

140

Appendix A

Total magnetic moment: 2µB Bond lengths [˚ A]

5

7

0.17

0.25

0.25

2 1

8

0.27

0.27

3 4

0.27

11111-

2: 4: 5: 6: 7:

2.87 2.61 2.78 2.67 2.58

2222-

3: 5: 6: 7:

2.61 2.78 2.67 2.58

3- 4: 2.87

0.27

3- 5: 2.78 3- 6: 2.67 3- 8: 2.58

4- 8: 2.58 5- 7: 2.66 5- 8: 2.66

4- 5: 2.78 4- 6: 2.67

Average bond length: 2.69 ˚ A 4Emag : 4 meV

6 0.27

Point symmetry: C2v - mm2

Total magnetic moment: 4µB Bond lengths [˚ A]

5

7

0.44

0.55

0.55

2 1

8

0.52

0.52

3 4

11111-

2: 4: 5: 6: 7:

2.74 2.69 2.79 2.67 2.59

0.52

2222-

3: 5: 6: 7:

2.69 2.79 2.67 2.59

3- 4: 2.74

0.52

3- 5: 2.79 3- 6: 2.67 3- 8: 2.59

4- 8: 2.59 5- 7: 2.58 5- 8: 2.58

4- 5: 2.79 4- 6: 2.67

Average bond length: 2.68 ˚ A 4Emag : - meV 4Estruct : 78 meV

6 0.38

Point symmetry: C2v - mm2

Total magnetic moment: 6µB Bond lengths [˚ A]

5

7

0.78

0.75

0.75

2 1

8

0.74

0.74

3 4 0.74

6 0.76

0.74

11111-

2: 4: 5: 6: 7:

2.66 2.68 2.79 2.66 2.60

2222-

3: 5: 6: 7:

2.68 2.79 2.66 2.60

3- 4: 2.66 Average bond length: 2.67 ˚ A 4Emag : 28 meV Point symmetry: C2v - mm2

3- 5: 2.79 3- 6: 2.66 3- 8: 2.60 4- 5: 2.79 4- 6: 2.66

4- 8: 2.60 5- 7: 2.56 5- 8: 2.56

Pd-Cluster

141

Pd8b∗: Bicapped Octahedron II Total magnetic moment: 0µB Bond lengths [˚ A]

5

7

0.00

0.17

2

-0.01

1

-0.17 4 -0.17

3

2: 4: 5: 6:

2.67 2.68 2.67 2.66

2- 5: 2.69 2- 6: 2.69

0.00

8

1111-

0.17

6

2- 7: 2.67 2- 8: 2.67 3333-

4: 5: 6: 7:

2.67 2.69 2.69 2.66

3- 8: 2.66

6- 8: 2.67

4- 5: 2.67 4- 6: 2.67

7- 8: 2.68

5- 7: 2.67

Average bond length: 2.67 ˚ A 4Emag : 18 meV

0.00

Point symmetry: C2v - mm2

Total magnetic moment: 2µB Bond lengths [˚ A]

5

7

0.28

0.22

2

0.28

1

0.22 4 0.22

3

2: 4: 5: 6:

2.67 2.68 2.67 2.67

2- 5: 2.69 2- 6: 2.69

0.28

8

1111-

0.22

6

0.28

2- 7: 2.67 2- 8: 2.67 3333-

4: 5: 6: 7:

2.67 2.69 2.69 2.67

3- 8: 2.67

6- 8: 2.67

4- 5: 2.67 4- 6: 2.67

7- 8: 2.68

5- 7: 2.67

A Average bond length: 2.67 ˚ 4Emag : - meV 4Estruct : - meV Point symmetry: C2v - mm2

Total magnetic moment: 4µB Bond lengths [˚ A]

5

7

0.51

1

0.49

2

0.51

0.49 4 0.49

3

6

0.51

2: 4: 5: 6:

2.72 2.67 2.64 2.64

2- 5: 2.67 2- 6: 2.67

0.51

8

1111-

0.49

2- 7: 2.65 2- 8: 2.65 3333-

4: 5: 6: 7:

2.72 2.67 2.67 2.65

Average bond length: 2.67 ˚ A 4Emag : 11 meV Point symmetry: C2v - mm2

3- 8: 2.65

6- 8: 2.71

4- 5: 2.64 4- 6: 2.64

7- 8: 2.66

5- 7: 2.71

142

Appendix A

Total magnetic moment: 6µB Bond lengths [˚ A]

5

7

0.77

0.65

2

1

0.81

0.77 4 0.77

3

1111-

2: 4: 5: 6:

2.76 2.81 2.59 2.59

2- 5: 2.66 2- 6: 2.66

0.81

8

0.65

6

2- 7: 2.68 2- 8: 2.68 3333-

4: 5: 6: 7:

2.76 2.66 2.66 2.68

3- 8: 2.68

6- 8: 2.65

4- 5: 2.59 4- 6: 2.59

7- 8: 2.61

5- 7: 2.65

A Average bond length: 2.67 ˚ 4Emag : 81 meV

0.77

Point symmetry: C2v - mm2

Pd9a: Capped Pentagonal Bipyramid Total magnetic moment: 0µB 8

Bond lengths [˚ A]

9

0.10

0.10

1111-

6 -0.01

4

3

-0.01 2 -0.09

0.17

5

-0.01

1

-0.09

7

2: 5: 6: 7:

2.66 2.67 2.64 2.66

2- 3: 2.67 2- 6: 2.64 2- 7: 2.66

3333-

4: 6: 7: 8:

2.72 2.80 2.72 2.67

4- 5: 2.72 4- 6: 3.01

4- 7: 2.73 4- 8: 2.65 4- 9: 2.65

6- 8: 2.64 6- 9: 2.64 8- 9: 2.66

5- 6: 2.80 5- 7: 2.72 5- 9: 2.67

Average bond length: 2.68 ˚ A 4Emag : 21 meV

-0.14

Point symmetry: C1h - m

Total magnetic moment: 2µB Bond lengths [˚ A]

8

9

0.30

0.30

1111-

6 0.28

3

0.09 2 0.30

4

0.17

5 1 0.30

7

0.16

0.09

2: 5: 6: 7:

2.68 2.68 2.64 2.65

2- 3: 2.68 2- 6: 2.64 2- 7: 2.65

3333-

4: 6: 7: 8:

2.73 2.81 2.73 2.67

4- 5: 2.73 4- 6: 2.95

Average bond length: 2.71 ˚ A 4Emag : 11 meV Point symmetry: C1h - m

4- 7: 2.70 4- 8: 2.65 4- 9: 2.65

6- 7: 2.95 6- 8: 2.64 6- 9: 2.64

5- 6: 2.81 5- 7: 2.73 5- 9: 2.67

8- 9: 2.68

Pd-Cluster

143

Total magnetic moment: 4µB Bond lengths [˚ A]

8

9

0.50

0.50

1111-

6 0.35

4

3

0.36 2 0.51

0.45

5 1 0.51

7

0.45

0.36

2: 5: 6: 7:

2.65 2.67 2.66 2.63

2- 3: 2.67 2- 6: 2.66 2- 7: 2.63

3333-

4: 6: 7: 8:

2.74 2.75 2.74 2.67

4- 5: 2.74 4- 6: 2.95

4- 7: 2.73 4- 8: 2.62 4- 9: 2.62

6- 7: 2.94 6- 8: 2.66 6- 9: 2.66

5- 6: 2.75 5- 7: 2.74 5- 9: 2.67

8- 9: 2.65

Average bond length: 2.71 ˚ A 4Emag : - meV 4Estruct : 10 meV Point symmetry: C1h - m

Pd9b∗: Doubled Trigonal Antiprism Total magnetic moment: 0µB 4 -0.01

9 -0.31

1

0.02

6 1111-

5 0.32

Bond lengths [˚ A]

-0.07

3 0.11

3: 4: 6: 9:

2.66 2.63 2.65 2.68

2- 5: 2.65 2- 7: 2.64

2- 8: 2.63 2- 9: 2.66 3333-

5: 6: 7: 8:

2.60 2.67 2.67 2.66

3- 9: 2.66 4- 5: 2.65 4- 6: 2.64 4- 9: 2.66

5- 7: 2.68 5- 9: 2.76 7- 8: 2.65 8- 9: 2.68

5- 6: 2.68

A Average bond length: 2.66 ˚

7

2 -0.01

8 0.02

-0.07

4Emag : 4 meV Point symmetry: C1h * - m

Total magnetic moment: 2µB 4 0.17

9 0.32

1

0.17

0.17

5 0.31

Bond lengths [˚ A]

6

3 0.39

1111-

3: 4: 6: 9:

2.66 2.64 2.64 2.67

2- 5: 2.66 2- 7: 2.64

2- 8: 2.64 2- 9: 2.66 3333-

5: 6: 7: 8:

2.65 2.67 2.67 2.67

Average bond length: 2.66 ˚ A

7

2 0.16

8 0.16

0.16

4Emag : - meV 4Estruct : - meV Point symmetry: C1h - m

3- 9: 2.65 4- 5: 2.66 4- 6: 2.64 4- 9: 2.66

5- 7: 2.67 5- 9: 2.69 7- 8: 2.64 8- 9: 2.67

5- 6: 2.67

144

Appendix A

Total magnetic moment: 4µB

4

6

1

0.61

Bond lengths [˚ A]

0.61

0.60

1111-

5

9

0.79

0.80

3 0.79

3: 4: 6: 9:

2.67 2.64 2.64 2.67

2- 5: 2.67 2- 7: 2.65

2- 8: 2.65 2- 9: 2.67 3333-

5: 6: 7: 8:

2.62 2.67 2.67 2.67

3- 9: 2.63 4- 5: 2.67 4- 6: 2.65 4- 9: 2.67

5- 7: 2.67 5- 9: 2.63 7- 8: 2.64 8- 9: 2.67

5- 6: 2.67

Average bond length: 2.67 ˚ A

7

2 0.60

0.60

8 0.60

4Emag : 5 meV Point symmetry: D3h - ¯ 62m

Pd10a:Tetragonal Antiprism with Capped Square Faces Total magnetic moment: 0µB

5

Bond lengths [˚ A]

-0.33

3

2

-0.07 1 -0.07

-0.07 4 -0.07

8 7

0.22

0.22

9 6

0.22

0.22

11111-

2: 4: 5: 6: 9:

2.66 2.66 2.68 2.71 2.71

2- 3: 2.66 2- 5: 2.68

2- 8: 2.71 2- 9: 2.71 3333-

4: 5: 7: 8:

2.66 2.68 2.71 2.71

A Average bond length: 2.68 ˚ 4Emag : 9 meV

10 0.05

Point symmetry: C4v - 4mm

4- 5: 2.68 4- 6: 2.71 4- 7: 2.71 6- 7: 2.65 6- 9: 2.65 6-10: 2.68

7- 8: 2.65 7-10: 2.68 8- 9: 2.65 8-10: 2.68 9-10: 2.68

Pd-Cluster

145

Total magnetic moment: 2µB 5

Bond lengths [˚ A]

0.10

3

2

0.23 4 0.23

0.23 1 0.23

8 7

0.22

0.22

9 6

0.22

0.22

11111-

2: 4: 5: 6: 9:

2.66 2.66 2.69 2.69 2.69

2- 3: 2.66 2- 5: 2.69

2- 8: 2.69 2- 9: 2.69 3333-

4: 5: 7: 8:

4- 5: 2.69 4- 6: 2.69 4- 7: 2.69

2.66 2.69 2.69 2.69

6- 7: 2.66 6- 9: 2.66 6-10: 2.69

7- 8: 2.66 7-10: 2.69 8- 9: 2.66 8-10: 2.69 9-10: 2.69

Average bond length: 2.68 ˚ A 4Emag : 12 meV

10 Point symmetry: C4v - 4mm

0.10

Total magnetic moment: 4µB Bond lengths [˚ A]

5

0.39

3

2

0.40 1 0.40

0.40 4 0.40

0.40

2: 4: 5: 6: 9:

2.66 2.66 2.68 2.71 2.71

2- 3: 2.66 2- 5: 2.68

8 7

11111-

0.40

2- 8: 2.71 2- 9: 2.71 3333-

4: 5: 7: 8:

2.66 2.68 2.71 2.71

4- 5: 2.68 4- 6: 2.71 4- 7: 2.71 6- 7: 2.66 6- 9: 2.66 6-10: 2.68

7- 8: 2.66 7-10: 2.68 8- 9: 2.66 8-10: 2.68 9-10: 2.68

9 6

0.40

10 0.39

0.40

A Average bond length: 2.68 ˚ 4Emag : - meV 4Estruct : 45 meV Point symmetry: C4v - 4mm

Isosurfaces of the magnetization densities for M=0µB to M=4µB for Pd10 tetragonal antiprism with capped square faces (from left to right). Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

146

Appendix A

Pd10b: Edge Sharing Double Octahedra Total magnetic moment: 2µB Bond lengths [˚ A]

8

7

0.33

6

5

3

0.15

2

0.15

1111-

0.33

4

0.04

1

0.04

0.15

0.15

2: 4: 7: 9:

2.61 2.63 2.66 2.66

2- 3: 2.74 2- 5: 2.61 2- 7: 2.74

2- 8: 2.74 2- 9: 2.74 2-10: 2.74 3- 4: 2.61 3- 6: 2.61 3- 7: 2.74 3- 8: 2.74 3- 9: 2.74

3-10: 2.74 4- 7: 2.66 4- 9: 2.66

6- 8: 2.66 6-10: 2.66 7- 8: 2.72

5- 6: 2.63 5- 8: 2.66 5-10: 2.66

9-10: 2.72

Average bond length: 2.68 ˚ A

10

0.33

9 0.33

4Emag : 10 meV Point symmetry: D2h - mmm

Total magnetic moment: 4µB Bond lengths [˚ A]

8

6

5

2

0.41

0.37

3

0.41

1111-

7

0.37

4

0.45

1

0.45

0.41

0.41

2: 4: 7: 9:

2.64 2.63 2.67 2.67

2- 3: 2.73 2- 5: 2.64 2- 7: 2.72

2- 8: 2.72 2- 9: 2.72 2-10: 2.72 3- 4: 2.64 3- 6: 2.64 3- 7: 2.72 3- 8: 2.72 3- 9: 2.72

3-10: 2.72 4- 7: 2.67 4- 9: 2.67

6- 8: 2.67 6-10: 2.67 7- 8: 2.71

5- 6: 2.63 5- 8: 2.67 5-10: 2.67

9-10: 2.71

Average bond length: 2.68 ˚ A

10

0.37

9

4Emag : - meV 4Estruct : - meV

0.37

Point symmetry: D2h - mmm

Total magnetic moment: 6µB Bond lengths [˚ A]

8

7

0.60

6

5

3

0.56

2

0.56

0.60

4

0.69

1

0.69

10

1111-

0.60

0.56

0.56

2: 4: 7: 9:

2.63 2.60 2.67 2.67

2- 3: 2.60 2- 5: 2.63 2- 7: 2.72

2- 8: 2.72 2- 9: 2.72 2-10: 2.72 3- 4: 2.63 3- 6: 2.63 3- 7: 2.72 3- 8: 2.72 3- 9: 2.72

Average bond length: 2.68 ˚ A

9 0.60

4Emag : 19 meV Point symmetry: D2h - mmm

3-10: 2.72 4- 7: 2.67 4- 9: 2.67

6- 8: 2.67 6-10: 2.67 7- 8: 2.73

5- 6: 2.60 5- 8: 2.67 5-10: 2.67

9-10: 2.73

Pd-Cluster

147

Pd10c : Trigonal Pyramid Total magnetic moment: 0µB Bond lengths [˚ A]

10

0.00

9

7

0.00

1

0.00

1- 2: 2.56 1- 4: 2.55 1- 7: 2.55

8

0.00

0.00

6

4

0.00

2

0.00

0.00

5

0.00

3

22222-

3: 4: 5: 7: 8:

2.56 2.79 2.79 2.79 2.79

3- 5: 2.55 3- 8: 2.55 4444-

5: 6: 7: 9:

2.79 2.55 2.79 2.79

5- 6: 2.55 5- 8: 2.79 5- 9: 2.79

8- 9: 2.79 8-10: 2.55

6- 9: 2.55

9-10: 2.55

7- 8: 2.79 7- 9: 2.79 7-10: 2.55

A Average bond length: 2.67 ˚ 4Emag : 13 meV

0.00

Point symmetry: T - 23

Total magnetic moment: 2µB Bond lengths [˚ A]

10

0.15

9

7

0.17

1

0.28

1- 2: 2.57 1- 4: 2.55 1- 7: 2.55

8

0.17

6

4

0.17

2

0.47

22222-

-0.01

0.15

5

0.17

3

3: 4: 5: 7: 8:

2.57 2.78 2.78 2.78 2.78

3- 5: 2.55 3- 8: 2.55 4444-

5: 6: 7: 9:

2.79 2.56 2.78 2.79

5- 6: 2.56 5- 8: 2.78 5- 9: 2.79

8- 9: 2.79 8-10: 2.56

6- 9: 2.55

9-10: 2.55

7- 8: 2.79 7- 9: 2.79 7-10: 2.56

Average bond length: 2.67 ˚ A 4Emag : - meV 4Estruct : 69 meV

0.28

Point symmetry: C2v - mm2

Total magnetic moment: 4µB Bond lengths [˚ A]

10

0.37

9

7

0.42

1

0.38

1- 2: 2.56 1- 4: 2.56 1- 7: 2.56

8

0.41

0.41

6

4

0.42

2

0.41

3

0.38

0.38

5

0.42

22222-

3: 4: 5: 7: 8:

2.56 2.77 2.77 2.78 2.78

3- 5: 2.56 3- 8: 2.56 4444-

5: 6: 7: 9:

2.78 2.56 2.77 2.78

A Average bond length: 2.67 ˚ 4Emag : 7 meV Point symmetry: T - 23

5- 6: 2.56 5- 8: 2.77 5- 9: 2.78

8- 9: 2.78 8-10: 2.56

6- 9: 2.56

9-10: 2.56

7- 8: 2.77 7- 9: 2.78 7-10: 2.56

148

Appendix A

Total magnetic moment: 6µB Bond lengths [˚ A]

