Scalable Properties of Metal Clusters · 2.1. Transition Metal Clusters – Experiment, Theory, and Applications Metal clusters occupy a unique position between molecules and extended

Master’s Thesis

in the joint international graduate program

Advanced Materials Science (AMS)

within the “Elitenetzwerk Bayern” (ENB)

offered by

Technische Universitat Munchen (TUM)

Ludwig-Maximilians-Universitat Munchen (LMU)

Universitat Augsburg (UA)

Scalable Properties of Metal Clusters:A Computational Study

submitted by

Ralph Koitz, BSc

First Advisor: Prof. Dr. Dr. h.c. Notker RoschSecond Advisor: Prof. Dr. Johannes A. LercherPlace: MunichDate of Submission: May 2nd, 2011

Acknowledgments

The research presented in this thesis was carried out from October 2010 to May 2011 atthe Fachgebiet fur Theoretische Chemie at TU Munchen within the international graduateprogram Advanced Materials Science.

I am grateful to my advisor Prof. Dr. N. Rosch for the opportunity to work in his researchgroup, his supervision, and his support in academic matters and beyond.

I thank Dr. Alexander Genest for overseeing my work, for steering my efforts in the rightdirection on numerous occasions, and for always being available with guidance andfeedback. I also thank Dr. Sven Kruger for many helpful and interesting discussions.

I truly appreciate the frequent support by the ParaGAUSS developer team, especiallyDr. Alexei Matveev and Thomas Soini, who were always ready to provide technicaladvice and whose work much of this thesis is built upon.

I thank the colleagues in my office, David Tittle and George Beridze, as well as theentire group for the enjoyable and productive atmosphere and many interesting con-versations.

Finally, I wholeheartedly thank my parents, my entire family, and my girlfriend Irina fortheir unwavering support in all matters. Without them the writing of this thesis wouldnot have been possible.

Contents

1. Introduction 1

2. Background and Theory 32.1. Transition Metal Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2. Cluster Models in this Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3. Exchange-Correlation Functionals . . . . . . . . . . . . . . . . . . . . . . . 9

3. Computational Details 17

4. Results and Discussion 204.1. Scalable Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2. Magnetism in Small Pd Clusters . . . . . . . . . . . . . . . . . . . . . . . . 41

5. Summary 55

A. Basis Sets 58

B. Validation of the VMT[sol] Implementation 61

References 63

iii

1Introduction

Nanoscience is one of the most important research fields of the current decade.1,2 Ma-terials with properties tailored at the nanoscale level are promising for a wide varietyof applications in energy technology,3 electronics,4 information technology,5 and indus-trial processes such as the utilization of renewable resources and advanced catalysis.6,7

Nanoclusters are especially interesting materials since they lie at the interface betweenmolecules and solids and exhibit a multitude of fascinating properties that most othermaterials lack. As the size of a material is reduced from the macro- to the nanoscale,its physical, chemical, electronic and magnetic properties change drastically — metalsbecome semiconductors, semiconductors become insulators,8 nanoclusters show super-paramagnetism9,10 and confined electronic states11 — coupled with a general increasein surface area per volume and in reactivity.

Recent advances in experimental methods for the preparation of nanoclusters,12 usingtechniques such as molecular beam epitaxy, chemical vapor deposition, and numeroussynthesis strategies, as well as for their characterization have made it possible to createand investigate nanostructures in many sizes and morphologies. Metal nanoclusters fallinto two size ranges, the scalable regime (clusters beyond about 100 atoms), where prop-erties smoothly scale with cluster size, and the non-scalable regime (clusters with tens ofatoms), where a single atom may significantly change the properties of the entire clus-ter.13 Sophisticated preparation and analysis techniques have made it possible to followentire catalytic reaction cycles occurring on the surface of clusters14,15 and nanocatalysisis steadily gaining importance for many applications in chemistry.16

Concomitant with the development of experimental techniques, a host of theoreti-cal methods has also been developed for the prediction and interpretation of materials’properties at all size scales. Density Functional Theory (DFT) is currently the most pop-ular method for the investigation of metal clusters,17 and has had tremendous successin reproducing experimental results with high accuracy18 and has helped to elucidateprocesses19 otherwise not accessible by experimental or computational methods. Mod-ern implementations of DFT and the favorable scaling behavior of the method20 make itpossible to do sophisticated quantum-mechanical calculations on comparatively large

1

Master’s Thesis Introduction

systems with heavy-metal atoms and large numbers of electrons, while keeping thecomputational cost at a manageable level. Clusters of more than 300 atoms21 have thusbeen successfully treated, a feat that one currently is not able to accomplish with otherab-initio methods of comparable quality.

In spite of its success, DFT has the fundamental drawback that it cannot be easilyimproved in a systematic way.22 The biggest challenge for density functional methodsis the correct treatment of the exchange-correlation energy, which comprises most of thechemically relevant energetics of a system — but whose exact expression is unknownand can only be approximated.23 During the last forty years, much work has gone intothe development of exchange-correlation functionals, ranging from chiefly empirical tohighly systematic approaches.24 Notwithstanding these efforts, a fairly small numberof functionals are commonly used in applications, and experience is necessary to selectthe most suitable functional, choosing according to desired accuracy, computational costand the problem at hand.

Comparative studies of exchange-correlation functionals,25–27 particularly for metalclusters, are scarce and a number of recently published functionals28,29 have yet to beevaluated. The present work aims to add to the available assessments by comparingten functionals, some of them well-established in the field, some very recent additions.The chief concern is the scaling behavior of metal clusters Mn, for M = Pd and Au wheren = 13, 19, 38, 55, 79, 147. The clusters are characterized with respect to their equilib-rium geometries, cohesive energies as well as ionization potentials and electron affini-ties with the entire set of functionals. Such a comparison allows for a more judiciousselection of functionals for cluster problems and also aims to rationalize the behavior ofparticular functionals and functional classes. Moreover, investigations are carried outon the magnetic properties of small palladium clusters, (i) to characterize and quantifythe dependence of magnetic properties on the cluster structure and (ii) to rationalize thedifferent behaviors of functionals with respect to this problem.

The initial chapters of this work are dedicated to the background and theory under-lying the subsequent research. The state of the art is introduced with regard to recentexperimental and theoretical work, followed by a description of the employed modelsand density functional methods, primarily focusing on the various exchange-correlationfunctionals. Subsequently the results of the study are presented and discussed, detailingthe comparative DFT study on energetic and structural scaling in palladium and goldclusters, followed by investigations of the magnetic behavior of Pd19 and Pd38 clusters.The thesis concludes with a summary and a collection of supplementary information inthe appendices.

2

2Background and Theory

The following chapter outlines a number of previous theoretical and experimental stud-ies on metal clusters and mentions a few applications of nanoclusters. Subsequently thecluster models used throughout the presented studies are introduced with respect totheir structures and general properties. The chapter concludes with a brief introduc-tion of the theoretical background of density functional theory, with an emphasis onexchange-correlation approximations and the various functionals investigated.

2.1. Transition Metal Clusters – Experiment, Theory, and

Applications

Metal clusters occupy a unique position between molecules and extended solids. Theterm “metal cluster” typically applies to compounds of two atoms to several dozenatoms with direct metal-metal bonds.30 Even larger assemblies of this type, with up toseveral thousand atoms, are more properly designated as “metal particles”, but some-times (as in this text) also referred to as clusters, for the sake of convenience. A numberof unique effects are observed in clusters due to their small sizes and unusual bond-ing.9–11,31 This section briefly outlines some experimental and theoretical research onclusters and a number of applications.