10

0.52

9

7

0.65

1

0.53

1- 2: 2.57 1- 4: 2.57 1- 7: 2.57

8

0.65

0.65

6

4

0.65

2

0.65

0.52

5

0.65

3

22222-

3: 4: 5: 7: 8:

2.57 2.76 2.76 2.76 2.76

3- 5: 2.57 3- 8: 2.57 4444-

5: 6: 7: 9:

2.76 2.57 2.76 2.76

5- 6: 2.57 5- 8: 2.76 5- 9: 2.76

8- 9: 2.76 8-10: 2.57

6- 9: 2.57

9-10: 2.57

7- 8: 2.76 7- 9: 2.76 7-10: 2.57

A Average bond length: 2.66 ˚ 4Emag : 6 meV

0.53

Point symmetry: T - 23

Isosurfaces of the magnetization densities for M=2µB to M=6µB for Pd10 trigonal pyramid (from left to right). Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

Pd11a: Polytetrahedral Cluster Total magnetic moment: 2µB Bond lengths [˚ A]

1

2

0.12

0.12

1111-

6 0.17 7 0.17

5

0.10

9

0.27

0.10

8

0.27

11

3 0.27

4 0.12

2: 5: 6: 7:

2.66 2.70 2.61 2.61

2- 3: 2.70 2- 6: 2.61 2- 7: 2.61 3- 4: 2.79

3- 6: 2.81 3- 7: 2.81 3- 8: 2.71 3-10: 2.71 44444-

5: 6: 7: 8: 9:

2.79 2.96 2.96 2.62 2.62

Average bond length: 2.71 ˚ A

10 0.27

4Emag : 15 meV Point symmetry: C2v * - mm2

4-10: 2.62 4-11: 2.62 5- 6: 2.81 5- 7: 2.81 5- 9: 2.71 5-11: 2.71

6- 9: 2.61 7-10: 2.61 7-11: 2.61 8- 9: 2.65 10-11: 2.65

6- 7: 2.98 6- 8: 2.61

Pd-Cluster

149

Total magnetic moment: 4µB Bond lengths [˚ A]

1

2

0.31

0.31

1111-

6 0.37 5

0.42

9 0.35

0.42

3- 4: 2.79

0.35

4

11

3 8

0.40

0.35

2.66 2.70 2.61 2.61

2- 3: 2.70 2- 6: 2.61 2- 7: 2.61

7 0.37

2: 5: 6: 7:

3- 6: 2.81 3- 7: 2.81 3- 8: 2.71 3-10: 2.71 44444-

5: 6: 7: 8: 9:

2.79 2.95 2.95 2.62 2.62

4-10: 2.62 4-11: 2.62 5- 6: 2.81 5- 7: 2.81 5- 9: 2.71 5-11: 2.71

6- 9: 2.61 7-10: 2.61 7-11: 2.61 8- 9: 2.65 10-11: 2.65

6- 7: 2.97 6- 8: 2.61

Average bond length: 2.71 ˚ A

10 0.35

4Emag : 4 meV Point symmetry: C2v - mm2

Total magnetic moment: 6µB Bond lengths [˚ A]

1

2

1111-

0.52

0.52

6 0.59 0.59

0.54

9 0.52

0.54

0.52

4

11

3 8

0.59

0.52

2.64 2.69 2.64 2.64

2- 3: 2.69 2- 6: 2.64 2- 7: 2.64

7 5

2: 5: 6: 7:

3- 4: 2.75 3- 6: 2.75 3- 7: 2.75 3- 8: 2.69 3-10: 2.69 4- 5: 2.75 4- 8: 2.64 4- 9: 2.64

4-10: 2.64 4-11: 2.64 5- 6: 2.75 5- 7: 2.75 5- 9: 2.69 5-11: 2.69

6- 9: 2.64 7-10: 2.64 7-11: 2.64 8- 9: 2.64 10-11: 2.64

6- 8: 2.64

Average bond length: 2.68 ˚ A

10

4Emag : - meV 4Estruct : 12 meV

0.52

Point symmetry: C2v - mm2

Pd11b∗: Edge Sharing Octahedra Plus Adatom Total magnetic moment: 2µB Bond lengths [˚ A]

11 8 0.10

0.10

7 0.24

6 5

0.09

0.14

3

2

0.31

0.13

4 1

0.22

10 0.28

0.18

1111-

2: 4: 7: 9:

2.61 2.63 2.64 2.66

22222-

3: 5: 7: 8: 9:

2.86 2.62 2.88 2.63 2.65

2-10: 2.72

4- 9: 2.72

3- 4: 2.62 3- 6: 2.65 3- 7: 2.62 3- 9: 2.66 3-10: 2.67 3-11: 2.65

5- 6: 2.64 5- 8: 2.70 5-10: 2.70

4- 7: 2.70

Average bond length: 2.68 ˚ A

9 0.23

4Emag : 2 meV Point symmetry: C1 - 1

7- 8: 2.68 7-11: 2.69 8-11: 2.63 9-10: 2.77

6- 8: 2.73 6-10: 2.71 6-11: 2.67

150

Appendix A

Total magnetic moment: 4µB Bond lengths [˚ A]

11 8 0.32

0.35

7 0.34

6 5

0.39

0.39

3

2

0.48

0.34

4 1

0.31

0.29

10

1111-

2: 4: 7: 9:

2.61 2.61 2.66 2.68

22222-

3: 5: 7: 8: 9:

2.75 2.60 2.83 2.65 2.64

2-10: 2.74

4- 9: 2.73

3- 4: 2.62 3- 6: 2.64 3- 7: 2.63 3- 9: 2.65 3-10: 2.69 3-11: 2.63

5- 6: 2.61 5- 8: 2.72 5-10: 2.67

7- 8: 2.71 7-11: 2.69 8-11: 2.62 9-10: 2.77

6- 8: 2.71 6-10: 2.72 6-11: 2.71

4- 7: 2.70

Average bond length: 2.68 ˚ A

0.45

9 4Emag : 4 meV

0.34

Point symmetry: C1 - 1

Total magnetic moment: 6µB Bond lengths [˚ A]

11 8 0.54

0.53

7 0.58

6 5

0.53

0.48

3

2

0.67

0.64

4 1

0.49

2: 4: 7: 9:

2.65 2.58 2.70 2.68

22222-

3: 5: 7: 8: 9:

2.68 2.63 2.70 2.68 2.68

2-10: 2.71

4- 9: 2.71

3- 4: 2.63 3- 6: 2.65 3- 7: 2.63 3- 9: 2.66 3-10: 2.70 3-11: 2.64

5- 6: 2.63 5- 8: 2.69 5-10: 2.67

4- 7: 2.73

Average bond length: 2.68 ˚ A

10 0.49

0.53

1111-

9 0.54

4Emag : - meV 4Estruct : - meV Point symmetry: C1 - 1

7- 8: 2.78 7-11: 2.70 8-11: 2.61 9-10: 2.77

6- 8: 2.65 6-10: 2.75 6-11: 2.74

Pd-Cluster

151

Pd12a: Capped Cube Total magnetic moment: 2µB 9 0.13

5 7 6

0.19

0.18

Bond lengths [˚ A] 1- 2: 2.60 1- 6: 2.68 1- 7: 2.60 1-10: 2.57 1-12: 2.68

4 8

0.18

0.19

0.13

3 10 1

11

0.19

2

0.18

0.13

0.19

2- 3: 2.68 2- 8: 2.60 2-11: 2.57 2-12: 2.68

3- 4: 2.68 3- 8: 2.68 3-11: 2.68 4- 5: 2.60 4- 8: 2.57 4- 9: 2.68 4-11: 2.60

5- 6: 2.68 5- 7: 2.57 5- 9: 2.68 5-10: 2.60 6- 7: 2.68 6-10: 2.68

8- 9: 2.68 10-11: 2.60 10-12: 2.68 11-12: 2.68

7- 8: 2.60 7- 9: 2.68

Average bond length: 2.64 ˚ A

0.18

4Emag : 11 meV

12

Point symmetry: D4h - 4/mmm

0.13

Total magnetic moment: 4µB 9

Bond lengths [˚ A]

0.25

5 7 6

0.37

0.37

1- 2: 2.61 1- 6: 2.68 1- 7: 2.61 1-10: 2.55 1-12: 2.68

4 8

0.37

0.37

0.25

3 10 1

11

0.37

2

0.37

0.37

12 0.25

0.37

0.25

2- 3: 2.68 2- 8: 2.61 2-11: 2.55 2-12: 2.68

3- 4: 2.68 3- 8: 2.68 3-11: 2.68 4- 5: 2.61 4- 8: 2.55 4- 9: 2.68 4-11: 2.61

5- 6: 2.68 5- 7: 2.55 5- 9: 2.68 5-10: 2.61 6- 7: 2.68 6-10: 2.68 7- 8: 2.61 7- 9: 2.68

Average bond length: 2.64 ˚ A 4Emag : - meV 4Estruct : 41 meV Point symmetry: D4h - 4/mmm

8- 9: 2.68 10-11: 2.61 10-12: 2.68 11-12: 2.68

152

Appendix A

Total magnetic moment: 6µB

9 0.38

5 7 6

0.56

0.56

Bond lengths [˚ A] 1- 2: 2.60 1- 6: 2.68 1- 7: 2.60 1-10: 2.59 1-12: 2.68

4 8

0.56

0.56

0.38

3 10 1

11

0.56

2

0.56

0.38

0.56

2- 3: 2.68 2- 8: 2.60 2-11: 2.59 2-12: 2.68

3- 4: 2.68 3- 8: 2.68 3-11: 2.68 4- 5: 2.60 4- 8: 2.59 4- 9: 2.68 4-11: 2.60

5- 6: 2.68 5- 7: 2.59 5- 9: 2.68 5-10: 2.60 6- 7: 2.68 6-10: 2.68

8- 9: 2.68 10-11: 2.60 10-12: 2.68 11-12: 2.68

7- 8: 2.60 7- 9: 2.68

Average bond length: 2.64 ˚ A

0.56

4Emag : 26 meV

12

Point symmetry: D4h - 4/mmm

0.38

Total magnetic moment: 8µB

9 0.51

5 7 6

0.75

0.75

Bond lengths [˚ A] 1- 2: 2.61 1- 6: 2.69 1- 7: 2.61 1-10: 2.61 1-12: 2.68

4 8

0.75

0.75

0.51

3 10 1

11

0.75

2

0.75

0.75

12 0.51

0.75

0.51

2- 3: 2.69 2- 8: 2.61 2-11: 2.61 2-12: 2.68

3- 4: 2.69 3- 8: 2.68 3-11: 2.68 4- 5: 2.61 4- 8: 2.61 4- 9: 2.68 4-11: 2.61

5- 6: 2.69 5- 7: 2.61 5- 9: 2.68 5-10: 2.61 6- 7: 2.68 6-10: 2.68 7- 8: 2.61 7- 9: 2.68

Average bond length: 2.65 ˚ A 4Emag : 49 meV Point symmetry: D4h - 4/mmm

8- 9: 2.68 10-11: 2.61 10-12: 2.68 11-12: 2.68

Pd-Cluster

153

Pd12b: Incomplete Icosahedron Total magnetic moment: 2µB 11

12 0.29

8

-0.07

0.41

1

0.07

2 0.30

9 0.29

0.40

6

10 -0.07

5

3

-0.15

0.30

4

0.07

1- 2: 2.85 1- 3: 2.85 1- 4: 2.85 1- 5: 2.85 1- 6: 2.85 1- 7: 2.91 1- 8: 2.56 1- 9: 2.58 1-10: 2.63 1-11: 2.63 1-12: 2.58

Bond lengths [˚ A] 2- 3: 2.81 2- 6: 2.77 2- 7: 2.68 2- 8: 2.65 2-12: 2.65 3333-

4: 7: 8: 9:

2.77 2.68 2.65 2.65

4- 5: 2.75

70.18

4- 7: 2.70 4- 9: 2.65 4-10: 2.64 5- 6: 2.75 5- 7: 2.73 5-10: 2.63 5-11: 2.63

8- 9: 2.97 8-12: 2.97 9-10: 2.98 10-11: 2.99 11-12: 2.98

6- 7: 2.70 6-11: 2.64 6-12: 2.65

Average bond length: 2.75 ˚ A 4Emag : 15 meV Point symmetry: C1h - m

Total magnetic moment: 4µB 11

12 0.25

0.32

8 0.20

0.34

2 0.30

9 0.25

1

0.49

6

10 0.32

5

3

0.36

0.30

4

0.34

1- 2: 2.84 1- 3: 2.84 1- 4: 2.85 1- 5: 2.85 1- 6: 2.85 1- 7: 2.90 1- 8: 2.63 1- 9: 2.60 1-10: 2.57 1-11: 2.57 1-12: 2.60

Bond lengths [˚ A] 2- 3: 2.75 2- 6: 2.76 2- 7: 2.70 2- 8: 2.64 2-12: 2.64 3333-

4: 7: 8: 9:

2.76 2.70 2.64 2.64

4- 5: 2.78

70.53

4- 7: 2.69 4- 9: 2.65 4-10: 2.65 5- 6: 2.78 5- 7: 2.67 5-10: 2.66 5-11: 2.66

8- 9: 2.99 8-12: 2.99 9-10: 2.97 10-11: 2.97 11-12: 2.97

6- 7: 2.69 6-11: 2.65 6-12: 2.65

Average bond length: 2.74 ˚ A 4Emag : 7 meV Point symmetry: C1h - m

Total magnetic moment: 6µB 11

12 0.46

0.46

8 0.46

0.53

0.53

9 0.46

1

0.50

6 2

10 0.46

5

3

0.53

70.55

0.53

4

0.53

1- 2: 2.85 1- 3: 2.85 1- 4: 2.85 1- 5: 2.85 1- 6: 2.85 1- 7: 2.88 1- 8: 2.58 1- 9: 2.57 1-10: 2.58 1-11: 2.58 1-12: 2.57

Bond lengths [˚ A] 2- 3: 2.76 2- 6: 2.76 2- 7: 2.66 2- 8: 2.69 2-12: 2.69 3333-

4: 7: 8: 9:

2.76 2.66 2.69 2.69

4- 5: 2.76 Average bond length: 2.74 ˚ A 4Emag : - meV 4Estruct : 17 meV Point symmetry: C5v - 5m

4- 7: 2.66 4- 9: 2.69 4-10: 2.69 5- 6: 2.76 5- 7: 2.66 5-10: 2.69 5-11: 2.69

8- 9: 2.94 8-12: 2.94 9-10: 2.94 10-11: 2.95 11-12: 2.94

6- 7: 2.66 6-11: 2.69 6-12: 2.69

154

Appendix A

Total magnetic moment: 8µB 11

12 0.63

0.63

8 0.64

9 0.63

1

0.71

6 2 0.66

10 0.63

5

0.66

0.66

3

4

0.66

0.66

Bond lengths [˚ A]

1- 2: 2.87 1- 3: 2.87 1- 4: 2.88 1- 5: 2.87 1- 6: 2.88 1- 7: 2.98 1- 8: 2.55 1- 9: 2.55 1-10: 2.55 1-11: 2.55 1-12: 2.55

2- 3: 2.72 2- 6: 2.72 2- 7: 2.65 2- 8: 2.70 2-12: 2.70 3333-

4: 7: 8: 9:

2.72 2.65 2.70 2.70

4- 5: 2.72

70.81

4- 7: 2.65 4- 9: 2.70 4-10: 2.70 5- 6: 2.72 5- 7: 2.65 5-10: 2.70 5-11: 2.70

8- 9: 2.93 8-12: 2.93 9-10: 2.92 10-11: 2.95 11-12: 2.92

6- 7: 2.65 6-11: 2.70 6-12: 2.70

A Average bond length: 2.74 ˚ 4Emag : 4 meV Point symmetry: C1h - m

Pd12c∗ : Edge Sharing Octahedra Plus Two Adatoms Total magnetic moment: 2µB 12 8

-0.06

0.02

0.25

0.01

3 0.30

2

0.31

7

-0.07

6

5

11

0.09

4

0.25

1 0.31

10 0.30

1111-

Bond lengths [˚ A] 2: 4: 7: 9:

2.66 2.65 2.70 2.65

2- 3: 2.85 2- 5: 2.66 2- 7: 2.69 2- 8: 2.70 2- 9: 2.69 2-10: 2.69

3- 4: 2.69 3- 6: 2.69 3- 7: 2.84 3- 8: 2.84 3- 9: 2.68 3-10: 2.68 3-11: 2.69 3-12: 2.69

4- 9: 2.71 4-11: 2.68

7- 8: 2.63 7-11: 2.58

5- 6: 2.65 5- 8: 2.70 5-10: 2.65

8-12: 2.58

6- 8: 2.68 6-10: 2.71 6-12: 2.68

11-12: 2.60

9-10: 2.66

4- 7: 2.68

9

Average bond length: 2.69 ˚ A

0.30

4Emag : 20 meV Point symmetry: C1h - m

Total magnetic moment: 4µB 12 8

0.21

6

5 0.44

0.32

0.26

11 7

0.21

0.26

3 2

0.37

10 0.34

0.46

4 1 0.44

0.33

1111-

Bond lengths [˚ A] 2: 4: 7: 9:

2.65 2.65 2.70 2.65

2- 3: 2.84 2- 5: 2.65 2- 7: 2.69 2- 8: 2.70 2- 9: 2.69 2-10: 2.69

3- 4: 2.69 3- 6: 2.69 3- 7: 2.84 3- 8: 2.84 3- 9: 2.68 3-10: 2.68 3-11: 2.69 3-12: 2.69 4- 7: 2.68