2.1.1. Preparation and Characterization of Clusters

As mentioned in the introduction, one typically distinguishes between the scalable andnon-scalable size regimes of clusters. Within the former, the properties of a cluster arechiefly determined by its size and scale accordingly. The latter is characterized by prop-erties not proportional to the size, where adding or removing a single atom may dras-tically change the properties of the entire cluster.13 In order to prepare clusters of adefined size, sophisticated techniques for cluster synthesis and size selection are neces-sary. To produce clusters, one usually evaporates the desired material by laser pulses,32

electrical discharges33 or ion bombardment.13 The resulting extremely hot gas is then

3

Master’s Thesis Transition Metal Clusters

Figure 2.1.: Supported Pd particle on Al2O3, ref. 34

allowed to expand through a nozzle so that it is cooled and condenses to form a well-defined beam of clusters. Depending on the desired element, cluster sizes, and proper-ties, different sources are chosen, and different measurements can then be carried outon the clusters in the gas phase or after their deposition.13

The generated molecular beam of clusters usually needs to be filtered to select clustersof the desired size. Typical sources produce particles with fairly wide size distributions,which are commonly separated either by a quadrupole mass analyzer or by a time-of-flight analyzer. If supported clusters are desired, the clusters subsequently need to bedecelerated to soft-landing velocities.13 For this purpose, a repulsive potential is appliedto the substrate, which retards the incoming ions. Depending on the velocity distribu-tion, some are backscattered, while others are implanted below the surface, so that onlya fraction actually lands undamaged and is thus deposited.13

A number of spectroscopic and diffractive techniques can be employed to character-ize clusters in the gas phase. These techniques have made it possible to follow entirecatalytic reaction cycles on the surface of free clusters.14,15 When clusters are depositedon a support, powerful imaging techniques such as transmission electron microscopyand atomic force microscopy can be used to visualize their structures. Fig. 2.1 shows acrystalline palladium particle supported on aluminum oxide.

2.1.2. Theoretical Studies on Clusters

Numerous theoretical studies have been carried out on the structure, reactivity and scal-able behavior of transition metal clusters. Because of the metallic character of theseclusters it is essential that the employed method offers good treatment of electronic cor-relation, and, especially for heavy metal atoms, an adequate relativistic treatment is alsocrucial. Due to the fairly large sizes and thus high electron counts of “realistic” clusters,methods based on density functional theory (DFT) are currently the only suitable firstprinciples approach for treating clusters of more than 5–10 atoms,17 because more com-

4

Master’s Thesis Transition Metal Clusters

plex wavefunction methods become computationally too demanding.As metal clusters typically lack the directed bonds characteristic of molecules, they

can exist as a large number of isomers. Resolving these structural differences exper-imentally is very difficult but theoretical investigations on cluster structures can givevaluable insight. The stability of various isomers of small to medium-sized palladiumclusters was investigated∗ in a study by Ahlrichs et al.21 They showed that the moststable structures of small Pdn (n=4–7,9,13) correspond to those of Lennard-Jones clus-ters, but the energy differences between isomers were determined to be very small, giv-ing very flexible and “floppy” structures.21 They concluded that in general Pd clustersare three-dimensional for n>4 and tend toward dense packing. Beyond nuclearities ofPd100, the authors observed a preference for octahedral (i.e. fcc-derived) structures, butgenerally noted small energy differences between various isomers.17

Zanti et al.35 recently published a related study on the structures of small bimetallicPdnAum (n+m<14) clusters. They reported that the structure preferences for the twoatom types follow somewhat opposing trends, where pure Au and Au-rich clustersprefer planar structures and Pd and Pd-rich clusters tend towards three-dimensionalarrangements. Moreover, the Pd-rich clusters showed high spin states (i.e. a high num-ber of unpaired electrons), while the Au clusters remained at the lowest multiplicitythroughout.

As the investigated cluster sizes increase beyond a certain threshold, one enters thescalable regime where certain properties of the cluster depend on the cluster size. Thismakes it possible to extrapolate to infinite cluster sizes, i.e. the bulk material. Severalstudies have investigated the scaling properties of such systems, related to structure, en-ergetics, and reactivity. In addition to small Pd clusters, Ahlrichs et al.21 also examinedthe scaling behavior of Pd38–Pd309, thereby extending and complementing a previousstudy by Kruger et al.36 The authors extrapolated the average Pd-Pd bond length inthe cluster as a function of the average coordination number, finding a bulk value of281.6 pm compared to 274.8 pm of the experiment. The cohesive energy of the clustercould be extrapolated to 346.4 kJ·mol-1 (exp. 376.3 kJ·mol-1).21 When comparing the rel-ative stabilities of icosahedral and octahedral structures, the authors found the latter tobe more stable, as would be expected from fcc-palladium.

A general trend for the cluster structures is the contraction of bond lengths as thecluster gets smaller, which has also been observed experimentally.37 A recent investiga-tion focused on the details of this “buckling” of the structure and showed that most ofthe variability of bond lengths is contained within the bonds to surface atoms, and thatsubsurface atoms show little buckling and narrow distributions of bond lengths.38

Further studies on clusters have investigated their energetics and reactivity. As a

∗Using the TURBOMOLE DFT code with the BP functional and a slightly modified SVP basis set

5

Master’s Thesis Cluster Models in this Study

model for catalytic reactions, the adsorption of CO on cluster surfaces has receivedparticular attention. Both two-dimensional (“flat”) clusters39 and highly symmetricpolyhedral clusters40 have been used as adsorption models. As a general result it wasfound that some observables (vibration frequencies, adsorption bond lengths and an-gles) are fairly insensitive to the cluster size,40,41 whereas others (particularly adsorp-tion energies) show oscillations with cluster size. Only when the size of the (three-dimensional) cluster is increased beyond about 80 atoms, do adsorption energies start toconverge.40 Indeed, such adsorption models were successful in reproducing the exper-imental CO/cluster binding energy with chemical accuracy.40 Various other techniquessuch as artificially straining the core geometry of an otherwise relaxed cluster, have alsobeen employed to study adsorption behavior on clusters.18

2.1.3. Applications and Reactivity

Homogeneous catalysts based on small metal clusters have been applied in a number ofimportant reactions and processes.42 They have the advantage of being tunable by de-sign of the ligands, which influence reactivity and selectivity, and by modifying themetal-metal interactions in the cluster.42 Cluster catalysts are particularly suited formulti-electron reactions, such as the reduction of nitrophenol to aniline and varioushydrogenation reactions.42,43

Solid metal clusters have also been extensively used as catalysts. By reducing theparticle size down to a few atoms, even inert materials such as gold become catalyticallyactive.31 Gold clusters, both free and supported, have successfully been used for COcombustion, even at low temperatures, and various other reactions.

Interestingly, supported gold clusters are characterized by a distinctly size-dependentreactivity, where particles smaller than Au8 exhibit no reactivity toward CO oxidation,whereas larger clusters show oscillations of the reactivity with cluster size, presumablydue to different adsorption sites on clusters of different size.31 Supported Pd clustersshow similar effects, e.g. an increase of the heat of adsorption of CO with particle size.44

2.2. Cluster Models in this Study

In order to model a range of cluster sizes for an investigation of the scalability of prop-erties, six clusters M13 . . . M147 of octahedral symmetry with Pd and Au as the metal Mwere chosen for this study. This section outlines the structures of the clusters and theirgeneral properties.

6


(a) M13 (b) M19 (c) M38 (d) M55 (e) M79 (f) M147

Figure 2.4.: Structures of the cluster models M13–M147

Figure 2.5.: Atom types and faces in an M79 cluster

2.2.1. Structures of the Clusters

Six clusters of the nuclearities M13, M19, M38, M55, M79, and M147 with (truncated) oc-tahedral and cuboctahedral shapes were chosen for this study. The clusters are shownin Figure 2.4, and a summary of their properties is given in Table 2.1. Compared to aninfinite fcc crystal, where each M atom is coordinated by 12 neighbors, various coor-dination environments exist in a cluster, depending on the location of each atom. Inparticular, all atoms on the surface of the cluster are coordinatively unsaturated, thecorner atoms more so than edge atoms and edge atoms more strongly than atoms onthe (100) and (111) facets. These exposed atoms are subject to an inward-directed force,which distorts the cluster structure from the ideal bulk geometry to a more contractedshape that reduces the surface energy. The magnitude of this buckling increases withthe number of unsaturated bonds and decreases with cluster size. Fig. 2.5 shows thedifferent atom types in an M79 cluster.

As the size of the cluster increases, the fraction of surface atoms decreases until, in the

7


limit of an infinitely large cluster, an extended solid results. In the present models thefraction of surface atoms decreases from 92% to (M13) to 63% (M147). As more than halfof all simulated atoms remain at the surface even in the largest cluster, surface effectscan be expected to have a significant influence on the cluster properties and buckling ofthe structure is an effect that is present in all models.

In the present study all models were optimized in all degrees of freedom within oc-tahedral symmetry. With an increase in cluster size the number of degrees of freedomalso increases with the number of unique atoms and their positions.

Table 2.1.: Structural properties of the metal clustersused in this study

nAa Shape nu

b CNavc d.o.f.d xs

e

13 trunc. octahedr. 2 5.538 1 0.92

19 octahedron 3 6.316 6 0.95

38 trunc. octahedr. 3 7.579 9 0.84

55 cuboctahedron 5 7.855 12 0.76

79 trunc. octahedr. 6 8.509 15 0.75

147 cuboctahedron 9 8.980 24 0.63

a Total number of atoms in the cluster;b Number of unique atoms;c Average coordination number; d Total number of

degrees of freedom within octahedral symmetry;e ratio of surface atoms to total number of atoms;

2.2.2. Choice of Model

The choice of these clusters for the study of scalable properties, and for solid-state mod-eling in general has a number of advantages compared to other cluster geometries andother systems. Firstly, experimental findings suggest that such structures are indeedfavored by transition metal clusters, which tend to adopt very high symmetries (pointgroups Ih and Oh) and preferentially occur in “magic number” stoichiometries (n=6,13, 19, 38, 40, 44, 55, 85, 147).45,46 The structures of the octahedral isomers correspondto fragments of an ideal fcc lattice, which the metals palladium and gold adopt understandard conditions.47

Secondly, the use of highly symmetric structures is necessary in order to reduce thecomputational cost to a feasible level. By using symmetry constraints, very large clus-ters can be reduced to a small number of (unique) atoms, which are replicated bysymmetry operations to yield the final cluster. Thus, an M147 cluster is formally re-duced to 9 atoms and calculations of systems as large as 309 atoms and beyond become

8

Master’s Thesis Exchange-Correlation Functionals

tractable.21,36 Table 2.1 shows the number of unique atoms in each of the studied clus-ters.

Moreover, the selected models span a representative range of the scalable regime asdemonstrated by previous studies,36 and provide the possibility to extrapolate proper-ties to bulk values, thus constituting a link between molecules and extended solids. Ex-trapolated bulk values can be compared to experimental data, which is otherwise ratherscarce for well-defined small clusters, and can be used to judge the quality of calcula-tions. Furthermore, the series of collected data can be used to extrapolate properties forclusters of arbitrary size, which may otherwise be inaccessible computationally and/orexperimentally.18

In the present study the clusters were treated as single molecules in the gas phasewithout considering further effects such as ligands, solvation or surface support. Whilethis approach obviously neglects some components in typical experiments, it also re-duces the possible sources of error in the calculation and ensures that all observed ef-fects are truly effects of the metal cluster and its DFT treatment. Moreover, recent exper-iments with bare metal clusters in a molecular beam show that such isolated particlescan be prepared and additionally support this choice of model.13,14,48

2.3. Exchange-Correlation Functionals

According to the first Hohenberg-Kohn Theorem,49 the electronic ground state of a sys-tem is uniquely given from the system’s electron density ρ(~r). Furthermore, the sec-ond Hohenberg-Kohn Theorem provides a variational approach to actually obtain theground-state density by minimization of a trial density, giving the following expressionfor the ground-state energy:

E0 = minρ→N

(F [ρ] +

∫ρ(~r)VNed~r

)(2.1)

where ~r refers to a variable point in space, VNe is the potential of electron-nuclei in-teraction, and minρ→N refers to a minimization under the constraint of preserving theelectron number as the total density.23 F [ρ] is a so-called universal functional, given by

F [ρ] = T [ρ(~r)] + J [ρ(~r)] + EXC [ρ(~r)] (2.2)

where, within the Kohn-Sham approach, T [ρ(~r)] is the exact kinetic energy of the elec-trons in a non-interacting reference system and J [ρ(~r)] is the classical Coulomb re-pulsion of the electrons.23 Both T and J are straightforward to compute, whereas theexchange-correlation energy EXC , containing the remaining components of the energysuch as self-interaction correction, exchange and correlation as well as a minor part of

9


the kinetic energy, is not known. EXC is frequently written as an integral (Eq. 2.3), whereεXC is the XC energy density per particle. XC functionals are categorized according tothe functional dependence in the integral kernel (vide infra).

EXC =

∫ρ(~r)εXC(~r) (2.3)

The most difficult aspect of determining the ground-state energy is computing EXC ,which can only be approximated. In fact, without a suitable approximation toEXC [ρ(~r)],density functional theory could not be practically implemented at all. Therefore, vari-ous approximate exchange-correlation functionals of different degrees of accuracy andcomplexity have been developed since the advent of DFT.

A number of different strategies are employed for the design of exchange-correlationfunctionals.24 On the one hand, very empirical procedures may be used to fit a certainanalytic form of EXC to a set of experimental values.24 On the other hand, fully non-empirical functionals may be constructed by requiring the functional to satisfy funda-mental constraints, such as the sum rules for the exchange and correlation holes, scalingrelations, and correct asymptotic behavior.23 In between those two extremes, some func-tionals are developed to satisfy a number of constraints, while at the same time contain-ing fitted parameters.50 No universal “best” functional or “best” construction approachhas yet been found, and the choice of what functional should be used depends on manyfactors.

According to a classification by J. P. Perdew,51 the various density functionals can begrouped in five classes or rungs of a “Jacob’s Ladder” by the functional dependence ofεXC on ρ(~r). The lowest rung of the ladder is occupied by functionals where εXC de-pends only on the local value of the density, ρ(~r), constituting the Local Density Approxi-mation (LDA). Located at the second rung of the ladder are those, where εXC , in additionto the local density also depends on its gradient, ∇ρ(~r); they are collectively referred toas Generalized Gradient Approximations (GGA). meta-GGA functionals rely additionally onthe positive orbital kinetic energy density τ as variable of εXC . The topmost rungs of theladder are occupied by functionals with fully non-local components. hyper-GGA func-tionals add the exact exchange energy density, and the random-phase approximationadds unoccupied orbitals.51 The complexity of the functionals as well as their compu-tational cost increases as the ladder is ascended, and in many cases the accuracy of theresults improves as well.

The present study evaluates ten functionals on the first through third rungs of theladder, which are summarized in Table 2.2. In order to provide some background on theconstruction of those functionals and their forms, the remainder of this section brieflyoutlines the functional groups and the functionals studied in the present work.

10


Table 2.2.: Overview of the Exchange-Correlation Functionals used inthis study.

Name Authors Type Ref.

VWN Vosko, Wilk, Nusair LDA 52

BP Becke, Perdew GGA 53,54

PW91 Perdew, Wang GGA 55

PBE Perdew, Burke, Ernzerhof GGA 56

PBEsol Perdew, Ruzsinsky, Csonka GGA 57

VMT[sol] Vela, Medel, Trickey GGAa 28

VT{84}[sol] Vela, Trickey GGAa 29

M06-L Zhao, Truhlar meta-GGA 50

a Only treats the exchange part of EXC and uses PBE correlation.

2.3.1. Local Density Functionals: VWN

In the LDA only the local value ρ(~r) is used to compute the exchange-correlation en-ergy, which corresponds to the exchange-correlation energy of the homogenous electrongas (HEG), which is known exactly.23 LDA is an exact method in the limit of a HEG, andthus is well-suited for the treatment of extended systems (metals) where the (valence)electron density indeed resembles that of a uniform “gas”. For molecular systems, how-ever, LDA methods have been found to underestimate the exchange energy by about10% and overestimate correlation by up to a factor of 2.22 This results in significant“overbinding”, i.e. too short bond lengths and too high binding energies.