9

Average bond length: 2.69 ˚ A

0.35

4Emag : 15 meV Point symmetry: C1h - m

4- 9: 2.72 4-11: 2.68

7- 8: 2.63 7-11: 2.59

5- 6: 2.66 5- 8: 2.70 5-10: 2.65

8-12: 2.59

6- 8: 2.68 6-10: 2.72 6-12: 2.68

11-12: 2.60

9-10: 2.67

Pd-Cluster

155

Total magnetic moment: 6µB 12 8

0.53

0.54

0.48

0.54

3 0.45

2

0.48

7

0.53

6

5

11

0.53

4

0.49

1 0.48

10

2: 4: 7: 9:

2.64 2.64 2.68 2.66

2- 3: 2.82 2- 5: 2.64 2- 7: 2.72 2- 8: 2.72 2- 9: 2.72 2-10: 2.72

3- 4: 2.68 3- 6: 2.68 3- 7: 2.87 3- 8: 2.86 3- 9: 2.68 3-10: 2.69 3-11: 2.69 3-12: 2.69

4- 9: 2.72 4-11: 2.68

7- 8: 2.66 7-11: 2.60

5- 6: 2.64 5- 8: 2.67 5-10: 2.66

8-12: 2.60

6- 8: 2.74 6-10: 2.72 6-12: 2.67

11-12: 2.61

9-10: 2.66

4- 7: 2.73

Average bond length: 2.69 ˚ A

9

0.48

1111-

Bond lengths [˚ A]

0.48

4Emag : - meV 4Estruct : - meV Point symmetry: C1h - m

Total magnetic moment: 8µB 12 8

0.73

6

5 0.65

0.62

0.64

11 7

0.73

0.64

3 2

0.75

10 0.64

0.70

4 1 0.64

0.62

1111-

Bond lengths [˚ A] 2: 4: 7: 9:

2.61 2.62 2.67 2.71

2- 3: 2.74 2- 5: 2.61 2- 7: 2.71 2- 8: 2.72 2- 9: 2.70 2-10: 2.70

3- 4: 2.67 3- 6: 2.67 3- 7: 2.95 3- 8: 2.93 3- 9: 2.67 3-10: 2.67 3-11: 2.68 3-12: 2.69 4- 7: 2.77

9

A Average bond length: 2.69 ˚

0.64

4Emag : 36 meV Point symmetry: C1h - m

4- 9: 2.69 4-11: 2.67

7- 8: 2.65 7-11: 2.60

5- 6: 2.62 5- 8: 2.66 5-10: 2.70

8-12: 2.60

6- 8: 2.78 6-10: 2.69 6-12: 2.66

11-12: 2.59

9-10: 2.69

156

Appendix A

Pd13a: Capped Cube with Central Atom Total magnetic moment: 2µB Bond lengths [˚ A]

10 0.27

6 8

5

0.29

9

0.01

0.29

0.01

7

1

-0.07

4

0.42

-0.07

11 2

12

0.01

3

0.29

0.01

1- 2: 2.55 1- 3: 2.55 1- 5: 2.55 1- 6: 2.55 1- 8: 2.52 1- 9: 2.52 1-11: 2.52 1-12: 2.52 2- 3: 2.89 2- 7: 2.69 2- 8: 2.90

2-11: 3.00 2-13: 2.67 3- 4: 2.69 3- 9: 2.90 3-12: 3.00 3-13: 2.67 4- 5: 2.69 4- 9: 2.66 4-12: 2.66

5- 6: 2.89 5- 9: 3.00 5-10: 2.67 5-12: 2.90

8- 9: 2.90 8-10: 2.68 9-10: 2.68

6- 7: 2.69 6- 8: 3.00 6-10: 2.67 6-11: 2.90

11-12: 2.90 11-13: 2.68 12-13: 2.68

7- 8: 2.66 7-11: 2.66

A Average bond length: 2.73 ˚

0.29

4Emag : 7 meV

13 0.27

Point symmetry: C2h - 2/m

Total magnetic moment: 4µB Bond lengths [˚ A]

10 0.26

6 8

5

0.28

9

0.28

0.28

0.28

7

1

0.26

4

0.69

0.26

11 2

12

0.28

3

0.28

0.28

13 0.26

0.28

1- 2: 2.54 1- 3: 2.54 1- 5: 2.54 1- 6: 2.54 1- 8: 2.54 1- 9: 2.54 1-11: 2.54 1-12: 2.54 2- 3: 2.88 2- 7: 2.68

2- 8: 2.88 2-13: 2.68 3- 4: 2.68 3- 9: 2.88 3-13: 2.68 4- 5: 2.68 4- 9: 2.67 4-12: 2.67

Average bond length: 2.69 ˚ A 4Emag : - meV 4Estruct : 39 meV Point symmetry: C4v - 4mm

5- 6: 2.88 5-10: 2.68 5-12: 2.88

8- 9: 2.89 8-10: 2.67 9-10: 2.67

6- 7: 2.68 6-10: 2.68 6-11: 2.88 7- 8: 2.67 7-11: 2.67

11-12: 2.89 11-13: 2.67 12-13: 2.67

Pd-Cluster

157

Total magnetic moment: 6µB

10

Bond lengths [˚ A]

0.43

6 8

5

0.44

9

0.44

0.44

0.44

7

1

0.42

4

0.77

0.42

11 2

12

0.44

3

0.44

0.44

1- 2: 2.55 1- 3: 2.55 1- 5: 2.55 1- 6: 2.55 1- 8: 2.56 1- 9: 2.56 1-11: 2.56 1-12: 2.56 2- 3: 2.79 2- 7: 2.68

2- 8: 2.80 2-13: 2.68 3- 4: 2.68 3- 9: 2.80 3-13: 2.68 4- 5: 2.68 4- 9: 2.69 4-12: 2.69

5- 6: 2.79 5-10: 2.68 5-12: 2.80

8- 9: 2.78 8-10: 2.68 9-10: 2.68

6- 7: 2.68 6-10: 2.68 6-11: 2.80 7- 8: 2.69 7-11: 2.69

11-12: 2.78 11-13: 2.68 12-13: 2.68

Average bond length: 2.68 ˚ A

0.44

4Emag : 5 meV

13 0.43

Point symmetry: C2h - 2/m

Total magnetic moment: 8µB

10

Bond lengths [˚ A]

0.61

6 8

5

0.60

9

0.60

0.60

0.60

7

1

0.61

4

0.74

0.61

11 2

12

0.60

3

0.60

0.60

13 0.61

0.60

1- 2: 2.55 1- 3: 2.55 1- 5: 2.55 1- 6: 2.55 1- 8: 2.55 1- 9: 2.55 1-11: 2.55 1-12: 2.55 2- 3: 2.90 2- 7: 2.68

2- 8: 2.89 2-13: 2.68 3- 4: 2.68 3- 9: 2.89 3-13: 2.68 4- 5: 2.68 4- 9: 2.68 4-12: 2.68

A Average bond length: 2.70 ˚ 4Emag : 19 meV Point symmetry: C2h - 2/m

5- 6: 2.90 5-10: 2.68 5-12: 2.89

8- 9: 2.90 8-10: 2.68 9-10: 2.68

6- 7: 2.68 6-10: 2.68 6-11: 2.89 7- 8: 2.68 7-11: 2.68

11-12: 2.90 11-13: 2.68 12-13: 2.68

158

Appendix A

Pd13b: Centred Icosahedron Total magnetic moment: 0µB Bond lengths [˚ A]

13

0.09

11

12

8

10

9

0.09 0.11 -0.33

-0.33 0.11

1

-0.55

6

2

-0.33 0.11

5

0.09

4

3 0.11

-0.33

1- 2: 2.67 1- 3: 2.67 1- 4: 2.58 1- 5: 2.70 1- 6: 2.58 1- 7: 2.70 1- 8: 2.70 1- 9: 2.58 1-10: 2.67 1-11: 2.67 1-12: 2.58 1-13: 2.70 2- 3: 2.75

2- 6: 2.80 2- 7: 2.81 2- 8: 2.81 2-12: 2.79 3333-

4: 7: 8: 9:

2.80 2.81 2.81 2.79

5- 6: 2.76 5- 7: 2.85 5-10: 2.81 5-11: 2.81 6- 7: 2.76 6-11: 2.79 6-12: 2.76

9-10: 2.80 9-13: 2.76 10-11: 2.75 10-13: 2.81 11-12: 2.80 11-13: 2.81 12-13: 2.76

4- 5: 2.76 4- 7: 2.76 4- 9: 2.76 4-10: 2.79

8- 9: 2.76 8-12: 2.76 8-13: 2.85

Average bond length: 2.75 ˚ A

7 0.09

4Emag : 31 meV Point symmetry: C2h - 2/m

Total magnetic moment: 2µB Bond lengths [˚ A]

13

0.24

11

12

8

10

0.24 0.06

0.11 0.06

9

0.11

1

0.36

6

0.11

2 0.06

5

0.24

7 0.24

3 0.06

4 0.11

1- 2: 2.71 1- 3: 2.71 1- 4: 2.67 1- 5: 2.58 1- 6: 2.67 1- 7: 2.58 1- 8: 2.58 1- 9: 2.67 1-10: 2.71 1-11: 2.71 1-12: 2.67 1-13: 2.58 2- 3: 2.87

2- 6: 2.81 2- 7: 2.76 2- 8: 2.76 2-12: 2.81 3333-

4: 7: 8: 9:

2.81 2.76 2.76 2.81

5- 6: 2.79 5- 7: 2.78 5-10: 2.76 5-11: 2.76 6- 7: 2.79 6-11: 2.81 6-12: 2.76

9-10: 2.81 9-13: 2.79 10-11: 2.87 10-13: 2.76 11-12: 2.81 11-13: 2.76 12-13: 2.79

4- 5: 2.79 4- 7: 2.79 4- 9: 2.76 4-10: 2.81

Average bond length: 2.75 ˚ A 4Emag : 21 meV Point symmetry: C2h - 2/m

8- 9: 2.79 8-12: 2.79 8-13: 2.78

Pd-Cluster

159

Total magnetic moment: 4µB

Bond lengths [˚ A]

13

0.40

11

12

8

10

0.40 0.21

0.26 0.21

9

0.26

1

0.51

6

0.26

5

2 0.21

0.40

4

3 0.21

0.26

1- 2: 2.71 1- 3: 2.71 1- 4: 2.67 1- 5: 2.58 1- 6: 2.67 1- 7: 2.58 1- 8: 2.58 1- 9: 2.67 1-10: 2.71 1-11: 2.71 1-12: 2.67 1-13: 2.58 2- 3: 2.88

2- 6: 2.81 2- 7: 2.75 2- 8: 2.75 2-12: 2.81 3333-

4: 7: 8: 9:

2.81 2.75 2.75 2.81

5- 6: 2.80 5- 7: 2.80 5-10: 2.75 5-11: 2.75 6- 7: 2.80 6-11: 2.81 6-12: 2.74

9-10: 2.81 9-13: 2.80 10-11: 2.88 10-13: 2.75 11-12: 2.81 11-13: 2.75 12-13: 2.80

4- 5: 2.80 4- 7: 2.80 4- 9: 2.74 4-10: 2.81

8- 9: 2.80 8-12: 2.80 8-13: 2.80

Average bond length: 2.75 ˚ A

7 0.40

4Emag : 17 meV Point symmetry: C2h - 2/m

Total magnetic moment: 6µB

Bond lengths [˚ A]

13

0.45

11

12

8

10

0.45 0.45

0.45 0.45

9

0.45

1

0.58

6

0.45

2 0.45

5

0.45

7 0.45

3 0.45

4 0.45

1- 2: 2.65 1- 3: 2.65 1- 4: 2.65 1- 5: 2.65 1- 6: 2.65 1- 7: 2.65 1- 8: 2.65 1- 9: 2.65 1-10: 2.65 1-11: 2.65 1-12: 2.65 1-13: 2.65 2- 3: 2.79

2- 6: 2.79 2- 7: 2.79 2- 8: 2.79 2-12: 2.79 3333-

4: 7: 8: 9:

2.79 2.79 2.79 2.79

5- 6: 2.79 5- 7: 2.79 5-10: 2.79 5-11: 2.79 6- 7: 2.79 6-11: 2.79 6-12: 2.79

9-10: 2.79 9-13: 2.79 10-11: 2.79 10-13: 2.79 11-12: 2.79 11-13: 2.79 12-13: 2.79

4- 5: 2.79 4- 7: 2.79 4- 9: 2.79 4-10: 2.79

Average bond length: 2.75 ˚ A 4Emag : 19 meV Point symmetry: Ih

8- 9: 2.79 8-12: 2.79 8-13: 2.79

160

Appendix A

Total magnetic moment: 8µB

Bond lengths [˚ A]

13

0.61

11

12

8

10

0.62 0.61

0.61 0.61

9

0.61

1

0.64

6

0.61

2 0.61

5

0.62

3 0.61

4 0.61

1- 2: 2.65 1- 3: 2.65 1- 4: 2.65 1- 5: 2.65 1- 6: 2.65 1- 7: 2.65 1- 8: 2.65 1- 9: 2.65 1-10: 2.65 1-11: 2.65 1-12: 2.65 1-13: 2.65 2- 3: 2.79

2- 6: 2.79 2- 7: 2.79 2- 8: 2.79 2-12: 2.79 3333-

4: 7: 8: 9:

2.79 2.79 2.79 2.79

5- 6: 2.79 5- 7: 2.79 5-10: 2.79 5-11: 2.79 6- 7: 2.79 6-11: 2.79 6-12: 2.79

9-10: 2.79 9-13: 2.79 10-11: 2.79 10-13: 2.79 11-12: 2.79 11-13: 2.79 12-13: 2.79

4- 5: 2.79 4- 7: 2.79 4- 9: 2.79 4-10: 2.79

8- 9: 2.79 8-12: 2.79 8-13: 2.79

A Average bond length: 2.75 ˚

7 0.61

4Emag : - meV 4Estruct : 20 meV Point symmetry: Ih

Isosurfaces of the magnetization densities of the S = 0 (left) and S = 4 (right) spin-isomers of a (nearly) icosahedral Pd13 cluster. Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

Pd-Cluster

161

Pd13c∗ : Cluster of Octahedra Total magnetic moment: 2µB Bond lengths [˚ A]

13 0.12

12

8

0.12

0.23

0.08

0.21

11

7

0.23

0.12

6

5

3 0.12

2

0.06

1111-

4 0.21

1

0.08

2: 4: 7: 9:

2.62 2.66 2.62 2.66

2- 3: 2.83 2- 5: 2.62 2- 7: 2.77 2- 8: 2.77 2- 9: 2.70 2-10: 2.70

3- 4: 2.68 3- 6: 2.68 3- 7: 2.83 3- 8: 2.83 3- 9: 2.68 3-10: 2.68 3-11: 2.68 3-12: 2.68 4- 7: 2.70 4- 9: 2.69 4-11: 2.67

5- 6: 2.66 5- 8: 2.62 5-10: 2.66

8-12: 2.70 8-13: 2.62 9-10: 2.67

6- 8: 2.70 6-10: 2.69 6-12: 2.67 7- 8: 2.77 7-11: 2.70 7-13: 2.62

11-12: 2.69 11-13: 2.66 12-13: 2.66

Average bond length: 2.69 ˚ A

10

9

0.22

4Emag : 1 meV

0.22

Point symmetry: C1h - m

Total magnetic moment: 4µB Bond lengths [˚ A]

13 0.28

12

8

0.33

6

5

0.28

0.30

0.29

11

7 0.33

3 2

0.33

10 0.30

0.38

1111-

0.29

4 1

0.28

0.30

2: 4: 7: 9:

2.64 2.66 2.63 2.66

2- 3: 2.81 2- 5: 2.64 2- 7: 2.76 2- 8: 2.76 2- 9: 2.70 2-10: 2.70

3- 4: 2.69 3- 6: 2.69 3- 7: 2.81 3- 8: 2.81 3- 9: 2.69 3-10: 2.69 3-11: 2.69 3-12: 2.69 4- 7: 2.70 4- 9: 2.69 4-11: 2.68

Average bond length: 2.69 ˚ A

9 0.30

4Emag : 3 meV Point symmetry: C1h - m

5- 6: 2.66 5- 8: 2.63 5-10: 2.66

8-12: 2.69 8-13: 2.63 9-10: 2.68

6- 8: 2.70 6-10: 2.69 6-12: 2.68 7- 8: 2.77 7-11: 2.69 7-13: 2.63

11-12: 2.69 11-13: 2.66 12-13: 2.66

162

Appendix A

Total magnetic moment: 6µB

Bond lengths [˚ A]

13 1111-

0.45

12

8

0.49

0.45

0.45

0.45

0.45

0.49

6

5

11

7

3 0.51

2

0.49

4 0.45

1

0.45

10

2.64 2.65 2.64 2.66

2- 3: 2.79 2- 5: 2.64 2- 7: 2.76 2- 8: 2.76 2- 9: 2.72 2-10: 2.72

3- 4: 2.69 3- 6: 2.69 3- 7: 2.79 3- 8: 2.79 3- 9: 2.69 3-10: 2.69 3-11: 2.69 3-12: 2.69

6- 8: 2.72 6-10: 2.69 6-12: 2.69

4- 7: 2.72 4- 9: 2.69 4-11: 2.69

7- 8: 2.76 7-11: 2.72 7-13: 2.64

5- 6: 2.65 5- 8: 2.64 5-10: 2.66

8-12: 2.72 8-13: 2.64 9-10: 2.69 11-12: 2.69 11-13: 2.66 12-13: 2.66

A Average bond length: 2.69 ˚ 4Emag : - meV 4Estruct : - meV

9

0.45

2: 4: 7: 9:

0.45

Point symmetry: C1h - m

Total magnetic moment: 8µB Bond lengths [˚ A]

13 0.59

12

8

0.67

6

5

0.59

0.59

0.59

11

7 0.67

3 2

0.68

10 0.59

0.66

1111-

0.59

4 1

0.59

0.59

2: 4: 7: 9:

2.62 2.66 2.62 2.67

2- 3: 2.81 2- 5: 2.62 2- 7: 2.73 2- 8: 2.73 2- 9: 2.73 2-10: 2.73

3- 4: 2.68 3- 6: 2.68 3- 7: 2.82 3- 8: 2.82 3- 9: 2.68 3-10: 2.68 3-11: 2.68 3-12: 2.68 4- 7: 2.73 4- 9: 2.68 4-11: 2.67