The most widely used LDA functional is the one constructed by Vosko, Wilk, and Nu-sair (VWN),52 which uses the Slater Xα exchange energy and an interpolation functionto fit the HEG correlation energy obtained from accurate Quantum Monte Carlo data.22

2.3.2. Gradient-Corrected functionals: BP, PBE, PBEsol, PW91

As a first improvement on LDA, the Generalized-Gradient Approximation methods use thegradient of ρ(~r) as an additional variable in the integral kernel. As was already the casefor LDA, EXC is typically split into EX and EC , which are determined independently(Eq. 2.4). The gradient typically enters the expressions in the form of the reduced densitygradient s(~ρ) (Eq. 2.5).

EGGAXC = EGGA

X [s] + EGGAC [s] (2.4)

s(~r) =|∇ρ(~r)|ρ4/3(~r)

(2.5)

11


Becke-Perdew (BP)The BP†functional is a combination of the gradient-dependent approximation of the ex-change energy suggested by Becke54 and a correlation term suggested by Perdew.53 TheBecke exchange is a corrective term added to the LDA exchange energy, which enforcesthe correct asymptotic behavior in the large-s limit of the energy density. Furthermore,it also recovers the fundamental lowest-order gradient correction in the small-s limit.The exchange term contains only one adjustable parameter, β = 0.0042, which was fit-ted to Hartree-Fock exchange energies of six noble gas atoms.54 Eq. 2.6 gives the Beckeexchange functional.

EB88X = ELDA

X [s]− βρ1/3 s2

1 + 6βs sinh−1 x(2.6)

The correlation term is an improvement on a previous functional by Langreth andMehl,58,59 which was designed to satisfy a number of systematic constraints. It is re-quired to reduce to the LDA in the limit of a uniform density, as well as to recover thesecond-order density gradient expansion in the limit of a slowly varying density.24 Thefunctional contains one parameter, which was fitted to recover an accurate value for thecorrelation energy of the neon atom.60

As can be seen from calculations on reference systems23,24 the BP functional notice-ably improves the LDA values especially with respect to energies of formation, whichis a general feature of gradient-corrected functionals. However, molecular geometriesare not improved compared to VWN, in fact, GGA functionals are generally prone tooverestimating bond lengths and BP shows this behavior somewhat more strongly thanother functionals.

Perdew-Wang (PW91)The PW91 functional is a fully non-empirical functional, where the exchange partEPW91

X

is based on Becke’s exchange functional (Eq. 2.6), but adds an additional Gaussian termin s and a s4 term.55 EPW91

X satisfies a large number of exact constraints, but abandonsthe constraint of reproducing the correct asymptotic exchange energy density.24

The correlation component EPW91C (Eq. 2.7) consists of a term for the LDA correlation

and a gradient correction term H(t),22 and is truncated in order to integrate to zero (anecessary property of the correlation hole).

EPW91C = ELDA

C +H(t) (2.7)

H(t) is a gradient correction term where t is a scaled density gradient similar to s. The

†In the literature the acronym BP86 is frequently used to refer to the Becke-Perdew functional. However,the current implementation of ParaGAUSS uses the acronym ‘BP’, which is also applied throughoutthe present work.

12


analytic form of H(t) is chosen such that two conditions are satisfied:24

1. H reduces to the second-order term in the gradient expansion in the limit of aslowly varying density.

2. In the limit of an infinitely fast varying density (t→∞) EPW91C → 0.

For molecular energies PW91 improves slightly upon the BP results,24 while mostother quantities remain fairly similar.

Perdew-Burke-Ernzerhof (PBE and PBEsol)The PBE functional was intended as the successor of PW91 and as such retains manyof the original principles as described above and is also fully non-empirical. As animprovement over PW91, PBE provides an accurate description of the linear responseof the HEG, correct behavior for uniform coordinate scaling and a smoother potential.56

However, while PW91 was designed to satisfy as many exact conditions as possible,PBE satisfies only those which are significant for calculating energies.

The exchange part of PBE is designed to obey four constraints, namely the recoveryof the HEG limit for a vanishing density gradient, a spin-scaling condition, the recoveryof the linear response of the HEG in LDA for s → 0 and the Lieb-Oxford bound56,61 (videinfra). In addition to the two important asymptotic constraints for EC listed above, thecorrelation part of PBE also satisfies the condition of uniform scaling, so that in the limitof an infinitely high density EPBE

C scales to a constant.24,56

The PBEsol functional57 is a very recent re-parametrization of the original PBE, de-signed especially for the accurate treatment of extended solids. In fact, the only dif-ference between PBE and PBEsol are the values of two parameters, so that most of theexact constraints remain fulfilled.

In particular, the gradient expansion behavior, which is violated in PW91 and PBE, isrestored in PBEsol. This means, that in the limit of s → 0 the exchange energy recoversthe gradient expansion

EX = ELDAX [ρ]− βX

∫ρ4/3s2 d~r (2.8)

which is equivalent to

EX =

∫ρεLDAX (ρ)

[1 + µXs

2]

d~r (2.9)

where µX is a parameter proportional to βX and the term [1 + µXs2] is usually referred

to as the exchange enhancement factor FX .In order to recover the HEG limit µ needs to have the value µ = µGE = 10/81, which is

violated in PBE (µ = 0.2195) but obeyed by PBEsol. This change of µ is expected to im-prove the description of the slowly varying densities in extended solids, at the expenseof worsening the energies of atoms.57 Thus, quantities dependent on atomic energies,

13


such as bond energies and cohesive energies are expected to be worse in PBEsol, whilebond lengths should improve.

The second change with respect to PBE concerns the correlation term. Once again theparameter µ of the PBE form is changed to µGE and another parameter, β is set to 0.046,which is a compromise between βGE (as in PBE) and β = 0.0375 in the LDA limit of theHEG response to a weak potential. The value of beta was chosen to fit the correlationpart to the surface XC energy of a jellium cluster of infinite size. Therefore, PBEsol isalso expected to improve surface energies compared to LDA and PBE.57

2.3.3. meta-GGA Functionals: M06-L

Meta-GGA functionals represent a step up from gradient-corrected functionals on theladder of exchange-correlation. This group of functionals has an additional functionaldependence on the Laplacian of the density,∇2ρ(~r), or, more commonly, the non-interactingkinetic energy density τ(~r).51

τ is the kinetic energy density of the Kohn-Sham (KS) system, and corresponds to theintegrand of the kinetic energy term62 T [ρ(~r)].

τ =1

2

∑σ

occ.∑j

|∇φjσ(~r)|2 (2.10)

where σ sums over both spins and φjσ denotes the KS orbitals. In the limit of a slowlyvarying density τ and∇2ρ can be expanded to give the following expression:22,63

τ =3

10(6π2)2/3ρ(~r)5/3 +

1

72

|∇ρ(~r)|2

ρ(~r)+

1

6∇2ρ(~r) +O(∇4) (2.11)

Accordingly, the two quantities τ and ∇2ρ contain a similar amount of chemical infor-mation, but it has been suggested that τ has some additional non-local character as itdepends on the KS orbitals, which in turn are influenced by integrals over all space.64

For this reason, and also due to numerical instabilities associated with evaluation ofthe Laplacian of the density, τ is the quantity commonly used in the construction ofmeta-GGA functionals.

The M06-L functional is a meta-GGA recently published by Zhao and Truhlar,50 andis reported to perform better for molecular systems than the highly popular B3LYP func-tional. It originated from the M05 series of functionals by the same group (in turn partlybased on PBE), which was extended to include terms of the VSXC meta-GGA func-tional62 and parametrized as a local functional (hence the L in M06-L). The functionalwas constructed with three paradigms in mind, obeying a number of constraints‡, mod-

‡Among others, M06-L satisfies the HEG limit and its correlation part satisfies the condition for one-

14


eling the XC hole and empirically fitting to parameters.50

The M06-L functional is highly parametrized, containing 42 parameters, 37 of whichwere fitted to minimize the root mean square (rms) deviation from 22 databases includ-ing bond lengths, ionization potentials, electron affinities, barrier heights, non-covalentinteraction energies, etc. In particular, a training function — the sum of rms errorsbetween the functional and experimental results, with added weight to non-covalentinteraction data — was minimized by varying the parameters in the functional expres-sion.

One of the remarkable features of M06-L is its good performance for non-covalent in-teractions, which are typically not well reproduced by local functionals. Calculations bythe functional’s authors show improved performance for quantities such as π-π stack-ing interactions, binding energies in charge-transfer complexes and hydrogen bonds,etc.50 In addition to this feature, good performance is also observed for ionization andatomization energies. That study also reports small errors for the bond lengths of 13main group molecules and 13 metal-ligand bond lengths, with very similar results asthe VSXC functional. This demonstrates the overall good performance of the M06-Lfunctional, as no geometry data were used in the initial parametrization.50

2.3.4. The VMT and VT{84} Families of Exchange Functionals

The recently suggested VMT and VT{84}§ exchange functionals28,29 comprise two some-what differing forms of the exchange enhancement factors FX with two different pa-rameter sets. The functionals are based on the PBE exchange, but modified to satisfysomewhat different constraints. The expressions for the exchange enhancement factorsof VMT and VT{84} are given in Eq. 2.12 and 2.13, respectively. In this context a slightly

different definition of the reduced density gradient is used, so that s′ =1

2(3π2)1/3s.

FVMTX (s′) = 1 +

µs′2e−αs′2

1 + µs′2(2.12)

FVT{mn}X (s′) = FVMT

X (s′) +(

1− e−αs′m/2) (s′−n/2 − 1

)(2.13)

Depending on the parameters µ and α, each of the functionals comes in two variants.The abbreviations VMT and VT{84} refer to those versions, which use the original µ ofthe PBE form, while VMTsol and VT{84}sol use µ = µGE as in the PBEsol functional.Table 2.3 summarizes the parameters used in the various functionals.

This family of functionals was designed to obey a number of fundamental constraints,

electron self-interation correction.§VT{84} is actually a member in a series of functionals VT{mn}, which differ slightly in the choice of

exponents.

15


1.0

1.2

1.4

1.6

1.8

0 10 20 30 40 50

s’

FX

VMTsolVMT

LO bound

(a) VMT and VMTsol

1.0

1.2

1.4

1.6

1.8

0 10 20 30 40 50

s’

FX

VT84sol

VT84

LO bound

(b) VT84 and VT{84}sol

Figure 2.8.: Exchange Enhancement Factor of the VMTsol and VT{84}sol functionals

namely to recover HEG behavior in the small-s and the large-s limits, to recover thegradient expansion and to satisfy the Lieb-Oxford (LO) bound.28 Lieb and Oxford61 gavea constraint for the exchange energy of electrons in Coulomic systems where

EXC [ρ]

ELDAX ρ

≤ λLO (2.14)

with a value of λLO = 2.2733. For the exchange enhancement factor this means that

F VMTX [s′] ≤ λLO

21/3≈ 1.804 (2.15)

which requires the parameter α to be adjusted accordingly, for the given values of µ =

µPBE or µGE . The graphs of FX [s′] are shown in Fig. 2.8, where it is evident that the LObound is reached at exactly one value of s′ in each case.

As these functionals only treat the exchange component, PBE correlation is used toevaluate the correlation part. The implementation of VMT and VMTsol in ParaGAUSSwas verified by comparison with a set of published reference energies:28 The mean ab-solute deviation (MAD) between ParaGAUSS and the original implementation in thecode deMon2k65 was found to be <0.13 pm for bond lengths, <4.1 kJ·mol-1 for total en-ergies and <9.5 kJ·mol-1 for ionization energies. Details of the validation procedure andtabulated results are provided in Appendix B.

Table 2.3.: Parameters for VMT[sol] and VT{84}[sol]

Param. VMT VMTsol VT{84} VT84sol

µ 0.2195164 10/81 0.2195164 10/81

α 0.002762 0.0015532 0.000069 0.000023

16

3Computational Details

All calculations of the present study were carried out with version 3.1.6 of the Para-GAUSS code. ParaGAUSS is a density functional code based on the linear combinationof Gaussian-type orbitals fitting-function method (LCGTO-FF-DF),66–68 The code is par-allelized for all computationally demanding steps and makes efficient use of symmetryin order to reduce computational cost. This combination of features is well-suited for thestudy of large systems with heavy metals, such as the clusters in the present study.36,38,40

The second-order Douglas-Kroll-Hess (DKH) approach was used for the treatmentof scalar relativistic effects.69,70 For the study of palladium, calculations were carriedout both with and without a scalar relativistic treatment, in order to quantify the rela-tivistic contribution to the investigated parameters. In the case of gold, only relativisticcalculations were performed. Calculations were done in spin-polarized fashion for allclusters with n≤55, whereas those beyond M55 were treated spin-restricted, with the ex-ception of some exploratory calculations on Pd79. For Pd the spin is typically quenchedrapidly with increasing cluster size, justifying this approximation. For Au79, the largestgold cluster studied, this results in an artificial closed-shell configuration, where oneunpaired electron is split evenly into spin-up and spin-down parts.

The Kohn-Sham orbitals were represented as linear combinations of Gaussian-typefunctions with flexible radial parts and angular parts in the form of harmonic polyno-mials (from spherical harmonics). For palladium, an all-electron GTO basis set by Huz-inaga71 was contracted in generalized form using atomic eigenvectors (18s,13p,9d)→[7s, 6p, 4d]. The atomic eigenvectors were determined with spin-averaged calculationsusing the VWN LDA functional in Ih symmetry, separately for the relativistic and non-relativistic cases. In order to assess the validity of the contraction coefficients, analogouscontraction coefficents were also calculated with the BP and PBE functionals and com-pared to the VWN-contracted bases in test calculations. The relative differences in totalenergy were found to be less than 10-5%, so that for better comparability the VWN-contracted basis set was chosen for all calculations. For gold, an all-electron basis set72

was contracted in an analogous way (21s, 17p, 11d, 7f)→ [8s, 7p, 5d, 3f].The LCGTO-FF-DF method uses an approximate electron density in order to evaluate

17

Master’s Thesis Computational Details

the classical Coulomb term of the electron-electron interaction.66,73 The method relies onan auxiliary Gaussian-type basis set and requires only three-center integrals instead ofthe computationally expensive four-center integrals. The exponents of s- and r2-typefunctions of the charge fit basis are based on a subset of the exponents of the orbitalbasis, scaled by a factor of two and augmented with sets of p and d polarization expo-nents. The auxiliary basis is not contracted. The exponents of the employed orbital andcharge fit basis sets are listed in Appendix A.

As metallic systems have numerous electronic states of very similar energy, conver-gence of the self-consistent field (SCF) procedure is frequently problematic. Thereforethe present calculations make use of the fractional occupation number (FON) technique,which allows non-integer occupation of orbitals. A Fermi-type broadening functionwith an energy range of 0.25 eV was used to average over low-lying excited states nearthe ground state.74 Thus electronic convergence was made possible for the larger clus-ters.

The metal clusters were optimized in all degrees of freedom, constrained to octa-hedral symmetry, with the built-in optimizer of ParaGAUSS.75 A quasi-Newton (QN)algorithm was employed for the minimization of analytical gradients, instead of the dy-namic Trust Radius Model (TRM) more recently included in the code. When using theFON population of electronic states, slight changes of the energy due to the changingoccupation prevent the TRM-based optimizer from finding the minimum, thus requir-ing the use of the QN algorithm. Each of the investigated functionals was used for thefull optimization of all clusters, starting either at the bulk geometry or at a pre-relaxedgeometry determined with a different functional.

For the convergence of the electronic SCF procedure the maximum density differencebetween cycles was set to 10-7 au and the displacement gradient vectors and step lengthin geometry optimizations were required to converge to 10-5. The numerical integrationgrid was constructed as a superposition of atom-centered grids of Lebedev-type. Thegrid parameters NRAD=70 and NANG=171 (a grid of 70 radial shells with 171 angularpoints) were set in ParaGAUSS, resulting in a symmetry-adapted grid size of about24000 points for an M55 cluster.