5- 6: 2.66 5- 8: 2.62 5-10: 2.67

8-12: 2.73 8-13: 2.62 9-10: 2.68

6- 8: 2.73 6-10: 2.68 6-12: 2.67 7- 8: 2.74 7-11: 2.73 7-13: 2.62

11-12: 2.68 11-13: 2.67 12-13: 2.67

Average bond length: 2.69 ˚ A

9 0.59

4Emag : 22 meV Point symmetry: C1h - m

Isosurfaces of the magnetization densities for M = 0µB to M = 6µB for Pd13 cluster with a polyoctahedral structure (from left to right). Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

B

Appendix Rh-Cluster Rh2: Dumbbell Total 0µB

magnetic

moment: Total 2µB

magnetic

2

2

-0.14

1.00

1

1

-0.14

1.00

Bond lengths [˚ A]

Bond lengths [˚ A]

1- 2: 2.24

1- 2: 2.18

Average bond length: 2.24 ˚ A

Average bond length: 2.18 ˚ A

4Emag : 398 meV

4Emag : 299 meV

Point symmetry: D∞h *

Point symmetry: D∞h

Total magnetic moment: 4µB Bond lengths [˚ A]

2

2.00

1- 2: 2.21 A Average bond length: 2.21 ˚

1

2.00

4Emag : - meV Point symmetry: D∞h

163

moment:

164

Appendix B

Rh4: Equilateral Triangle Total magnetic moment: 1µB Bond lengths [˚ A]

3 0.33

1- 2: 2.38 1- 3: 2.38

2- 3: 2.38

A Average bond length: 2.38 ˚

1

2

0.33

0.33

4Emag : 74 meV Point symmetry: D3h - ¯ 62m

Total magnetic moment: 3µB Bond lengths [˚ A]

3 1.00

1- 2: 2.38 1- 3: 2.38

2- 3: 2.38

Average bond length: 2.38 ˚ A

1

2

1.00

1.00

4Emag : - meV Point symmetry: D3h - ¯ 62m

Total magnetic moment: 5µB Bond lengths [˚ A]

3 1.67

1- 2: 2.43 1- 3: 2.43

2- 3: 2.43

Average bond length: 2.43 ˚ A

1

2

1.67

1.67

4Emag : 28 meV Point symmetry: D3h - ¯ 62m

Total magnetic moment: 7µB Bond lengths [˚ A]

3 2.33

1- 2: 2.43 1- 3: 2.43

2- 3: 2.43

Average bond length: 2.43 ˚ A

1

2.33

2

2.33

4Emag : 418 meV Point symmetry: D3h - ¯ 62m

Rh-Cluster

165

Rh4a: Square Total magnetic moment: 0µB 4

0.00

Bond lengths [˚ A]

3

0.00

1- 2: 2.31 1- 4: 2.31

2- 3: 2.31

3- 4: 2.31

A Average bond length: 2.31 ˚

1

0.00

2

0.00

4Emag : 113 meV Point symmetry: D4h - 4mmm

Total magnetic moment: 2µB 4

0.50

Bond lengths [˚ A]

3

0.50

1- 2: 2.32 1- 4: 2.32

2- 3: 2.32

3- 4: 2.32

Average bond length: 2.32 ˚ A

1

0.50

2

0.50

4Emag : 62 meV Point symmetry: D4h - 4mmm

Total magnetic moment: 4µB Bond lengths [˚ A]

4

1.00

3

1.00

1- 2: 2.35 1- 4: 2.35

2- 3: 2.35

3- 4: 2.35

A Average bond length: 2.35 ˚

1

1.00

2

1.00

4Emag : - meV 4Estruct : 26 meV Point symmetry: D4h - 4mmm

Total magnetic moment: 6µB 4

1.50

Bond lengths [˚ A]

3

1.50

1- 2: 2.38 1- 4: 2.38

2- 3: 2.38

3- 4: 2.38

Average bond length: 2.38 ˚ A

1

1.50

2

1.50

4Emag : 85 meV Point symmetry: D4h - 4mmm

166

Appendix B

Rh4b: Rhombus Total magnetic moment: 0µB Bond lengths [˚ A]

4

3

0.37

-0.39

1- 2: 2.44 1- 4: 2.44

2- 4: 2.35

3- 4: 2.44

2- 3: 2.44

Average bond length: 2.42 ˚ A

1

2

-0.39

4Emag : 156 meV

0.37

Point symmetry: D2h - mmm

Total magnetic moment: 2µB Bond lengths [˚ A]

4

3

1.05

-0.05

1- 2: 2.45 1- 4: 2.45

2- 4: 2.36

3- 4: 2.45

2- 3: 2.45

Average bond length: 2.43 ˚ A

1

2

-0.05

4Emag : 111 meV

1.05

Point symmetry: D2h - mmm

Total magnetic moment: 4µB Bond lengths [˚ A]

4

3

1.02

0.98

1- 2: 2.44 1- 4: 2.44

2- 4: 2.37

3- 4: 2.44

2- 3: 2.44

Average bond length: 2.43 ˚ A

1

2

0.98

4Emag : 47 meV

1.02

Point symmetry: D2h - mmm

Total magnetic moment: 6µB Bond lengths [˚ A]

4

3

1.49

1.50

1- 2: 2.41 1- 4: 2.41

2- 4: 2.78 2- 3: 2.41

Average bond length: 2.48 ˚ A

1

1.50

2

1.49

4Emag : - meV 4Estruct : 195 meV Point symmetry: D2h - mmm

3- 4: 2.41

Rh-Cluster

167

Rh4c: Tetrahedron Total magnetic moment: 0µB Bond lengths [˚ A]

4

0.00

1

1- 2: 2.45 1- 3: 2.45 1- 4: 2.45

3

0.00

0.00

2- 3: 2.45 2- 4: 2.45

3- 4: 2.45

Average bond length: 2.45 ˚ A 4Emag : - meV 4Estruct : - meV

2 0.00

Point symmetry: Td - ¯ 43m

Total magnetic moment: 2µB 4

Bond lengths [˚ A]

0.64 1- 2: 2.42 1- 3: 2.45 1- 4: 2.42

1

3

0.63

0.09

2- 3: 2.43 2- 4: 2.71

3- 4: 2.43

A Average bond length: 2.48 ˚ 4Emag : 73 meV

2 0.64

Point symmetry: C1h - m

Total magnetic moment: 4µB 4

Bond lengths [˚ A]

1.00 1- 2: 2.40 1- 3: 2.73 1- 4: 2.40

1

3

0.99

1.01

2 1.00

2- 3: 2.40 2- 4: 2.73

Average bond length: 2.51 ˚ A 4Emag : 57 meV Point symmetry: S4 - ¯ 4

3- 4: 2.39

168

Appendix B

Total magnetic moment: 6µB 4

Bond lengths [˚ A]

1.50 1- 2: 2.52 1- 3: 2.52 1- 4: 2.52

1

3

1.50

1.49

2- 3: 2.52 2- 4: 2.52

3- 4: 2.52

Average bond length: 2.52 ˚ A 4Emag : 104 meV

2 1.50

Point symmetry: Td - ¯ 43m

Isosurfaces of the magnetization densities for M=2µB to M=6µB of Rh4 tetrahedron (from left to right). Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

Rh5a: Square Pyramid Total magnetic moment: 3µB Bond lengths [˚ A]

5

0.90

4 1

0.52

0.52

1- 2: 2.40 1- 4: 2.40 1- 5: 2.53

3 2 0.52

0.52

2- 3: 2.40 2- 5: 2.53

Average bond length: 2.47 ˚ A 4Emag : 22 meV Point symmetry: C4v - 4mm

3- 4: 2.40 3- 5: 2.53

4- 5: 2.53

Rh-Cluster

169

Total magnetic moment: 5µB Bond lengths [˚ A]

5

1.13

1- 2: 2.39 1- 4: 2.39 1- 5: 2.58

4

0.97

1

0.97

3

0.97

2 0.97

2- 3: 2.39 2- 5: 2.58

3- 4: 2.39 3- 5: 2.58

4- 5: 2.58

Average bond length: 2.49 ˚ A 4Emag : - meV 4Estruct : - meV Point symmetry: C4v - 4mm

Total magnetic moment: 7µB Bond lengths [˚ A]

5

1.22

1- 2: 2.42 1- 4: 2.42 1- 5: 2.60

4

1.44

1

1.44

3

1.44

2 1.44

2- 3: 2.42 2- 5: 2.60

3- 4: 2.42 3- 5: 2.60

Average bond length: 2.51 ˚ A 4Emag : 11 meV Point symmetry: C4v - 4mm

Rh5b: Trigonal Bipyramid Total magnetic moment: 3µB Bond lengths [˚ A]

5 0.49 1- 2: 2.49 1- 3: 2.49 1- 4: 2.49

2 0.67

3

2- 3: 2.52 2- 4: 2.52 2- 5: 2.49

4- 5: 2.49

0.67 A Average bond length: 2.50 ˚

4 0.67

1

3- 4: 2.52 3- 5: 2.49

0.49

4Emag : 12 meV Point symmetry: D3h - ¯ 62m

Total magnetic moment: 5µB Bond lengths [˚ A]

5 0.92 1- 2: 2.50 1- 3: 2.50 1- 4: 2.50

2 1.05

3

2- 3: 2.57 2- 4: 2.57 2- 5: 2.50

4- 5: 2.50

1.05

4 1 0.92

3- 4: 2.57 3- 5: 2.50

1.05

Average bond length: 2.52 ˚ A 4Emag : 23 meV Point symmetry: D3h - ¯ 62m

4- 5: 2.60

170

Appendix B

Total magnetic moment: 7µB Bond lengths [˚ A]

5 1.35

1- 2: 2.50 1- 3: 2.50 1- 4: 2.50

2 1.43

3

2- 3: 2.64 2- 4: 2.64 2- 5: 2.50

3- 4: 2.64 3- 5: 2.50 4- 5: 2.50

1.43 Average bond length: 2.55 ˚ A

4 1.43

1

4Emag : - meV 4Estruct : 57 meV

1.35

Point symmetry: D3h - ¯ 62m

Rh5c: Flat Trigonal Bipyramid Total magnetic moment: 3µB 3

Bond lengths [˚ A]

0.70 1- 2: 2.40 1- 3: 2.40 1- 4: 2.40

1

0.46

5 4

0.70

1- 5: 2.82

4- 5: 2.40 3- 5: 2.40

2- 5: 2.40

Average bond length amounts to 2.46 ˚ A

0.45

4Emag : 11 meV

2 0.70

Point symmetry: D3h - ¯ 62m

Total magnetic moment: 5µB 3

Bond lengths [˚ A]

1.01 1- 2: 2.42 1- 3: 2.42 1- 4: 2.42

1 5 4

1.01

0.98

3- 5: 2.41 2- 5: 2.41 4- 5: 2.41

Average bond length amounts to 2.42 ˚ A

1.00

4Emag : 36 meV

2 1.01

Point symmetry: D3h - ¯ 62m

Rh-Cluster

171

Total magnetic moment: 7µB 3

Bond lengths [˚ A]

1.40 1- 2: 2.44 1- 3: 2.44 1- 4: 2.44

1

1.39

5 4

1.40

3- 5: 2.44 2- 5: 2.44 4- 5: 2.44

Average bond length amounts to 2.44 ˚ A

1.42

4Emag : - meV 4Estruct : 185 meV

2 1.40

Point symmetry: D3h - ¯ 62m

Rh6a: Pentagonal Pyramid Total magnetic moment: 2µB 5

Bond lengths [˚ A]

0.23

6 0.92

4 -0.13

1

1- 2: 2.34 1- 5: 2.48 1- 6: 2.59

2- 3: 2.48 2- 6: 2.59 3- 4: 2.36

0.38

3 0.23

2

0.38

3- 6: 2.56 5- 6: 2.56 4- 5: 2.36 4- 6: 2.56

A Average bond length: 2.49 ˚ 4Emag : 63 meV Point symmetry: C1h

Total magnetic moment: 4µB 5

Bond lengths [˚ A]

0.71

6 1.14

4 0.45

1

1- 2: 2.40 1- 5: 2.37 1- 6: 2.65

2- 3: 2.37 2- 6: 2.65 3- 4: 2.37

0.50

3 2

0.50

0.71

3- 6: 2.65 5- 6: 2.65 4- 5: 2.37 4- 6: 2.61

Average bond length amounts to 2.51 ˚ A 4Emag : 22 meV Point symmetry: C1h

172

Appendix B

Total magnetic moment: 6µB 5

Bond lengths [˚ A]

0.93

4

6 1.26

0.91

1

1- 2: 2.41 1- 5: 2.41 1- 6: 2.60

2- 3: 2.41 2- 6: 2.60 3- 4: 2.41

0.98

3 0.93

2

0.98

3- 6: 2.60 5- 6: 2.60 4- 5: 2.41 4- 6: 2.56

Average bond length: 2.50 ˚ A 4Emag : 12 meV Point symmetry: C5v * - 5m

Total magnetic moment: 8µB Bond lengths [˚ A]

5

1.33

4

6 1.16

1.31

1

1- 2: 2.43 1- 5: 2.43 1- 6: 2.59

2- 3: 2.43 2- 6: 2.59 3- 4: 2.43

1.43

3- 6: 2.59 5- 6: 2.59 4- 5: 2.43 4- 6: 2.59

Average bond length: 2.51 ˚ A

3 1.33

2

1.43

4Emag : - meV 4Estruct : 197 meV Point symmetry: C5v * - 5m

Rh6b: Octahedron Total magnetic moment: 0µB 5

Bond lengths [˚ A]

0.00

2 1

3

0.00

0.00

0.00

4 0.00

1111-

2: 4: 5: 6:

2.51 2.51 2.51 2.51

2- 3: 2.51 2- 5: 2.51 2- 6: 2.51

Average bond length: 2.51 ˚ A 4Emag : 14 meV

6 0.00

Point symmetry: Oh - m3m

3- 4: 2.51 3- 5: 2.51 3- 6: 2.51

4- 5: 2.51 4- 6: 2.51

Rh-Cluster

173

Total magnetic moment: 2µB 5

Bond lengths [˚ A]

-0.09

2 1

3

0.55

0.55

0.55

4

1111-

2: 4: 5: 6:

2.54 2.54 2.51 2.51

2- 3: 2.54 2- 5: 2.51 2- 6: 2.51

3- 4: 2.54 3- 5: 2.51 3- 6: 2.51

4- 5: 2.51 4- 6: 2.51

Average bond length: 2.52 ˚ A

0.55

4Emag : 51 meV

6

Point symmetry: D4h - 4/mmm

-0.09

Total magnetic moment: 4µB 5

Bond lengths [˚ A]

1.02

2 1

3

0.49

0.49

0.49

4

1111-

2: 4: 5: 6:

2.46 2.46 2.57 2.57

2- 3: 2.46 2- 5: 2.57 2- 6: 2.57

3- 4: 2.46 3- 5: 2.57 3- 6: 2.57

4- 5: 2.57 4- 6: 2.57

A Average bond length: 2.53 ˚

0.49

4Emag : 35 meV

6

Point symmetry: D4h - 4/mmm

1.02

Total magnetic moment: 6µB 5

Bond lengths [˚ A]

1.01

2 1

3

1.00

1.00

1.00

4 1.00

6 1.01

1111-

2: 4: 5: 6:

2.54 2.54 2.54 2.54

2- 3: 2.54 2- 5: 2.54 2- 6: 2.54

Average bond length: 2.54 ˚ A 4Emag : - meV 4Estruct : - meV Point symmetry: Oh - m3m

3- 4: 2.54 3- 5: 2.54 3- 6: 2.54

4- 5: 2.54 4- 6: 2.54

174

Appendix B

Total magnetic moment: 8µB 5

Bond lengths [˚ A]

1.35

2 3

1.33

1 1.33

1.33

4

1111-

2: 4: 5: 6:

2.57 2.57 2.56 2.56

2- 3: 2.57 2- 5: 2.56 2- 6: 2.56

3- 4: 2.57 3- 5: 2.56 3- 6: 2.56

4- 5: 2.56 4- 6: 2.56

Average bond length: 2.56 ˚ A

1.33

4Emag : 30 meV

6

Point symmetry: D4h - 4/mmm

1.35

Isosurfaces of the magnetization densities for M=2µB to M=8µB of the Rh6 octahedron (from left to right). Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

Rh6c: Incomplete Pentagonal Bipyramid Total magnetic moment: 4µB Bond lengths [˚ A]

6 0.65

5

3

0.37

1

0.37

2

0.98

0.98

4

0.65

1111-

2: 4: 5: 6:

2.46 2.58 2.49 2.58

4- 5: 2.46 2- 3: 2.49 2- 4: 2.58 2- 6: 2.58

Average bond length: 2.51 ˚ A 4Emag : - meV 4Estruct : 36 meV Point symmetry: C2v - mm2

3- 4: 2.46 3- 6: 2.46

5- 6: 2.46

Rh-Cluster

175

Total magnetic moment: 6µB Bond lengths [˚ A]

6 1.16

5

3

0.99

1

0.99

2

0.85

1111-

2: 4: 5: 6:

2.45 2.58 2.46 2.58

4- 5: 2.49 2- 3: 2.46 2- 4: 2.58 2- 6: 2.58

3- 4: 2.49 3- 6: 2.49

5- 6: 2.49

Average bond length: 2.51 ˚ A

0.85

4

4Emag : 45 meV

1.16

Point symmetry: C2v - mm2

Total magnetic moment: 8µB Bond lengths [˚ A]

6 1.27

5

3

1.32

1

1.32

2

1.41

1111-

2: 4: 5: 6:

2.48 2.62 2.46 2.62

4- 5: 2.51 2- 3: 2.46 2- 4: 2.62 2- 6: 2.62

3- 4: 2.51 3- 6: 2.51

5- 6: 2.51

A Average bond length: 2.54 ˚

1.41

4

4Emag : 140 meV

1.27

Point symmetry: C2v - mm2

Rh7a: Centred Hexagon Total magnetic moment: 5µB 6

5

0.58

7

4

1

0.64

Bond lengths [˚ A]