Some aspects of the computational approach were modified from the descriptionabove when exploring the magnetism of small Pd clusters as presented in Section 4.2.In that study, the stabilization of a series of spin states (number of unpaired electrons)was investigated for the clusters Pdn (n=19,38). Single-point calculations were carriedout at various optimized geometries and fixed multiplicities (FIXED SPIN DIFF op-tion in ParaGAUSS) and their total energy determined relative to the non-spinpolarized(closed-shell) case. For these calculations no broadening of electronic states was used, inorder to eliminate any influence of this procedure on the results. In rare cases, primarilyfor the closed-shell occupations, convergence was very difficult to obtain. In such situa-

18

Master’s Thesis Computational Details

tions a small magnitude of Fermi-broadening was allowed during the initial SCF cyclesand gradually reduced to zero or near-zero values in order to facilitate convergence.

All calculations were carried out either on the group’s 28 Quadcore 2.25 GHz Ne-halem cluster, located in-house, or the HLRB-II Itanium2 platform of the Leibniz-Rech-enzentrum. Calculations on the former cluster typically utilized 1–16 processor coreswith 4000 MB of memory per core, while in the latter case 24–64 processors with 3600 MBof memory were typically used.

In addition to the quantum mechanical calculations with ParaGAUSS, a small num-ber of auxiliary scripts and software was used for data collection and interpretation ofresults. In particular, the determination of average bond lengths and coordination num-bers was automated in a perl script, which was used to read the optimized structuresas coordinate files and then determined structural parameters such as M-M distances.This way a number of quantities such as the average bond length, the distribution ofbond lengths and the displacement of surface atoms were determined in an automaticand efficient way.

19

4Results and Discussion

Having established the background and methodological aspects, this chapter presentsthe results of the present study with special emphasis on two aspects. The first sectiondetails the comparative evaluation of a series of exchange-correlation functionals withrespect to the scalability of cluster properties such as interatomic distances and cohesiveenergies of Pd and Au clusters as well as the ionization potential and electron affinityof the Pd clusters. That section is followed by a related study of magnetic properties insmall Pd clusters, comparing the different behavior of various functionals.

4.1. Scalable Cluster Properties: Comparing Modern

Exchange-Correlation Functionals

4.1.1. Scaling of Bond Lengths

The first part of the study examined the average bond length in the cluster, dav as afunction of cluster size, using this quantity as a model for the scalability of structuralfeatures of the cluster. The average bond length is defined as the arithmetic mean of allnearest-neighbor M-M distances in an Mn cluster according to

dav =

n∑i

CN i∑j

dMi−Mj

n∑i

CN i

(4.1)

where CN i is the number of nearest neighbors of the i-th atom and dMi−Mjis the distance

between the i-th and j-th atom. The linear scaling relation is given as a fit of dav as afunction of the average coordination number CN av:

dav = k · CN av + b (4.2)

20

Master’s Thesis Scalable Properties

264

268

272

276

280

6 8 10 12

dav,

pm

CNav

dbulkexp = 275 pm

VWNBP

PW91PBE

M06LPBEsol

VMTVT84

VT84solVMTsol

Figure 4.1.: Average bond lengths (dav, pm) in Pd clusters vs. average coordination num-ber (CN av), from scalar relativistic calculations with 10 functionals

The bulk interatomic distance can be estimated by extrapolating to CN av = 12, whichmay be compared to the experimental bulk values (dexpav (Pd) = 275 pm,37 dexpav (Au) =288 pm76) in order to assess the quality of the results.

Average Bond Lengths in Pd Clusters

Tables 4.1 and 4.2 list the parameters of the linear fit of dav vs. CN av determined for allinvestigated functionals with and without relativistic treatment, and Fig. 4.1 shows aplot of the scaling behavior for the relativistic case.

Table 4.1.: Linear fit of average bond length dav (pm) vs. CN av for Pdn clusters, from non-relativisticcalculations.

VWN BP PW91 PBE PBEsol M06-L VMT VMTsol VT{84} VT{84}sol

r2 a 0.982 0.963 0.962 0.956 0.968 0.925 0.946 0.945 0.945 0.945kb 1.302 1.231 1.221 1.221 1.210 0.945 1.136 1.161 1.132 1.160bc 260.8 269.9 269.9 270.6 264.9 274.1 272.2 265.1 272.3 265.1dbulkav

d 276.4 284.7 284.6 285.2 279.5 285.4 285.9 279.0 285.9 279.0

a Correlation coefficient of linear fit; b Slope; c Axis intercept;d Extrapolated bulk Pd-Pd distance in pm (dexpav = 275 pm37);

21


In general the calculated results show a very good linear fit of the scaling relation,with r2 ? 0.95 for the non-relativistic calculations and r2 ? 0.99 in the relativistic case.In both instances the M06-L functional exhibits a marginally worse fit than the otherfunctionals with r2 = 0.93 and r2 = 0.98 for the non-relativistic and scalar relativisticcalculations, respectively. The slopes of the fitted lines k fall in the rather narrow rangeof 1.15. . . 1.30 (non-relativistic) and 1.43. . . 1.55 (scalar relativistic), from which the M06-L functional deviates in the non-relativistic (k ≈ 0.95) and scalar relativistic (k ≈ 1.38)cases; also the VT{84}sol functional deviates in the scalar relativistic case (k ≈ 1.28). Ascan be seen also in Fig. 4.1, the fitted lines are almost parallel with the exception of M06-L and VT{84}, which have a smaller slope, hence cross a number of others. The axisintercepts b span the range of 260. . . 275 pm for the non-relativistic fits and 254. . . 265 pmfor the relativistic fits. In both cases the VWN functional yields the lowest value of band M06-L the highest, with the other functionals clustering in between those extrema.

The extrapolated bulk bond lengths dbulkav determined by the non-relativistic calcula-tions span a range of 10 pm from 276 pm to 286 pm, while the relativistic values rangefrom 272 pm to 281 pm. The PBEsol functional shows the best agreement with the ex-perimental value, followed by VMTsol and VT{84}sol, which all lie within 1 pm of theexperiment with relativistic treatment. The geometries obtained with the M06-L meta-GGA are among the most expanded, with dbulkav some 6 pm longer than the experimentand almost 10 pm longer than the VWN value. Furthermore, optimization of the Pd147

cluster did not succeed in the relativistic case, as the cluster expanded to average bondlengths beyond 300 pm, and convergence of the optimized structure was not possible.

The average bond lengths change most between the smallest clusters of the set, typ-ically by 1 to 1.3 pm between Pd13 and Pd19, while the change between Pd38 and Pd55

amounts to only 0.5 pm. Because the Pd55 cluster is a perfect cuboctahedron, containingonly 12 corner atoms, and its average coordination number (7.86) is very similar to thatof Pd38 (7.58), the change in average bond length accordingly is small.

With respect to average bond lengths, one observes a contraction of ∼5 pm with thescalar relativistic treatment, compared to the non-relativistic results. Without relativis-tics, even the shortest dbulkav as computed with the VWN functional is 1.4 pm longer thanthe experimental value, while all other GGA functionals overestimate dbulkav by 4. . . 10 pm.It is only with scalar relativistic treatment that good agreement with the experiment canbe obtained.

22


Table 4.2.: Average bond lengths dav (pm) and linear fit parameters of dav vs. CN av for Pdn clusters fromscalar relativistic calculations.


Pd13 262.5 270.2 270.1 270.4 265.6 272.0 271.5 265.3 271.5 266.0Pd19 264.2 271.6 271.4 271.6 267.0 273.4 272.6 266.6 272.7 266.7Pd38 265.8 273.2 273.1 273.3 268.6 275.2 274.3 268.2 274.4 268.3Pd55 266.2 273.7 273.6 273.8 269.1 275.0 274.8 268.6 274.9 268.7Pd79 267.4 274.9 274.8 275.0 270.3 276.3 275.9 269.8 276.0 269.8Pd147 267.9 275.4 275.3 275.4 270.7 –a 276.4 270.2 276.5 270.2

r2 b 0.991 0.996 0.996 0.997 0.996 0.982 0.997 0.996 0.997 0.989kc 1.548 1.525 1.512 1.482 1.487 1.379 1.452 1.426 1.449 1.282bd 254.1 261.8 261.8 262.2 257.5 264.5 263.4 257.5 263.5 258.7dbulkav

e 272.7 280.1 279.9 280.0 275.3 281.1 280.9 274.6 281.0 274.1

a No optimized structure available; b Correlation coefficient of linear fit; c Slope;d Axis intercept; e Extrapolated bulk Pd-Pd distance in pm (dexpav = 275 pm37);

The 10 exchange-correlation functionals studied fall into four groups according to thepredicted relativistic dbulkav :

1. VWN gives the shortest dav of the functionals studied, deviating 2.3 pm from theexperimental value.

2. The three functionals PBEsol, VMTsol and VT{84}sol constitute the group of second-shortest bond lengths, which agree best with the experimental value. The groupspans the range of 274.1. . . 275.3 pm, bracketing dexpav .

3. The GGA functionals BP, PBE, and PW91 show a spread of only 0.2 pm for theextrapolated bulk value, from 279.9 to 280.1 pm. These functionals overestimatethe reference value by 5 pm.

4. VMT, VT{84}, and M06-L extrapolate to the largest estimate of dav between 280.9and 281.1 pm, about 6 pm above the reference, 1 pm longer than the next-lowergroup.

The presented results agree with previous studies. Using the BP functional and theTURBOMOLE code, a study by Ahlrichs et al.21 found an extrapolated dbulkav of 280.9 pmfor a series of fcc Pd clusters. This value agrees somewhat better with the experimentalvalue than a previous study,36 which found dbulkav of 290.5 pm with BP and 281.2 pm withVWN, but did not consider relativistic effects and only optimized the breathing modeof the clusters. A more recent study of similar Pd clusters found a dbulkav of 275.2 pmwith the VWN functional.38 While the results cannot be precisely compared for vari-ous methodological differences (basis sets, auxiliary basis sets, codes and/or different

23


clusters used), the agreement is generally good, with a deviation of 0.8 pm in the mostfavorable case.

The tendency of LDA functionals to underestimate bond lengths is a common ob-servation;22,23 it is also reproduced in the present study, as are the typically long bondlengths of the BP-type GGAs (Group 3, above). The “best” cluster geometries are ob-tained with those functionals explicitly designed for extended solids (sol-type function-als, Group 2). Thus it seems that, with respect to geometries, the studied clusters indeedresemble solids. Due to the strong expansion of bonds and the problematic optimizationof the largest cluster, the M06-L functional represents no improvement on the regularsecond-rung GGA methods for equilibrium geometries.

The possibility of enabling and disabling the relativistic treatment allows one to quan-tify the magnitude of relativistic effects. As the relativistic contraction of bond lengthsamounts to about 5 pm for dbulkav , accounting for (scalar) relativistics is clearly of great im-portance for accurate calculations. Relativistic effects due to heavy nuclei cause a con-traction of the s- and p- orbitals, concomitant with an increase of d-orbital energies.77

Thus the contracted bond lengths can be rationalized with the contraction of atomicorbitals, which also accounts for the increase in bond energies. While palladium withatomic number 46 is not an extremely heavy element, the relativistic effects are stillrather pronounced, contributing about 2.2% to the overall bond length, and reducingthe error in extrapolated bond length by over 50%.

Four very recent functionals (VMT, VMTsol, VT{84}, VT{84}sol) are included in thepresent study and fall into two distinct categories of predicted dbulkav .∗ The sol-variants ofthe functionals keep some parameters from the PBEsol functional, which may rational-ize the similarity of the results for these functionals. On the other hand, the functionalsVMT and VT{84} are related to PBE, and thus produce expanded bond lengths. TheVMTsol and VT{84}sol functionals reproduce dexpav with good accuracy; they show thesame good agreement as PBEsol for the studied systems with respect to geometries.The other two functionals show no improvement in comparison with the Group 3-GGAfunctionals — in fact the predicted dbulkav is slightly worsened (by ∼1 pm). This observa-tion is to some extent in line with previously published results for small molecules,where virtually no change of bond lengths was found between PBE and VMT, andPBEsol and VMTsol.28

While the improvement of VMTsol on PBEsol is vanishingly small, the comparisonof these functionals still allows for some insight into the effect of EXC on the clusterstructure. Only the exchange parts of VMT and VMTsol are specific, combined with thePBE correlation functional, while PBEsol differs from PBE in both EX and EC (by oneparameter each, see Section 2.3.2). Thus, the exchange contribution of VMT is reflected

∗All of the following values refer to the scalar relativistic calculations.

24


in the difference between VMT and PBE, with a magnitude of about 1 pm. On the otherhand, the difference between PBE and VMTsol amounts to 5.4 pm, which should bechiefly contained within the exchange contribution, as it is the only difference betweenthe two functionals. This result is consistent with the fact that the exchange energy istypically a much larger contribution to EXC than the correlation energy,22 which is alsoreflected in the bond lengths. For the systems at hand the data suggest that the mostsignificant improvements of XC functionals can be made by improving the EX term.

Average Bond Lengths in Au Clusters

In the same spirit, a study of the scaling of average bond lengths was carried out forgold clusters Au13. . . Au79. Table 4.3 lists the calculated average bond lengths and sum-marizes the linear fit and extrapolated bulk average bond lengths determined for therelativistic case. The trends observed for the palladium clusters are partially repro-duced also for gold. With the exception of the M06-L functional, all linear fits are verygood with r2 > 0.99. In contrast to Pd, in this case the clusters exhibit a somewhatgreater variability of the fitted slopes, ranging from 1.68 (M06-L) to 1.96 (BP). The rangeof about 10 pm for the axis intercept is similar to the one found for palladium and is alsothe dominant contribution to the variability among the functionals.

The extrapolated values of dbulkav range range from 286.2 to 294.8 pm, spanning ap-proximately 9 pm. These results essentially confirm those of a previous study38 wheredbulkav = 286.3 pm at the LDA level. The PBEsol, VMTsol, and VT{84}sol functionals givethe closest agreement with the experimental bond length, 288 pm,76 while VWN under-estimates dbulkav by about 2 pm and the remaining GGA functionals overestimate it by upto 6 pm. The grouping of functionals established for Pd is partially valid also for Au.The VWN functional forms a group of its own, while the sol-functionals group togetheraround the experimental value. The somewhat distinct Groups 3 and 4 that were notedfor Pd, merge for the Au case and comprise the remaining GGA functionals and M06-L.

25


Table 4.3.: Linear fit of average bond length dav (pm) vs CN av for Aun clusters (n=13. . . 79), from scalarrelativistic calculations.


Au13 273.7 282.2 281.9 282.0 283.0 276.8 283.2 276.2 283.2 276.2Au19 275.0 283.5 283.1 283.1 283.4 277.8 284.2 277.1 284.3 277.1Au38 277.5 286.0 285.6 285.5 285.9 280.3 286.6 279.5 286.7 279.5Au55 278.3 286.7 286.3 286.3 286.8 281.1 287.3 280.3 287.4 280.3Au79 279.4 288.0 287.3 287.4 287.6 282.1 288.4 281.3 288.5 281.3

r2 a 0.997 0.999 0.996 0.996 0.996 0.969 0.995 0.993 0.995 0.993kb 1.953 1.956 1.896 1.896 1.862 1.681 1.818 1.778 1.815 1.776bc 262.8 271.3 271.3 271.6 266.3 273.3 272.9 266.2 273.0 266.2dbulkav

d 286.2 294.5 294.0 294.3 288.6 293.5 294.8 287.5 294.8 287.5

a Correlation coefficient of linear fit; b Slope; c Axis intercept;d Extrapolated bulk Au-Au distance in pm (dexpav = 288 pm76);

In general, the gold and palladium clusters exhibit very similar trends. The most ac-curate functionals are the same in both cases (PBEsol, VMTsol, VT{84}sol); they predictthe bulk bond length with equally small errors. The novel functionals evaluated in thisstudy show equal performance with respect to geometries as the well-established GGAfunctionals. Very similar results are found for the VMT and VT{84} functionals andVMTsol and VT{84}sol, which can be rationalized with the rather similar expressionsfor the exchange enhancement factor. The VMT and VT{84} families of functionalsdiffer solely by a small additive term in the expression for EX that appears to affectgeometry results in a minor way only.