0.58

1.41

0.64

11111-

2: 3: 4: 5: 6:

2.43 2.43 2.54 2.43 2.43

1- 7: 2.54

3- 4: 2.49

2- 3: 2.43 2- 7: 2.49

4- 5: 2.49

6- 7: 2.49

Average bond length: 2.47 ˚ A

2

3

0.58

0.58

4Emag : 39 meV Point symmetry: D2h - mmm

5- 6: 2.43

176

Appendix B

Total magnetic moment: 7µB 6

5

1.05

7

4

1

0.69

Bond lengths [˚ A]

1.05

1.40

0.69

11111-

2: 3: 4: 5: 6:

2.41 2.41 2.65 2.41 2.41

1- 7: 2.65

3- 4: 2.50

2- 3: 2.48 2- 7: 2.50

4- 5: 2.50

6- 7: 2.50

5- 6: 2.48

Average bond length: 2.49 ˚ A

2

3

1.05

1.05

4Emag : 38 meV Point symmetry: D2h - mmm

Total magnetic moment: 9µB 6 1.36

7

1.36

4

1

1.09

Bond lengths [˚ A]

5

1.39

1.09

11111-

2: 3: 4: 5: 6:

2.41 2.41 2.72 2.41 2.41

1- 7: 2.72

3- 4: 2.50

2- 3: 2.56 2- 7: 2.50

4- 5: 2.50

6- 7: 2.50

5- 6: 2.56

Average bond length: 2.51 ˚ A 4Emag : 31 meV

2

3

1.36

1.36

Point symmetry: D2h - mmm

Total magnetic moment: 11µB 6

5

1.61

7

4

1

1.57

Bond lengths [˚ A]

1.61

1.41

1.57

11111-

2: 3: 4: 5: 6:

2.43 2.43 2.68 2.43 2.43

1- 7: 2.68

3- 4: 2.45

2- 3: 2.64 2- 7: 2.45

4- 5: 2.45

6- 7: 2.45

Average bond length: 2.52 ˚ A

2

3

1.61

1.61

4Emag : 5 meV Point symmetry: D2h - mmm

5- 6: 2.64

Rh-Cluster

177

Total magnetic moment: 13µB 6 1.89

7

Bond lengths [˚ A]

5 1.89

4

1

1.88

1.69

1.88

11111-

2: 3: 4: 5: 6:

2.54 2.54 2.50 2.54 2.54

1- 7: 2.50

3- 4: 2.53

2- 3: 2.51 2- 7: 2.53

4- 5: 2.53

6- 7: 2.53

5- 6: 2.51

Average bond length: 2.52 ˚ A 4Emag : - meV 4Estruct : 377 meV

2

3

1.89

1.89

Point symmetry: D2h - mmm

Total magnetic moment: 15µB 6

5

2.08

7

4

1

2.32

Bond lengths [˚ A]

2.08

2.03

2.32

11111-

2: 3: 4: 5: 6:

2.49 2.49 2.65 2.49 2.49

1- 7: 2.65

3- 4: 2.51

2- 3: 2.61 2- 7: 2.51

4- 5: 2.51

6- 7: 2.51

5- 6: 2.61

Average bond length: 2.54 ˚ A 4Emag : 31 meV

2

3

2.08

2.08

Point symmetry: D2h - mmm

Rh7b∗: Pentagonal Bipyramid Total magnetic moment: 5µB Bond lengths [˚ A]

6 0.63

5

1

0.78

4

0.72

2

3

0.72

0.78

7 0.63

0.75

1111-

2: 5: 6: 7:

2.60 2.59 2.55 2.55

2- 3: 2.59

2- 6: 2.55 2- 7: 2.55 3- 4: 2.59 3- 6: 2.55 3- 7: 2.55

Average bond length: 2.57 ˚ A 4Emag : 124 meV Point symmetry: C5h * - 5/m

5- 7: 2.55 4- 5: 2.59 4- 6: 2.55 4- 7: 2.55 5- 6: 2.55

6- 7: 2.57

178

Appendix B

Total magnetic moment: 7µB Bond lengths [˚ A]

6 1.11

5

1

1.04

4

0.83

2

1.03

3

0.83

1.04

1111-

2: 5: 6: 7:

2.60 2.57 2.56 2.56

2- 3: 2.57

2- 6: 2.56 2- 7: 2.56 3- 4: 2.59 3- 6: 2.58 3- 7: 2.58

5- 7: 2.58 4- 5: 2.59 4- 6: 2.54 4- 7: 2.54

6- 7: 2.64

5- 6: 2.58

Average bond length: 2.58 ˚ A

7

4Emag : 91 meV

1.11

Point symmetry: C2v - mm2

Total magnetic moment: 9µB Bond lengths [˚ A]

6 1.36

5

1

1.26

4

1.25

2

1.24

3

1.25

1.26

1111-

2: 5: 6: 7:

2.59 2.59 2.57 2.57

2- 3: 2.59

2- 6: 2.57 2- 7: 2.57 3- 4: 2.58 3- 6: 2.58 3- 7: 2.58

5- 7: 2.58 4- 5: 2.58 4- 6: 2.58 4- 7: 2.58

6- 7: 2.68

5- 6: 2.58

A Average bond length: 2.58 ˚

7

4Emag : 45 meV

1.36

Point symmetry: C2v - mm2

Total magnetic moment: 11µB Bond lengths [˚ A]

6 1.61

5

1

1.60

4

1.44

2

3

1.44

1.60

7 1.61

1.70

1111-

2: 5: 6: 7:

2.49 2.69 2.58 2.58

2- 3: 2.69

2- 6: 2.58 2- 7: 2.58 3- 4: 2.55 3- 6: 2.55 3- 7: 2.55

Average bond length: 2.59 ˚ A 4Emag : 19 meV Point symmetry: C2v - mm2

5- 7: 2.55 4- 5: 2.55 4- 6: 2.61 4- 7: 2.61 5- 6: 2.55

6- 7: 2.66

Rh-Cluster

179

Total magnetic moment: 13µB

Bond lengths [˚ A]

6 1.86

5

1

1.86

4

1.85

2

3

1.85

1.86

7 1.86

1.85

1111-

2: 5: 6: 7:

2.60 2.59 2.57 2.57

2- 3: 2.59

2- 6: 2.57 2- 7: 2.57 3- 4: 2.59 3- 6: 2.57 3- 7: 2.57

5- 7: 2.57 4- 5: 2.59 4- 6: 2.57 4- 7: 2.57

6- 7: 2.65

5- 6: 2.57

Average bond length: 2.58 ˚ A 4Emag : - meV 4Estruct : - meV Point symmetry: C5h * -5/m

Isosurfaces of the magnetization densities for M=5µB to M=13µB of the Rh7 pentagonal bipyramid (from left to right). Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

180

Appendix B

Rh8a: Bicapped Octahedron I Total magnetic moment: 0µB Bond lengths [˚ A]

5

7

0.43

-0.12

-0.12

2 1

8

-0.02

-0.02

3 4

-0.02

-0.02

11111-

2: 4: 5: 6: 8:

2.55 2.51 2.65 2.53 2.47

2222-

3: 5: 6: 8:

2.51 2.65 2.53 2.47

3- 4: 2.55

3- 5: 2.65 3- 6: 2.53 3- 7: 2.47

4- 7: 2.47 5- 7: 2.46 5- 8: 2.46

4- 5: 2.65 4- 6: 2.53

Average bond length: 2.53 ˚ A 4Emag : 12 meV

6 0.01

Point symmetry: C2v - mm2

Total magnetic moment: 2µB Bond lengths [˚ A]

5

7

0.58

0.02

0.02

2 1

8

0.21

0.21

3 4

0.21

11111-

2: 4: 5: 6: 8:

2.57 2.52 2.66 2.53 2.46

2222-

3: 5: 6: 8:

2.52 2.66 2.53 2.46

3- 4: 2.57

0.21

3- 5: 2.66 3- 6: 2.53 3- 7: 2.46

4- 7: 2.46 5- 7: 2.45 5- 8: 2.45

4- 5: 2.66 4- 6: 2.53

Average bond length: 2.54 ˚ A 4Emag : 2 meV

6 0.53

Point symmetry: C2v - mm2

Total magnetic moment: 4µB Bond lengths [˚ A]

5

7

1.31

0.48

0.48

2 1

8

0.46

0.46

3 4 0.46

6 -0.10

0.46

11111-

2: 4: 5: 6: 8:

2.61 2.51 2.58 2.54 2.46

2222-

3: 5: 6: 8:

2.51 2.58 2.54 2.46

3- 4: 2.61 Average bond length: 2.55 ˚ A 4Emag : 6 meV Point symmetry: C2v - mm2

3- 5: 2.58 3- 6: 2.55 3- 7: 2.46 4- 5: 2.58 4- 6: 2.54

4- 7: 2.46 5- 7: 2.64 5- 8: 2.64

Rh-Cluster

181

Total magnetic moment: 6µB Bond lengths [˚ A]

5

7

1.07

0.89

8

0.89

2 1

0.67

0.67

3 0.67

4

11111-

2: 4: 5: 6: 8:

2.50 2.56 2.56 2.56 2.49

2222-

3: 5: 6: 8:

2.56 2.56 2.56 2.49

3- 4: 2.50

0.67

3- 5: 2.56 3- 6: 2.56 3- 7: 2.49

4- 7: 2.49 5- 7: 2.59 5- 8: 2.59

4- 5: 2.56 4- 6: 2.56

A Average bond length: 2.54 ˚ 4Emag : 4 meV

6 0.47

Point symmetry: C2v - mm2

Total magnetic moment: 8µB Bond lengths [˚ A]

5

7

0.76

0.68

8

0.68

2 1

1.17

1.17

3

11111-

2: 4: 5: 6: 8:

2.46 2.57 2.60 2.59 2.54

1.17

4 1.17

2222-

3: 5: 6: 8:

2.57 2.60 2.59 2.54

3- 4: 2.46

3- 5: 2.60 3- 6: 2.59 3- 7: 2.54

4- 7: 2.54 5- 7: 2.45 5- 8: 2.45

4- 5: 2.60 4- 6: 2.59

Average bond length: 2.55 ˚ A 4Emag : - meV 4Estruct : 11 meV

6 1.18

Point symmetry: C2v - mm2

Rh8b∗: Bicapped Octahedron II Total magnetic moment: 4µB Bond lengths [˚ A]

5

7

-0.14

1

1.14

2

-0.14

1.14 4 1.14

3

6

-0.14

2: 4: 5: 6:

2.55 2.59 2.57 2.57

2- 5: 2.49 2- 6: 2.49

-0.14

8

1111-

1.14

2- 7: 2.57 2- 8: 2.57 3333-

4: 5: 6: 7:

2.55 2.49 2.49 2.57

Average bond length: 2.55 ˚ A 4Emag : 48 meV Point symmetry: C2v - mm2

3- 8: 2.57

6- 8: 2.55

4- 5: 2.57 4- 6: 2.57

7- 8: 2.59

5- 7: 2.55

182

Appendix B

Total magnetic moment: 6µB Bond lengths [˚ A]

5

7

0.73

0.77

2

0.71

1

0.79 4 0.79

3

2: 4: 5: 6:

2.56 2.54 2.56 2.56

2- 5: 2.51 2- 6: 2.51

0.71

8

1111-

0.77

6

2- 7: 2.56 2- 8: 2.56 3333-

4: 5: 6: 7:

2.56 2.51 2.51 2.56

3- 8: 2.56

6- 8: 2.56

4- 5: 2.56 4- 6: 2.56

7- 8: 2.54

5- 7: 2.56

Average bond length: 2.55 ˚ A 4Emag : 29 meV

0.73

Point symmetry: C2v - mm2

Total magnetic moment: 8µB Bond lengths [˚ A]

5

7

1.03

0.97

2

1.03

1

0.97 4 0.97

3

2: 4: 5: 6:

2.59 2.51 2.56 2.56

2- 5: 2.52 2- 6: 2.52

1.03

8

1111-

0.97

6

2- 7: 2.56 2- 8: 2.56 3333-

4: 5: 6: 7:

2.59 2.52 2.52 2.56

3- 8: 2.56

6- 8: 2.59

4- 5: 2.56 4- 6: 2.56

7- 8: 2.50

5- 7: 2.59

A Average bond length: 2.55 ˚ 4Emag : 4 meV

1.03

Point symmetry: C2v - mm2

Total magnetic moment: 10µB Bond lengths [˚ A]

5

7

1.35

1

1.15

2

1.35

1.14 4 1.14

3

6

1.35

2: 4: 5: 6:

2.60 2.51 2.56 2.56

2- 5: 2.55 2- 6: 2.55

1.35

8

1111-

1.15

2- 7: 2.56 2- 8: 2.56 3333-

4: 5: 6: 7:

2.60 2.55 2.55 2.56

A Average bond length: 2.56 ˚ 4Emag : - meV 4Estruct : - meV Point symmetry: C2v - mm2

3- 8: 2.56

6- 8: 2.60

4- 5: 2.56 4- 6: 2.56

7- 8: 2.51

5- 7: 2.60

Rh-Cluster

183

Rh9a: Capped Pentagonal Bipyramid Total magnetic moment: 9µB Bond lengths [˚ A]

8

9

0.79

0.79

1111-

6 1.12

4

3

1.32 2 0.68

1.33

5

1.32

1 0.68

2: 5: 6: 7:

2.54 2.51 2.59 2.51

2- 3: 2.51 2- 6: 2.59 2- 7: 2.51

3333-

4: 6: 7: 8:

2.55 2.58 2.75 2.55

4- 5: 2.55 4- 6: 2.63

4- 7: 2.60 4- 8: 2.51 4- 9: 2.51

6- 7: 2.96 6- 8: 2.57 6- 9: 2.57

5- 6: 2.58 5- 7: 2.75 5- 9: 2.55

8- 9: 2.61

Average bond length: 2.59 ˚ A

7

4Emag : 12 meV

0.97

Point symmetry: C1h

Total magnetic moment: 11µB Bond lengths [˚ A]

8

9

0.73

0.73

1111-

6 1.13

4

3

1.57 2 1.17

1.30

5

1.57

1 1.17

2: 5: 6: 7:

2.53 2.53 2.58 2.57

2- 3: 2.53 2- 6: 2.58 2- 7: 2.57

3333-

4: 6: 7: 8:

2.53 2.60 2.69 2.65

4- 5: 2.53 4- 6: 2.62

4- 7: 2.61 4- 8: 2.51 4- 9: 2.51

6- 7: 2.96 6- 8: 2.58 6- 9: 2.58

5- 6: 2.60 5- 7: 2.69 5- 9: 2.65

8- 9: 2.53

Average bond length: 2.60 ˚ A

7

4Emag : - meV 4Estruct : 25 meV

1.62

Point symmetry: C1h

Total magnetic moment: 13µB Bond lengths [˚ A]

8

9

1.29

1.29

1111-

6 1.26

3

1.67 2 1.20

4

1.78

5 1 1.20

7

1.66

1.67

2: 5: 6: 7:

2.55 2.56 2.58 2.56

2- 3: 2.56 2- 6: 2.58 2- 7: 2.56

3333-

4: 6: 7: 8:

2.57 2.60 2.66 2.61

4- 5: 2.57 4- 6: 2.69

Average bond length: 2.60 ˚ A 4Emag : 2 meV Point symmetry: C1h

4- 7: 2.67 4- 8: 2.54 4- 9: 2.54

6- 7: 2.90 6- 8: 2.57 6- 9: 2.57

5- 6: 2.60 5- 7: 2.66 5- 9: 2.61

8- 9: 2.62

184

Appendix B

Rh9b∗: Doubled Trigonal Antiprism Total magnetic moment: 9µB 4 0.63

9 1.16

1

0.91

6 0.66 1111-

5 1.37

Bond lengths [˚ A]

3 1.13

3: 4: 6: 9:

2.57 2.52 2.51 2.56

2- 5: 2.55 2- 7: 2.55

2- 8: 2.52 2- 9: 2.58 3333-

5: 6: 7: 8:

2.59 2.59 2.58 2.56

3- 9: 2.56 4- 5: 2.56 4- 6: 2.53 4- 9: 2.59

5- 7: 2.55 5- 9: 2.58 7- 8: 2.52 8- 9: 2.55

5- 6: 2.55

Average bond length: 2.56 ˚ A

7

2 0.99

8 1.13

1.02

4Emag : 19 meV Point symmetry: C1 - 1

Total magnetic moment: 11µB 4 1.11

9 1.45

1

1.11

1.11

5 1.44

Bond lengths [˚ A]

6

3 1.45

1111-

3: 4: 6: 9:

2.56 2.55 2.55 2.56

2- 5: 2.56 2- 7: 2.55

2- 8: 2.55 2- 9: 2.56 3333-

5: 6: 7: 8:

2.61 2.56 2.56 2.56

3- 9: 2.61 4- 5: 2.56 4- 6: 2.55 4- 9: 2.56

5- 7: 2.56 5- 9: 2.61 7- 8: 2.55 8- 9: 2.56

5- 6: 2.56

Average bond length: 2.56 ˚ A

7

2 1.11

8 1.11

1.11

4Emag : - meV 4Estruct : - meV Point symmetry: D3h - ¯ 62m

Total magnetic moment: 13µB 4 1.36

9 1.63

1

1.36

6 1.35 1111-

5 1.62

Bond lengths [˚ A]

3 1.62

3: 4: 6: 9:

2.58 2.54 2.54 2.58

2- 5: 2.58 2- 7: 2.54

2- 8: 2.54 2- 9: 2.58 3333-

5: 6: 7: 8:

2.55 2.58 2.58 2.58

Average bond length: 2.56 ˚ A

7

2 1.35

8 1.35

1.36

4Emag : 10 meV Point symmetry: C1 - 1

3- 9: 2.56 4- 5: 2.58 4- 6: 2.54 4- 9: 2.58

5- 7: 2.58 5- 9: 2.55 7- 8: 2.54 8- 9: 2.58

5- 6: 2.58

Rh-Cluster

185

Rh10a:Tetragonal Antiprism with Capped Square Faces Total magnetic moment: 0µB 5

0.00

3

2

0.00 4 0.00

0.00 1 0.00

8 7

0.00

Bond lengths [˚ A]

0.00

9 6

0.00

0.00

11111-

2: 4: 5: 6: 9:

2.59 2.59 2.52 2.55 2.55

2- 3: 2.59 2- 5: 2.52

2- 8: 2.55 2- 9: 2.55 3333-

4: 5: 7: 8:

2.59 2.52 2.55 2.55

4- 5: 2.52 4- 6: 2.55 4- 7: 2.55 6- 7: 2.49 6- 9: 2.49 6-10: 2.59

7- 8: 2.49 7-10: 2.59 8- 9: 2.49 8-10: 2.59 9-10: 2.59

Average bond length: 2.55 ˚ A 4Emag : 141 meV Point symmetry: C4v - 4mm

10 0.00

Total magnetic moment: 6µB 5

0.13

3

2

0.72 1 0.72

0.72 4 0.72

8 7

0.72

Bond lengths [˚ A]

0.72

9 6

0.72

10 0.11

0.72

11111-

2: 4: 5: 6: 9:

2.53 2.53 2.56 2.56 2.56

2- 3: 2.53 2- 5: 2.56

2- 8: 2.56 2- 9: 2.56 3333-

4: 5: 7: 8:

2.53 2.56 2.56 2.56

Average bond length: 2.55 ˚ A 4Emag : 101 meV Point symmetry: C4v - 4mm

4- 5: 2.56 4- 6: 2.56 4- 7: 2.56 6- 7: 2.53 6- 9: 2.53 6-10: 2.56

7- 8: 2.53 7-10: 2.56 8- 9: 2.53 8-10: 2.56 9-10: 2.56

186

Appendix B

Total magnetic moment: 8µB

5

-0.05

3

2

1.01 4 1.01

1.01 1 1.01

8 7

1.02

Bond lengths [˚ A]

1.02

9 6

1.01

1.01

11111-

2: 4: 5: 6: 9:

2.51 2.51 2.58 2.57 2.57

2- 3: 2.51 2- 5: 2.58

2- 8: 2.57 2- 9: 2.57 3333-

4: 5: 7: 8:

2.51 2.58 2.57 2.57

4- 5: 2.58 4- 6: 2.57 4- 7: 2.57 6- 7: 2.51 6- 9: 2.51 6-10: 2.58

7- 8: 2.51 7-10: 2.58 8- 9: 2.51 8-10: 2.58 9-10: 2.58

Average bond length: 2.55 ˚ A 4Emag : 81 meV Point symmetry: C4v - 4mm

10 -0.06

Total magnetic moment: 10µB

5

0.67

3

2

1.08 1 1.08

1.08 4 1.08

8 7

1.08

Bond lengths [˚ A]

1.08

9 6

1.08

10 0.67

1.08

11111-

2: 4: 5: 6: 9:

2.51 2.51 2.56 2.59 2.59

2- 3: 2.51 2- 5: 2.56

2- 8: 2.59 2- 9: 2.59 3333-

4: 5: 7: 8:

2.51 2.56 2.59 2.59

Average bond length: 2.55 ˚ A 4Emag : 61 meV Point symmetry: C4v - 4mm

4- 5: 2.56 4- 6: 2.59 4- 7: 2.59 6- 7: 2.51 6- 9: 2.51 6-10: 2.56

7- 8: 2.51 7-10: 2.56 8- 9: 2.51 8-10: 2.56 9-10: 2.56

Rh-Cluster

187

Total magnetic moment: 12µB

5

0.99

3

2

1.25 4 1.25

1.25 1 1.25

8 7

1.25

Bond lengths [˚ A]

1.25

9 6

1.25

1.25

11111-

2: 4: 5: 6: 9:

2.50 2.50 2.56 2.60 2.60

2- 3: 2.50 2- 5: 2.56

2- 8: 2.60 2- 9: 2.60 3333-

4: 5: 7: 8:

2.50 2.56 2.60 2.60

4- 5: 2.56 4- 6: 2.60 4- 7: 2.60 6- 7: 2.50 6- 9: 2.50 6-10: 2.56

7- 8: 2.50 7-10: 2.56 8- 9: 2.50 8-10: 2.56 9-10: 2.56

Average bond length: 2.55 ˚ A 4Emag : 28 meV Point symmetry: C4v - 4mm

10 0.99

Total magnetic moment: 14µB

5

1.24

3

2

1.44 1 1.44

1.44 4 1.44

1.44

11111-

2: 4: 5: 6: 9:

2.49 2.49 2.58 2.61 2.61

2- 3: 2.49 2- 5: 2.58

8 7

Bond lengths [˚ A]

1.44

9 6

1.44

10 1.25

1.44

2- 8: 2.61 2- 9: 2.61 3333-

4: 5: 7: 8:

2.49 2.58 2.61 2.61

Average bond length: 2.56 ˚ A 4Emag : - meV 4Estruct : - meV Point symmetry: C4v - 4mm

4- 5: 2.58 4- 6: 2.61 4- 7: 2.61 6- 7: 2.49 6- 9: 2.49 6-10: 2.58

7- 8: 2.49 7-10: 2.58 8- 9: 2.49 8-10: 2.58 9-10: 2.58

188

Appendix B

Total magnetic moment: 16µB 5

1.56

3

2

1.61 1 1.61

1.61 4 1.61

8 7

1.61

Bond lengths [˚ A]

1.61

9 6

1.61

10

1.61

11111-

2: 4: 5: 6: 9:

2.50 2.50 2.60 2.61 2.61

2- 3: 2.50 2- 5: 2.60

2- 8: 2.61 2- 9: 2.61 3333-

4: 5: 7: 8:

2.50 2.60 2.61 2.61

4- 5: 2.60 4- 6: 2.61 4- 7: 2.61 6- 7: 2.50 6- 9: 2.50 6-10: 2.60

7- 8: 2.50 7-10: 2.60 8- 9: 2.50 8-10: 2.60 9-10: 2.60

Average bond length: 2.57 ˚ A 4Emag : 16 meV Point symmetry: C4v - 4mm

1.56

Isosurfaces of the magnetization densities for M=6µB to M=16µB of the Rh10 tetragonal antiprism with capped square faces (from left to right). Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

Rh-Cluster

189

Rh10b: Edge Sharing Double Octahedra Total magnetic moment: 10µB Bond lengths [˚ A]

8

7

1.19

6

5

3

0.81

2

0.81

1.19

4

1.02

1

1.02

0.81

0.81

10

1.19

1111-

2: 4: 7: 9:

2.55 2.51 2.55 2.55

2222-

3: 5: 7: 8:

2.64 2.55 2.59 2.59

2- 9: 2.59 2-10: 2.59 3- 4: 2.55 3- 6: 2.55 3- 7: 2.59 3- 8: 2.59 3- 9: 2.59 3-10: 2.59

6-10: 2.55 4- 7: 2.55 4- 9: 2.55 5- 6: 2.51 5- 8: 2.55 5-10: 2.55

7- 8: 2.55 9-10: 2.55

6- 8: 2.55

A Average bond length: 2.56 ˚

9 1.19

4Emag : 7 meV Point symmetry: D2h - mmm

Total magnetic moment: 12µB Bond lengths [˚ A]

8

7

1.47

6

5

3

0.91

2

0.91

1.47

4

1.25

1

1.25

0.91

0.91

1111-

2: 4: 7: 9:

2.56 2.47 2.57 2.57

2222-

3: 5: 7: 8:

2.56 2.56 2.61 2.61

2- 9: 2.61 2-10: 2.61 3- 4: 2.56 3- 6: 2.56 3- 7: 2.61 3- 8: 2.61 3- 9: 2.61 3-10: 2.61

6-10: 2.57 4- 7: 2.57 4- 9: 2.57 5- 6: 2.47 5- 8: 2.57 5-10: 2.57

7- 8: 2.61 9-10: 2.61

6- 8: 2.57

A Average bond length: 2.57 ˚

10

1.47

9 4Emag : - meV 4Estruct : 27 meV

1.47

Point symmetry: D2h - mmm

Total magnetic moment: 14µB Bond lengths [˚ A]

8

7

1.57

6

5

3

1.20

2

1.20

1.57

1.46

1

1.46

10

1.57

4 1.20

9 1.57

1.20

1111-

2: 4: 7: 9:

2.57 2.52 2.56 2.56

2222-

3: 5: 7: 8:

2.55 2.57 2.61 2.61

2- 9: 2.61 2-10: 2.61 3- 4: 2.57 3- 6: 2.57 3- 7: 2.61 3- 8: 2.61 3- 9: 2.61 3-10: 2.61

A Average bond length: 2.58 ˚ 4Emag : 10 meV Point symmetry: D2h - mmm

6-10: 2.56 4- 7: 2.56 4- 9: 2.56 5- 6: 2.52 5- 8: 2.56 5-10: 2.56 6- 8: 2.56

7- 8: 2.65 9-10: 2.65

190

Appendix B

Total magnetic moment: 16µB 8

7

1.66

1.66

6

3

1.50

5

4

1.69

2

1.50

1.50

1

1.69

1.50

10

1.66

Bond lengths [˚ A]

1111-

2: 4: 7: 9:

2.60 2.54 2.57 2.57

2222-

3: 5: 7: 8:

2.56 2.60 2.61 2.61

2- 9: 2.61 2-10: 2.61 3- 4: 2.60 3- 6: 2.60 3- 7: 2.61 3- 8: 2.61 3- 9: 2.61 3-10: 2.61

6-10: 2.57 4- 7: 2.57 4- 9: 2.57 5- 6: 2.54 5- 8: 2.57 5-10: 2.57

7- 8: 2.67 9-10: 2.67

6- 8: 2.57

A Average bond length: 2.59 ˚

9 1.66

4Emag : 3 meV Point symmetry: D2h - mmm

Rh10c : Trigonal Pyramid Total magnetic moment: 0µB Bond lengths [˚ A]

10

-0.04

9

7

0.01

1

-0.01

1- 2: 2.44 1- 4: 2.45 1- 7: 2.44

8

0.06

0.01

6

4

0.01

2

-0.01

-0.03

5

0.01

3

22222-

3: 4: 5: 7: 8:

2.44 2.64 2.64 2.64 2.64

3- 5: 2.45 3- 8: 2.44 4444-

5: 6: 7: 9:

2.67 2.44 2.59 2.60

5- 6: 2.44 5- 8: 2.59 5- 9: 2.60

8- 9: 2.59 8-10: 2.44

6- 9: 2.48

9-10: 2.49

7- 8: 2.67 7- 9: 2.59 7-10: 2.44

Average bond length: 2.53 ˚ A 4Emag : 15 meV

-0.01

Point symmetry: C1 - 1

Total magnetic moment: 2µB Bond lengths [˚ A]

10

0.25

9

7

0.37

1

-0.12

1- 2: 2.42 1- 4: 2.46 1- 7: 2.45

8

0.37

6

4

0.37

2

-0.27

22222-

0.58

3

-0.12

0.20

5

0.37

3: 4: 5: 7: 8:

2.42 2.65 2.65 2.65 2.65

3- 5: 2.46 3- 8: 2.45 4444-

5: 6: 7: 9:

2.70 2.44 2.57 2.59

Average bond length: 2.54 ˚ A 4Emag : - meV 4Estruct : 68 meV Point symmetry: C1 - 1

5- 6: 2.44 5- 8: 2.57 5- 9: 2.59

8- 9: 2.58 8-10: 2.44

6- 9: 2.49

9-10: 2.51

7- 8: 2.70 7- 9: 2.58 7-10: 2.44

Rh-Cluster

191

Total magnetic moment: 4µB Bond lengths [˚ A]

10

0.31

9

7

0.37

1

0.41

1- 2: 2.41 1- 4: 2.58 1- 7: 2.43

8

0.87

0.37

6

4

0.50

2

0.34

-0.08

5

0.50

3

22222-

3: 4: 5: 7: 8:

2.41 2.60 2.60 2.69 2.69

3- 5: 2.58 3- 8: 2.43 4444-

5: 6: 7: 9:

2.62 2.45 2.56 2.62

5- 6: 2.45 5- 8: 2.56 5- 9: 2.62

8- 9: 2.62 8-10: 2.42

6- 9: 2.48

9-10: 2.57

7- 8: 2.67 7- 9: 2.62 7-10: 2.42

Average bond length: 2.54 ˚ A 4Emag : 21 meV

0.41

Point symmetry: C1 - 1

Total magnetic moment: 6µB Bond lengths [˚ A]

10

0.38

9

7

0.56

1

0.45

1- 2: 2.42 1- 4: 2.56 1- 7: 2.44

8

1.07

0.56

6

4

0.80

2

0.66

3

0.45

0.25

5

0.80

22222-

3: 4: 5: 7: 8:

2.42 2.61 2.61 2.70 2.70

3- 5: 2.56 3- 8: 2.44 4444-

5: 6: 7: 9:

2.61 2.46 2.57 2.62

5- 6: 2.46 5- 8: 2.57 5- 9: 2.62

8- 9: 2.62 8-10: 2.43

6- 9: 2.49

9-10: 2.56

7- 8: 2.66 7- 9: 2.62 7-10: 2.43

A Average bond length: 2.55 ˚ 4Emag : 48 meV Point symmetry: C1 - 1

Isosurfaces of the magnetization densities for M=0µB to M=6µB of the Rh10 trigonal pyramid (from left to right). Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

192

Appendix B

Rh11a: Polytetrahedral Cluster Total magnetic moment: 3µB Bond lengths [˚ A]

1

2

-0.14

-0.14

1111-

6 0.40 7 0.40

5

0.71

9

0.18

0.71

8

0.18

11

3

2: 5: 6: 7:

2.45 2.51 2.60 2.60

2- 3: 2.51 2- 6: 2.60 2- 7: 2.60

0.18

4 0.32

3- 4: 2.55 3- 6: 2.51 3- 7: 2.51 3- 8: 2.49 3-10: 2.49 4- 5: 2.55 4- 8: 2.58 4- 9: 2.58

4-10: 2.58 4-11: 2.58 5- 6: 2.51 5- 7: 2.51 5- 9: 2.49 5-11: 2.49

6- 9: 2.61 7-10: 2.61 7-11: 2.61 8- 9: 2.47 10-11: 2.47

6- 8: 2.61

Average bond length: 2.54 ˚ A

10

4Emag : 9 meV

0.18

Point symmetry: C2v - mm2

Total magnetic moment: 5µB Bond lengths [˚ A]

1

2

1111-

0.78

0.78

6 0.44 7 0.44

5

0.90

9

0.09

0.90

8

0.09

11

3 0.09

4 0.40

2: 5: 6: 7:

2.57 2.49 2.58 2.58

2- 3: 2.49 2- 6: 2.58 2- 7: 2.58

3- 4: 2.54 3- 6: 2.53 3- 7: 2.53 3- 8: 2.48 3-10: 2.48 4- 5: 2.54 4- 8: 2.58 4- 9: 2.58

4-10: 2.58 4-11: 2.58 5- 6: 2.53 5- 7: 2.53 5- 9: 2.48 5-11: 2.48

6- 9: 2.59 7-10: 2.59 7-11: 2.59 8- 9: 2.51 10-11: 2.51

6- 8: 2.59

Average bond length: 2.55 ˚ A

10

4Emag : - meV 4Estruct : 7 meV

0.09

Point symmetry: C2v - mm2

Total magnetic moment: 7µB Bond lengths [˚ A]

1

2

0.13

0.13

1111-

6 0.48 7 0.48

5

1.05

9

0.82

1.05

8

0.82

11

3 0.82

4 0.42

2: 5: 6: 7:

2.54 2.49 2.59 2.59

2- 3: 2.49 2- 6: 2.59 2- 7: 2.59

3- 4: 2.54 3- 6: 2.52 3- 7: 2.52 3- 8: 2.49 3-10: 2.49 4- 5: 2.54 4- 8: 2.58 4- 9: 2.58

Average bond length: 2.55 ˚ A

10 0.82

4Emag : 1 meV Point symmetry: C2v - mm2

4-10: 2.58 4-11: 2.58 5- 6: 2.52 5- 7: 2.52 5- 9: 2.49 5-11: 2.49

6- 9: 2.61 7-10: 2.61 7-11: 2.61 8- 9: 2.55 10-11: 2.55

6- 8: 2.61

Rh-Cluster

193

Total magnetic moment: 9µB Bond lengths [˚ A]

1

2

0.80

0.80

1111-

6 0.59 7 0.59

5

1.21

9

0.80

1.21

8

0.80

11

3

2: 5: 6: 7:

2.54 2.50 2.59 2.59

2- 3: 2.50 2- 6: 2.59 2- 7: 2.59

0.80

4 0.60

3- 4: 2.52 3- 6: 2.52 3- 7: 2.52 3- 8: 2.50 3-10: 2.50 4- 5: 2.52 4- 8: 2.59 4- 9: 2.59

4-10: 2.59 4-11: 2.59 5- 6: 2.52 5- 7: 2.52 5- 9: 2.50 5-11: 2.50

6- 9: 2.59 7-10: 2.59 7-11: 2.59 8- 9: 2.54 10-11: 2.54

6- 8: 2.59

Average bond length: 2.55 ˚ A

10

4Emag : 8 meV

0.80

Point symmetry: C2v - mm2

Total magnetic moment: 11µB Bond lengths [˚ A]

1

2

0.83

0.83

1111-

6 0.95 7 0.95

5

1.47

9

0.88

1.47

8

0.88

11

3

2: 5: 6: 7:

2.51 2.52 2.59 2.59

2- 3: 2.52 2- 6: 2.59 2- 7: 2.59

0.88

4 0.98

3- 4: 2.55 3- 6: 2.55 3- 7: 2.55 3- 8: 2.51 3-10: 2.51 4- 5: 2.55 4- 8: 2.59 4- 9: 2.59

4-10: 2.59 4-11: 2.59 5- 6: 2.55 5- 7: 2.55 5- 9: 2.51 5-11: 2.51

6- 9: 2.59 7-10: 2.59 7-11: 2.59 8- 9: 2.52 10-11: 2.52

6- 8: 2.59

Average bond length: 2.56 ˚ A

10

4Emag : 14 meV

0.88

Point symmetry: C2v - mm2

Total magnetic moment: 13µB Bond lengths [˚ A]

1

2

1.41

1.41

1111-

6 1.07 7 1.07

5

1.42

9

0.94

1.42

8

0.94

11

3 0.94

4 1.44

2: 5: 6: 7:

2.52 2.51 2.61 2.61

2- 3: 2.51 2- 6: 2.61 2- 7: 2.61

3- 4: 2.56 3- 6: 2.56 3- 7: 2.56 3- 8: 2.51 3-10: 2.51 4- 5: 2.56 4- 8: 2.61 4- 9: 2.61

Average bond length: 2.57 ˚ A

10 0.94

4Emag : 13 meV Point symmetry: C2v - mm2

4-10: 2.61 4-11: 2.61 5- 6: 2.56 5- 7: 2.56 5- 9: 2.51 5-11: 2.51

6- 9: 2.61 7-10: 2.61 7-11: 2.61 8- 9: 2.52 10-11: 2.52

6- 8: 2.61

194

Appendix B

Rh11b∗ : Edge Sharing Double Octahedra Plus Adatom Total magnetic moment: 5µB Bond lengths [˚ A]

11 8 0.98

0.15

7 -0.31

6 5

0.68

-0.91

3

2

0.44

1.00

4 1

0.70

0.39

10

1111-

2: 4: 7: 9:

2.49 2.46 2.58 2.66

22222-

3: 5: 7: 8: 9:

2.61 2.49 2.55 2.61 2.58

2-10: 2.59

4- 9: 2.59

3- 4: 2.53 3- 6: 2.55 3- 7: 2.48 3- 9: 2.56 3-10: 2.54 3-11: 2.56

5- 6: 2.51 5- 8: 2.57 5-10: 2.62

7- 8: 2.62 7-11: 2.57 8-11: 2.52 9-10: 2.54

6- 8: 2.55 6-10: 2.74 6-11: 2.53

4- 7: 2.60

Average bond length: 2.56 ˚ A

1.11

9 4Emag : 26 meV

0.78

Point symmetry: C1 - 1

Total magnetic moment: 7µB Bond lengths [˚ A]

11 8 0.26

0.81

7 0.53

6 5

0.02

0.91

3

2

0.31

1.37

4 1

0.40

0.94

10

1111-

2: 4: 7: 9:

2.50 2.53 2.56 2.63

22222-

3: 5: 7: 8: 9:

2.62 2.50 2.57 2.53 2.61

2-10: 2.56 3- 4: 2.57 3- 6: 2.53 3- 8: 2.47 3- 9: 2.55 3-10: 2.58 3-11: 2.54

4- 9: 2.60 4-11: 2.64

7- 8: 2.65 7-11: 2.52

5- 6: 2.46 5- 8: 2.58 5-10: 2.61

8-11: 2.57 9-10: 2.62

6- 8: 2.59 6-10: 2.55

4- 7: 2.50

Average bond length: 2.56 ˚ A

0.90

9 4Emag : 20 meV

0.55

Point symmetry: C1 - 1

Total magnetic moment: 9µB Bond lengths [˚ A]

11 8 1.18

1.31

7 0.97

6 5

0.83

1.34

3

2

0.52

1.42

4 1

-0.36

10 0.56

1.25

1111-

2: 4: 7: 9:

2.52 2.49 2.57 2.56

22222-

3: 5: 7: 8: 9:

2.69 2.54 2.55 2.58 2.60

2-10: 2.57

4- 9: 2.52

3- 4: 2.53 3- 6: 2.53 3- 7: 2.48 3- 9: 2.56 3-10: 2.55 3-11: 2.59

5- 6: 2.56 5- 8: 2.64 5-10: 2.56

4- 7: 2.59

Average bond length: 2.57 ˚ A

9 -0.02

4Emag : 12 meV Point symmetry: C1 - 1

7- 8: 2.58 7-11: 2.58 8-11: 2.56 9-10: 2.69

6- 8: 2.53 6-10: 2.62 6-11: 2.61

Rh-Cluster

195

Total magnetic moment: 11µB Bond lengths [˚ A]

11 8 1.24

1.22

7 0.82

6 5

0.93

1.31

3

2

0.57

1.35

4 1

0.37

1.13

10

1111-

2: 4: 7: 9:

2.52 2.52 2.60 2.60

22222-

3: 5: 7: 8: 9:

2.70 2.56 2.56 2.55 2.55

2-10: 2.58

4- 9: 2.57

3- 4: 2.50 3- 6: 2.52 3- 7: 2.66 3- 9: 2.57 3-10: 2.54 3-11: 2.53

5- 6: 2.58 5- 8: 2.55 5-10: 2.57

7- 8: 2.62 7-11: 2.53 8-11: 2.53 9-10: 2.67

6- 8: 2.49 6-10: 2.58 6-11: 2.73

4- 7: 2.53

Average bond length: 2.57 ˚ A

1.29

9 4Emag : 7 meV

0.78

Point symmetry: C1 - 1

Total magnetic moment: 13µB Bond lengths [˚ A]

11 8 1.33

1.17

7 1.02

6 5

1.23

1.19

3

2

1.17

1.41

4 1

1.02

2: 4: 7: 9:

2.52 2.52 2.59 2.58

22222-

3: 5: 7: 8: 9:

2.71 2.55 2.61 2.57 2.56

2-10: 2.60

4- 9: 2.59

3- 4: 2.51 3- 6: 2.49 3- 7: 2.64 3- 9: 2.59 3-10: 2.54 3-11: 2.56

5- 6: 2.58 5- 8: 2.59 5-10: 2.56

4- 7: 2.54

Average bond length: 2.58 ˚ A

10 1.46

1.02

1111-

9 0.99

4Emag : - meV 4Estruct : - meV Point symmetry: C1 - 1

7- 8: 2.60 7-11: 2.55 8-11: 2.54 9-10: 2.72

6- 8: 2.50 6-10: 2.59 6-11: 2.66

196

Appendix B

Rh12a: Capped Cube Total magnetic moment: 12µB 9 0.92

5 7 6

1.05

1.05

Bond lengths [˚ A] 1- 2: 2.48 1- 6: 2.57 1- 7: 2.57 1-10: 2.48 1-12: 2.58

4 8

1.05

1.05

0.86

3 10 1

11

1.05

2

1.05

0.86

1.05

2- 3: 2.57 2- 8: 2.57 2-11: 2.48 2-12: 2.58

3- 4: 2.57 3- 8: 2.57 3-11: 2.57 4- 5: 2.48 4- 8: 2.48 4- 9: 2.58 4-11: 2.57

5- 6: 2.57 5- 7: 2.48 5- 9: 2.58 5-10: 2.57 6- 7: 2.57 6-10: 2.57

8- 9: 2.58 10-11: 2.48 10-12: 2.58 11-12: 2.58

7- 8: 2.48 7- 9: 2.58

Average bond length: 2.55 ˚ A

1.05

4Emag : 22 meV

12

Point symmetry: C2h - 2/m

0.92

Total magnetic moment: 14µB 9

Bond lengths [˚ A]

0.98

5 7 6

1.26

1.26

1- 2: 2.51 1- 6: 2.57 1- 7: 2.52 1-10: 2.50 1-12: 2.57

4 8

1.26

1.26

0.99

3 10 1

11

1.26

2

1.26

1.26

12 0.98

1.26

0.99

2- 3: 2.57 2- 8: 2.52 2-11: 2.50 2-12: 2.57

3- 4: 2.57 3- 8: 2.57 3-11: 2.57 4- 5: 2.51 4- 8: 2.50 4- 9: 2.57 4-11: 2.52

Average bond length: 2.54 ˚ A 4Emag : - meV 4Estruct : 60 meV Point symmetry: C2h - 2/m

5- 6: 2.57 5- 7: 2.50 5- 9: 2.57 5-10: 2.52 6- 7: 2.57 6-10: 2.57 7- 8: 2.52 7- 9: 2.56

8- 9: 2.56 10-11: 2.52 10-12: 2.56 11-12: 2.56

Rh-Cluster

197

Total magnetic moment: 16µB

9 1.03

Bond lengths [˚ A]

5 7 1.49

6

1.49

1- 2: 2.50 1- 6: 2.57 1- 7: 2.50 1-10: 2.56 1-12: 2.57

4

1.49

8 1.49

1.02

3 10

1.49

1

1.49

2

1.49

1.02

11

2- 3: 2.57 2- 8: 2.50 2-11: 2.56 2-12: 2.57

3- 4: 2.57 3- 8: 2.57 3-11: 2.57 4- 5: 2.50 4- 8: 2.56 4- 9: 2.57 4-11: 2.50

5- 6: 2.57 5- 7: 2.56 5- 9: 2.57 5-10: 2.50 6- 7: 2.57 6-10: 2.57

8- 9: 2.57 10-11: 2.50 10-12: 2.57 11-12: 2.57

7- 8: 2.50 7- 9: 2.57

Average bond length: 2.55 ˚ A

1.49

4Emag : 3 meV

12

Point symmetry: C4h - 4/m

1.03

Rh12b: Incomplete Icosahedron Total magnetic moment: 8µB Bond lengths [˚ A]

11

12 1.28

8

-0.37

1.17

1

2 1.26

1.06

9 1.28

0.58

6

10 -0.37

5

3

1.26

70.94

-1.13

4

1.06

1- 2: 2.57 1- 3: 2.57 1- 4: 2.78 1- 5: 2.58 1- 6: 2.78 1- 7: 2.98 1- 8: 2.49 1- 9: 2.54 1-10: 2.48 1-11: 2.48 1-12: 2.54

2- 3: 2.60 2- 6: 2.54 2- 7: 2.64 2- 8: 2.59 2-12: 2.56 3333-

4: 7: 8: 9:

2.54 2.64 2.59 2.56

4- 5: 2.62 Average bond length: 2.64 ˚ A 4Emag : 35 meV Point symmetry: C1h - m

4- 7: 2.68 4- 9: 2.57 4-10: 2.64 5- 6: 2.62 5- 7: 2.58 5-10: 2.54 5-11: 2.54

8- 9: 2.97 8-12: 2.97 9-10: 2.73 10-11: 2.85 11-12: 2.73

6- 7: 2.68 6-11: 2.64 6-12: 2.57

198

Appendix B

Total magnetic moment: 10µB 11

12 0.69

0.92

8 1.24

0.56

2 0.83

9 0.69

1

0.45

6

10 0.92

5

3

1.16

0.83

4

0.56

1- 2: 2.60 1- 3: 2.60 1- 4: 2.52 1- 5: 2.75 1- 6: 2.52 1- 7: 2.90 1- 8: 2.54 1- 9: 2.46 1-10: 2.51 1-11: 2.51 1-12: 2.46

Bond lengths [˚ A] 2- 3: 2.59 2- 6: 2.56 2- 7: 2.61 2- 8: 2.68 2-12: 2.59 3333-

4: 7: 8: 9:

2.56 2.61 2.68 2.59

4- 5: 2.55

71.17

4- 7: 2.61 4- 9: 2.69 4-10: 2.56 5- 6: 2.55 5- 7: 2.73 5-10: 2.64 5-11: 2.64

8- 9: 2.68 8-12: 2.68 9-10: 2.91 10-11: 2.70 11-12: 2.91

6- 7: 2.61 6-11: 2.56 6-12: 2.69

Average bond length: 2.63 ˚ A 4Emag : 40 meV Point symmetry: C1h - m

Total magnetic moment: 12µB 11

12 0.81

1.17

8 1.16

0.76

1.08

9 0.81

1

0.76

6 2

10 1.17

5

3

1.17

1.08

4

0.76

1- 2: 2.65 1- 3: 2.65 1- 4: 2.53 1- 5: 2.73 1- 6: 2.53 1- 7: 2.97 1- 8: 2.50 1- 9: 2.47 1-10: 2.51 1-11: 2.51 1-12: 2.47

Bond lengths [˚ A] 2- 3: 2.57 2- 6: 2.55 2- 7: 2.71 2- 8: 2.58 2-12: 2.67 3333-

4: 7: 8: 9:

2.55 2.71 2.58 2.67

4- 5: 2.54

71.30

4- 7: 2.57 4- 9: 2.58 4-10: 2.68 5- 6: 2.54 5- 7: 2.65 5-10: 2.67 5-11: 2.67

8- 9: 2.85 8-12: 2.85 9-10: 2.82 10-11: 2.60 11-12: 2.82

6- 7: 2.57 6-11: 2.68 6-12: 2.58

Average bond length: 2.63 ˚ A 4Emag : 27 meV Point symmetry: C1h - m

Total magnetic moment: 14µB 11

12 1.15

1.15

8 1.13

2 1.10

1.10

9 1.15

1

1.35

6

10 1.15

5

3

1.10

71.44

1.07

4

1.10

1- 2: 2.57 1- 3: 2.57 1- 4: 2.56 1- 5: 2.56 1- 6: 2.56 1- 7: 2.58 1- 8: 2.51 1- 9: 2.52 1-10: 2.52 1-11: 2.52 1-12: 2.52

Bond lengths [˚ A] 2- 3: 2.64 2- 6: 2.65 2- 7: 2.64 2- 8: 2.66 2-12: 2.69 3333-

4: 7: 8: 9:

2.65 2.64 2.66 2.69

4- 5: 2.66 Average bond length: 2.63 ˚ A 4Emag : 32 meV Point symmetry: C1h - m

4- 7: 2.63 4- 9: 2.66 4-10: 2.70 5- 6: 2.66 5- 7: 2.63 5-10: 2.68 5-11: 2.68

8- 9: 2.70 8-12: 2.70 9-10: 2.67 10-11: 2.66 11-12: 2.67

6- 7: 2.63 6-11: 2.70 6-12: 2.66

Rh-Cluster

199

Total magnetic moment: 16µB 11

12 1.26

1.42

8 1.28

9 1.26

1

1.45

6 1.37

1.42

5

1.21

2

10

1.23

3

4

1.21

1.37

Bond lengths [˚ A]

1- 2: 2.60 1- 3: 2.60 1- 4: 2.59 1- 5: 2.58 1- 6: 2.59 1- 7: 2.58 1- 8: 2.52 1- 9: 2.53 1-10: 2.52 1-11: 2.52 1-12: 2.53

2- 3: 2.70 2- 6: 2.65 2- 7: 2.68 2- 8: 2.63 2-12: 2.62 3333-

4: 7: 8: 9:

2.65 2.68 2.63 2.62

4- 5: 2.69

71.51

4- 7: 2.63 4- 9: 2.69 4-10: 2.68 5- 6: 2.69 5- 7: 2.62 5-10: 2.60 5-11: 2.60

8- 9: 2.66 8-12: 2.66 9-10: 2.83 10-11: 2.66 11-12: 2.83

6- 7: 2.63 6-11: 2.68 6-12: 2.69

A Average bond length: 2.64 ˚ 4Emag : 19 meV Point symmetry: C1h - m

Total magnetic moment: 18µB 11

12 1.46

1.49

8 1.43

9 1.46

1

1.44

6 2 1.47

10 1.49

5

1.50

1.54

3

4

1.50

1.47

Bond lengths [˚ A]

1- 2: 2.58 1- 3: 2.58 1- 4: 2.59 1- 5: 2.61 1- 6: 2.59 1- 7: 2.62 1- 8: 2.54 1- 9: 2.54 1-10: 2.55 1-11: 2.55 1-12: 2.54

2- 3: 2.73 2- 6: 2.72 2- 7: 2.72 2- 8: 2.57 2-12: 2.59 3333-

4: 7: 8: 9:

2.72 2.72 2.57 2.59

4- 5: 2.67

4- 7: 2.69 4- 9: 2.59 4-10: 2.62 5- 6: 2.67 5- 7: 2.70 5-10: 2.62 5-11: 2.62

8- 9: 2.81 8-12: 2.81 9-10: 2.75 10-11: 2.70 11-12: 2.75

6- 7: 2.69 6-11: 2.62 6-12: 2.59

A Average bond length: 2.64 ˚

71.76

4Emag : - meV 4Estruct : 33 meV Point symmetry: C1h - m

Rh12c∗ : Edge Sharing Octahedra Plus Two Adatoms Total magnetic moment: 12µB 12 8

1.10

6

5 0.89

0.94

1.10

11 7

1.10

1.10

3 2

0.94

10 1.04

0.93

4 1 0.89

0.94

1111-

Bond lengths [˚ A] 2: 4: 7: 9:

2.51 2.53 2.57 2.60

2- 3: 2.73 2- 5: 2.51 2- 7: 2.67 2- 8: 2.67 2- 9: 2.56 2-10: 2.56

3- 4: 2.60 3- 6: 2.60 3- 7: 2.73 3- 8: 2.72 3- 9: 2.57 3-10: 2.57 3-11: 2.61 3-12: 2.61 4- 7: 2.60

9

Average bond length: 2.59 ˚ A

1.04

4Emag : 9 meV Point symmetry: C1h - m

4- 9: 2.52 4-11: 2.69

7- 8: 2.48 7-11: 2.49

5- 6: 2.53 5- 8: 2.57 5-10: 2.60

8-12: 2.48

6- 8: 2.60 6-10: 2.52 6-12: 2.69

11-12: 2.41

9-10: 2.66

200

Appendix B

Total magnetic moment: 14µB 12 8

1.16

1.25

1.15

1.25

3 1.04

2

1.12

7

1.16

6

5

11

1.03

4

1.15

1 1.12

10 1.28

1111-

Bond lengths [˚ A] 2: 4: 7: 9:

2.50 2.54 2.56 2.63

2- 3: 2.72 2- 5: 2.50 2- 7: 2.70 2- 8: 2.70 2- 9: 2.57 2-10: 2.57

3- 4: 2.58 3- 6: 2.58 3- 7: 2.76 3- 8: 2.76 3- 9: 2.55 3-10: 2.55 3-11: 2.64 3-12: 2.64

4- 9: 2.53 4-11: 2.64

7- 8: 2.49 7-11: 2.49

5- 6: 2.54 5- 8: 2.56 5-10: 2.63

8-12: 2.49

6- 8: 2.56 6-10: 2.53 6-12: 2.64

11-12: 2.46

9-10: 2.68

4- 7: 2.56

9

Average bond length: 2.59 ˚ A

1.28

4Emag : 4 meV Point symmetry: C1h - m

Total magnetic moment: 16µB 12 8

1.40

1.28

1.26

1.28

3 1.29

2

1.34

7

1.40

6

5

11

1.19

4

1.26

1 1.34

10

2: 4: 7: 9:

2.51 2.58 2.56 2.64

2- 3: 2.68 2- 5: 2.50 2- 7: 2.67 2- 8: 2.67 2- 9: 2.57 2-10: 2.57

3- 4: 2.55 3- 6: 2.55 3- 7: 2.77 3- 8: 2.77 3- 9: 2.59 3-10: 2.59 3-11: 2.66 3-12: 2.66

4- 9: 2.54 4-11: 2.63

7- 8: 2.59 7-11: 2.49

5- 6: 2.58 5- 8: 2.56 5-10: 2.64

8-12: 2.49

6- 8: 2.53 6-10: 2.54 6-12: 2.63

11-12: 2.49

9-10: 2.69

4- 7: 2.53

Average bond length: 2.59 ˚ A

9

1.49

1111-

Bond lengths [˚ A]

1.49

4Emag : - meV 4Estruct : - meV Point symmetry: C1h - m

Total magnetic moment: 18µB 12 8

1.43

6

5 1.60

1.52

1.33

11 7

1.43

1.33

3 2

1.52

10 1.57

1.57

4 1 1.60

1.52

1111-

Bond lengths [˚ A] 2: 4: 7: 9:

2.57 2.56 2.60 2.63

2- 3: 2.68 2- 5: 2.57 2- 7: 2.65 2- 8: 2.65 2- 9: 2.55 2-10: 2.55

3- 4: 2.61 3- 6: 2.61 3- 7: 2.76 3- 8: 2.76 3- 9: 2.60 3-10: 2.60 3-11: 2.63 3-12: 2.63 4- 7: 2.56

9

Average bond length: 2.60 ˚ A

1.57

4Emag : 1 meV Point symmetry: C1h - m

4- 9: 2.58 4-11: 2.65

7- 8: 2.54 7-11: 2.49

5- 6: 2.56 5- 8: 2.60 5-10: 2.63

8-12: 2.49

6- 8: 2.56 6-10: 2.58 6-12: 2.65

11-12: 2.47

9-10: 2.67

Rh-Cluster

201

Rh13a: Capped Cube with Central Atom Total magnetic moment: 5µB

10

Bond lengths [˚ A]

0.83

6 8

5

0.19

9

0.20

0.19

0.20

7

1

0.84

4

0.10

0.84

11 2

12

0.20

3

0.19

0.20

1- 2: 2.47 1- 3: 2.47 1- 5: 2.47 1- 6: 2.47 1- 8: 2.47 1- 9: 2.47 1-11: 2.47 1-12: 2.47 2- 7: 2.57

2-11: 2.44 2-13: 2.57 3- 4: 2.57 3-12: 2.44 3-13: 2.57 4- 5: 2.57 4- 9: 2.57 4-12: 2.57

5- 9: 2.44 5-10: 2.57

8-10: 2.57 9-10: 2.57

6- 7: 2.57 6- 8: 2.44 6-10: 2.57

11-13: 2.57 12-13: 2.57

7- 8: 2.57 7-11: 2.57

Average bond length: 2.52 ˚ A

0.19

4Emag : - meV 4Estruct : 99 meV

13 0.83

Point symmetry: C4v * - 4mm

Total magnetic moment: 7µB

10

Bond lengths [˚ A]

0.98

6 8

5

0.33

9

0.33

0.33

0.33

7

1

1.16

4

0.09

1.16

11 2

12

0.33

3

0.33

0.33

13 0.98

0.33

1- 2: 2.46 1- 3: 2.46 1- 5: 2.46 1- 6: 2.46 1- 8: 2.46 1- 9: 2.46 1-11: 2.46 1-12: 2.46 2- 3: 2.97 2- 7: 2.57

2-11: 2.48 2-13: 2.57 3- 4: 2.57 3-12: 2.48 3-13: 2.57 4- 5: 2.57 4- 9: 2.57 4-12: 2.57

A Average bond length: 2.58 ˚ 4Emag : 6 meV Point symmetry: C2h - 2/m

5- 6: 2.97 5- 9: 2.48 5-10: 2.57

8- 9: 2.97 8-10: 2.57 9-10: 2.57

6- 7: 2.57 6- 8: 2.48 6-10: 2.57 7- 8: 2.57 7-11: 2.57

11-12: 2.97 11-13: 2.57 12-13: 2.57

202

Appendix B

Total magnetic moment: 9µB

Bond lengths [˚ A]

10 1.29

6 8

5

0.47

9

0.47

0.47

0.47

7

1

1.29

4

0.07

1.29

11 2

12

0.47

3

0.47

0.47

1- 2: 2.45 1- 3: 2.45 1- 5: 2.45 1- 6: 2.45 1- 8: 2.45 1- 9: 2.45 1-11: 2.45 1-12: 2.45 2- 3: 2.98 2- 7: 2.58 2- 8: 2.98

2-11: 2.51 2-13: 2.58 3- 4: 2.58 3- 9: 2.98 3-12: 2.51 3-13: 2.58 4- 5: 2.58 4- 9: 2.58 4-12: 2.58

5- 6: 2.98 5- 9: 2.51 5-10: 2.58 5-12: 2.98

8- 9: 2.98 8-10: 2.58 9-10: 2.58

6- 7: 2.58 6- 8: 2.51 6-10: 2.58 6-11: 2.98

11-12: 2.98 11-13: 2.58 12-13: 2.58

7- 8: 2.58 7-11: 2.58

Average bond length: 2.63 ˚ A

0.47

4Emag : 1 meV

13 1.29

Point symmetry: C4v - 4mm

Total magnetic moment: 11µB

Bond lengths [˚ A]

10 0.96

6 8

5

0.79

9

0.79

0.79

0.79

7

1

1.32

4

0.10

1.32

11 2

12

0.79

3

0.79

0.79

13 0.96

0.79

1- 2: 2.48 1- 3: 2.48 1- 4: 2.92 1- 5: 2.48 1- 6: 2.48 1- 7: 2.92 1- 8: 2.48 1- 9: 2.48 1-11: 2.48 1-12: 2.48

2- 7: 2.61 2-11: 2.52 2-13: 2.56

5- 6: 2.69 5- 9: 2.52 5-10: 2.56

8-10: 2.56

3- 4: 2.61 3-12: 2.52 3-13: 2.56

6- 7: 2.61 6- 8: 2.52 6-10: 2.56

11-12: 2.69 11-13: 2.56

4- 5: 2.61 4- 9: 2.61 4-12: 2.61

7- 8: 2.61 7-11: 2.61

9-10: 2.56

12-13: 2.56

2- 3: 2.69 Average bond length: 2.58 ˚ A 4Emag : 6 meV Point symmetry: C2h - 2/m

8- 9: 2.69

Rh-Cluster

203

Rh13b: Centred Icosahedron Total magnetic moment: 7µB Bond lengths [˚ A]

13

1.09

11

12

8

10

9

0.07 1.16 -0.22

-0.22 1.16

1

0.88

6

2

-0.22 1.16

5

0.07

4

3 1.16

-0.22

1- 2: 2.60 1- 3: 2.60 1- 4: 2.51 1- 5: 2.49 1- 6: 2.51 1- 7: 2.58 1- 8: 2.49 1- 9: 2.51 1-10: 2.60 1-11: 2.60 1-12: 2.51 1-13: 2.58 2- 3: 2.60

2- 6: 2.72 2- 7: 2.59 2- 8: 2.70 2-12: 2.74 3333-

4: 7: 8: 9:

2.72 2.59 2.70 2.74

5- 6: 2.65 5- 7: 2.75 5-10: 2.70 5-11: 2.70 6- 7: 2.74 6-11: 2.74 6-12: 2.56

9-10: 2.72 9-13: 2.74 10-11: 2.60 10-13: 2.59 11-12: 2.72 11-13: 2.59 12-13: 2.74

4- 5: 2.65 4- 7: 2.74 4- 9: 2.56 4-10: 2.74

8- 9: 2.65 8-12: 2.65 8-13: 2.75

Average bond length: 2.64 ˚ A

7 1.09

4Emag : 53 meV Point symmetry: C2h - 2/m

Total magnetic moment: 15µB Bond lengths [˚ A]

13

1.13

11

12

8

10

1.12 1.12

1.13 1.12

9

1.13

1

1.51

6

1.13

2 1.12

5

1.12

7 1.13

3 1.12

4 1.13

1- 2: 2.55 1- 3: 2.55 1- 4: 2.55 1- 5: 2.55 1- 6: 2.55 1- 7: 2.55 1- 8: 2.55 1- 9: 2.55 1-10: 2.55 1-11: 2.55 1-12: 2.55 1-13: 2.55 2- 3: 2.68

2- 6: 2.68 2- 7: 2.68 2- 8: 2.68 2-12: 2.68 3333-

4: 7: 8: 9:

2.68 2.68 2.68 2.68

5- 6: 2.68 5- 7: 2.68 5-10: 2.68 5-11: 2.68 6- 7: 2.68 6-11: 2.68 6-12: 2.68

9-10: 2.68 9-13: 2.68 10-11: 2.68 10-13: 2.68 11-12: 2.68 11-13: 2.68 12-13: 2.68

4- 5: 2.68 4- 7: 2.68 4- 9: 2.68 4-10: 2.68

Average bond length: 2.65 ˚ A 4Emag : 6 meV Point symmetry: Ih - m5m

8- 9: 2.68 8-12: 2.68 8-13: 2.68

204

Appendix B

Total magnetic moment: 17µB Bond lengths [˚ A]

13

1.30

11

12

8

10

1.29 1.29

1.29 1.29

9

1.29

1

1.48

6

1.29

5

2 1.29

1.29

4

3 1.29

1.29

1- 2: 2.56 1- 3: 2.56 1- 4: 2.56 1- 5: 2.56 1- 6: 2.56 1- 7: 2.56 1- 8: 2.56 1- 9: 2.56 1-10: 2.56 1-11: 2.56 1-12: 2.56 1-13: 2.56 2- 3: 2.69

2- 6: 2.69 2- 7: 2.69 2- 8: 2.69 2-12: 2.69 3333-

4: 7: 8: 9:

2.69 2.69 2.69 2.69

5- 6: 2.69 5- 7: 2.69 5-10: 2.69 5-11: 2.69 6- 7: 2.69 6-11: 2.69 6-12: 2.69

9-10: 2.69 9-13: 2.69 10-11: 2.69 10-13: 2.69 11-12: 2.69 11-13: 2.69 12-13: 2.69

4- 5: 2.69 4- 7: 2.69 4- 9: 2.69 4-10: 2.69

8- 9: 2.69 8-12: 2.69 8-13: 2.69

Average bond length: 2.65 ˚ A

7 1.30

4Emag : 3 meV Point symmetry: Ih - m5m

Total magnetic moment: 21µB Bond lengths [˚ A]

13

1.67

11

12

8

10

1.62 1.63

1.63 1.63

9

1.63

1

1.38

6

1.63

2 1.63

5

1.62

3 1.63

4 1.63

1- 2: 2.58 1- 3: 2.58 1- 4: 2.58 1- 5: 2.58 1- 6: 2.58 1- 7: 2.56 1- 8: 2.58 1- 9: 2.58 1-10: 2.58 1-11: 2.58 1-12: 2.58 1-13: 2.56 2- 3: 2.68

2- 6: 2.69 2- 7: 2.67 2- 8: 2.76 2-12: 2.78 3333-

4: 7: 8: 9:

2.69 2.67 2.76 2.78

1.67

6- 7: 2.67 6-11: 2.78 6-12: 2.76

9-10: 2.69 9-13: 2.67 10-11: 2.68 10-13: 2.67 11-12: 2.69 11-13: 2.67 12-13: 2.67

4- 5: 2.70 4- 7: 2.67 4- 9: 2.76 4-10: 2.78

Average bond length: 2.67 ˚ A

7

5- 6: 2.70 5- 7: 2.67 5-10: 2.76 5-11: 2.76

4Emag : - meV 4Estruct : 45 meV Point symmetry: C2h - 2/m

8- 9: 2.70 8-12: 2.70 8-13: 2.67

Rh-Cluster

205

Isosurfaces of the magnetization densities for M=7µB (left) and M=21µB (right) of the Rh13 centred icosahedron. Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

Rh13c∗ : Cluster of Octahedra

Total magnetic moment: 13µB

Bond lengths [˚ A]

13 0.94

12

8

1.03

6

5

0.93

1.02

1.02

11

7 1.03

3 2

1.02

10 1.01

1.03

1111-

1.02

4 1

0.93

1.02

2: 4: 7: 9:

2.54 2.58 2.53 2.58

2- 3: 2.77 2- 5: 2.54 2- 7: 2.59 2- 8: 2.59 2- 9: 2.58 2-10: 2.58

3- 4: 2.58 3- 6: 2.58 3- 7: 2.77 3- 8: 2.77 3- 9: 2.58 3-10: 2.58 3-11: 2.58 3-12: 2.58 4- 7: 2.58 4- 9: 2.52 4-11: 2.62

Average bond length: 2.59 ˚ A

9 1.01

4Emag : 2 meV Point symmetry: C1h - m

5- 6: 2.58 5- 8: 2.53 5-10: 2.58

8-12: 2.58 8-13: 2.53 9-10: 2.62

6- 8: 2.58 6-10: 2.52 6-12: 2.62 7- 8: 2.60 7-11: 2.58 7-13: 2.53

11-12: 2.52 11-13: 2.58 12-13: 2.58

206

Appendix B

Total magnetic moment: 15µB

Bond lengths [˚ A]

13 1111-

1.03

12

8

1.09

1.27

1.03

1.27

1.27

1.09

6

5

11

7

3 1.04

2

1.09

4 1.27

1

1.03

10

2.53 2.61 2.52 2.61

2- 3: 2.78 2- 5: 2.53 2- 7: 2.64 2- 8: 2.64 2- 9: 2.55 2-10: 2.55

3- 4: 2.59 3- 6: 2.59 3- 7: 2.78 3- 8: 2.78 3- 9: 2.59 3-10: 2.59 3-11: 2.59 3-12: 2.59

6- 8: 2.56 6-10: 2.54 6-12: 2.62

4- 7: 2.56 4- 9: 2.54 4-11: 2.62

7- 8: 2.64 7-11: 2.56 7-13: 2.52

5- 6: 2.61 5- 8: 2.52 5-10: 2.61

8-12: 2.56 8-13: 2.52 9-10: 2.62 11-12: 2.54 11-13: 2.61 12-13: 2.61

Average bond length: 2.59 ˚ A 4Emag : - meV 4Estruct : - meV

9

1.27

2: 4: 7: 9:

1.27

Point symmetry: C1h - m

Total magnetic moment: 17µB Bond lengths [˚ A]

13 1.16

12

8

1.30

6

5

1.20

1.31

1.40

11

7 1.30

3 2

1.34

10 1.47

1.14

1111-

1.40

4 1

1.20

1.31

2: 4: 7: 9:

2.56 2.58 2.56 2.57

2- 3: 2.70 2- 5: 2.56 2- 7: 2.65 2- 8: 2.65 2- 9: 2.58 2-10: 2.58

3- 4: 2.58 3- 6: 2.58 3- 7: 2.69 3- 8: 2.69 3- 9: 2.62 3-10: 2.62 3-11: 2.60 3-12: 2.60 4- 7: 2.56 4- 9: 2.58 4-11: 2.58

5- 6: 2.58 5- 8: 2.56 5-10: 2.57

8-12: 2.62 8-13: 2.57 9-10: 2.66

6- 8: 2.56 6-10: 2.58 6-12: 2.58 7- 8: 2.64 7-11: 2.62 7-13: 2.57

11-12: 2.61 11-13: 2.58 12-13: 2.58

A Average bond length: 2.60 ˚

9 1.47

4Emag : 11 meV Point symmetry: C1h - m

Isosurfaces of the magnetization densities for M=13µB to M=17µB of the Rh13 cluster with a polyoctahedral structure (from left to right). Darkblue surfaces surround regions of negative, green surfaces of positive magnetization.

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Curriculum vitae Address

Mag. Tanja Futschek Center for Computational Materials Science Materials Physics Institute, Univ. of Vienna Sensengasse 8/12 A-1090 Vienna mailto: [email protected] phone: ++43(1)4277 51402 fax: ++43(1)4277 9514

Home address

Simmeringer Hauptstrasse 100a/22 A-1110 Wien

Date of birth Place of birth Nationality Marital status

December 31st , 1978 Vienna, Austria Austria single

Education Sept. 1985 - June 1989 Sept. 1989 - June 1997 June 1997 Sept. 1997 March 1998 - Dec. 2001 Dec. 2000 - Dec. 2001

Dec. 2001 Jan. 2002 - April 2005

Volksschule, Alberndorf im Pulkautal, Lower Austria Gymnasium, Laa an der Thaya, Lower Austria Graduation, Study of Meteorology at University of Vienna Study of physics at University of Vienna Diploma Thesis ”Dynamisches Feuchtewachstum von organischen und anorganischen Aerosolsubstanzen” at Institute for Experimental Physics, University of Vienna Academic degree ”Magistra rerum naturalium” awarded by University of Vienna PhD Thesis ”Structural, Electronic, and Magnetic Properties of Transition Metal Clusters”

215