In addition to the scalability of bond lengths of the clusters the size-dependence ofthe cohesive energies was also examined. The next section presents the results.

4.1.2. Scaling of Cohesive Energies

The cohesive energy of an Mn cluster is defined as the total binding energy per atomaccording to eq. 4.3,

Ecoh =∆Etotn

(4.3)

where ∆Etot is given by

∆Etot = Etot (Mn)− nEtot (M1) . (4.4)

The ground-state energies of the free atoms M1 were determined with parametersequivalent to those of the cluster calculations with the symmetry of the system also re-stricted to Oh and a [Kr]4d10 configuration. Accordingly, the cohesive energies were

26


determined for the model set of Pdn (n=13. . . 147) and Aun (n=13. . . 79) clusters for theentire set of 10 investigated functionals. From the individual results a regression analy-sis was performed to fit the cohesive energies as a linear function of the average coordi-nation number CN av (Eq. 4.5).

Ecoh = k · CN av + b (4.5)

Cohesive Energy of Pd clusters

Tables 4.4 and 4.5 summarize the results of the linear fit for all investigated functionalsfor palladium in the relativistic and non-relativistic cases.

Table 4.4.: Linear fit of cohesive energies vs CN av for Pdn clusters, from non-relativistic calculations.


r2 a 0.994 0.991 0.991 0.991 0.993 0.994 0.991 0.994 0.991 0.990kb -29.1 -19.7 -20.7 -20.5 -25.3 -26.3 -19.0 -25.1 -18.9 -25.1bc -61.4 -38.9 -41.2 -38.9 -46.1 -13.0 -34.3 -40.7 -33.9 -40.5Ebulk

cohd -410.1 -275.7 -289.0 -284.6 -349.3 -328.9 -262.7 -342.0 -261.0 -341.3

a Correlation coefficient of linear fit; b Slope; c Axis intercept;d Extrapolated bulk cohesive energy in kJ·mol-1 (Eexp

coh = -376 kJ·mol-1 78);

Table 4.5.: Cohesive Energies (Ecoh) and linear fit of Ecoh vs. CN av for Pdn clusters, from scalar relativis-tic calculations.


Pd13 -293.0 -210.7 -219.1 -215.5 -254.5 -214.2 -202.0 -249.0 -200.9 -248.5Pd19 -330.8 -240.6 -249.3 -245.4 -288.8 -247.9 -230.6 -283.3 -229.4 -282.8Pd38 -366.2 -263.6 -273.6 -269.1 -319.6 -279.0 -253.0 -314.5 -251.7 -313.9Pd55 -378.8 -273.4 -283.2 -279.9 -331.2 -291.7 -262.7 -326.2 -261.4 -325.5Pd79 -394.8 -283.9 -294.7 -290.7 -344.9 -304.9 -273.4 -340.6 -272.0 -340.0Pd147 -412.8 -297.3 -308.7 -304.7 -361.8 –a -286.8 -357.3 -285.3 -356.6

r2 b 0.991 0.984 0.986 0.986 0.995 0.985 0.986 0.991 0.996 0.991kc -33.5 -23.8 -24.7 -24.6 -29.8 -31.6 -23.4 -30.1 -23.2 -30.1bd -113.1 -83.8 -87.2 -84.0 -94.3 -52.0 -77.2 -86.9 -76.7 -86.7Ebulk

cohe -514.6 -369.7 -383.4 -379.2 -452.6 -413.3 -357.4 -448.6 -355.6 -447.7

a No optimized structure available; b Correlation coefficient of linear fit; c Slope;d Axis intercept; e Extrapolated bulk cohesive energy in kJ·mol-1 (Eexp

coh = -376 kJ·mol-1 78);

The plotted data show a linear increase (more negative values) of the cohesive energywith increasing average coordination number. As smaller clusters contain more under-coordinated atoms, fewer bonds can be formed, leading to an overall decrease of bondenergies, and thus a lower Ecoh. Conversely, the higher the average coordination in the

27


-500

-400

-300

-200

6 8 10 12

Ecoh,

kJ m

ol-1

CNav

Ecohbulk

= -376 kJ mol-1

VWNBP

PW91PBE

M06LPBEsol

VMTVT84

VT84solVMTsol

Figure 4.2.: Cohesive Energy (Ecoh, kJ·mol-1) in Pd clusters vs. average coordination num-ber (CN av), from scalar relativistic calculations with various functionals

cluster, the greater (more negative) the cohesive energy. The most pronounced change inEcoh is found between the Pd13 and Pd19 clusters, and the difference between Ecoh of twoneighboring clusters progressively decreases with increasing cluster size. This behaviorcan be rationalized with the decreasing fraction of undercoordinated (surface-)atoms inlarger clusters, causing the absolute difference in Ecoh between clusters to shrink.

The cohesive energy of Pd19 is somewhat higher (more negative) than would corre-spond to the expected linear trend, making the data point for Pd19 the one with thelargest deviation from the linear fit. One reason for this systematic deviation may be thechoice of the cluster model. As Pd19 is the only full octahedron in the series of clusters,it has a set of corner atoms of CN 4, which the other clusters lack. These atoms seem tocontribute more to the binding energy than would be expected by their contribution tothe average coordination number.

A very good linear fit with r2 ≈ 0.99 is observed with all functionals, for both therelativistic and non-relativistic calculations, although the latter values are marginallysmaller (r2>0.98). The slope of the fitted line, k, ranges from -18.9 kJ·mol-1 to -29.1 kJ·mol-1

(non-relativistic) and from -23.2 kJ·mol-1 to -33.5 kJ·mol-1 (relativistic), amounting to arelative spread of about 35% and 29%, respectively. The axis intercepts, b, have valuesin the range -61.4. . . -13.0 kJ·mol-1 (non-relativistic) and -113.1 kJ·mol-1. . . -42.4 kJ·mol-1.With respect to b, the M06-L functional lies somewhat outside the range of the other

28


functionals, showing the smallest |b|, which differs from the nearest other functional byup to a factor of 3.

The extrapolated bulk cohesive energies Ebulkcoh are found in the range of -410 kJ·mol-1

. . . -261 kJ·mol-1 in the non-relativistic case and -514 kJ·mol-1. . . -355 kJ·mol-1 with rela-tivistic treatment. With respect to the experimental value -376 kJ·mol-1,78 the calculateddata have range of deviations from 9% stronger (more negative) to 30% weaker Ebulk

coh

(non-relativistic) and 37% stronger to 6% weaker Ebulkcoh (relativistic). The PBE, BP and

PW91 functionals yield the closest agreement with the experiment; these functionalsreproduce the reference value to within 7 kJ·mol-1, PBE being the best with an error of3 kJ·mol-1.

With respect to cohesive energies, the exchange-correlation functionals fall into threegroups:

1. The VWN functional shows the highest cohesive energies, separated from theother functionals by >60 kJ·mol-1, 138 kJ·mol-1 greater (by absolute value) than theexperimental value.

2. PBEsol, VMTsol, and VT{84}sol give smaller values of Ebulkcoh than VWN, which are

still up to 76 kJ·mol-1 too large (by absolute value).

3. The GGA functionals BP, PW91, PBE, VMT, VT{84} as well as the M06-L func-tional span a range of 26 kJ·mol-1 around the experimental value, giving the bestagreement.

The relativistic contribution to Ebulkcoh amounts to an increase of the cohesive energies

by 90 to 100 kJ·mol-1, typically >33%. It is only upon inclusion of scalar relativisticeffects that the good agreement of the Group 3-functionals is obtained, reducing therelative errors one order of magnitude in the best case. Owing to the shorter relativisticbond lengths, the binding in the clusters is also strengthened by relativistic effects, thusincreasing the cohesive energy.

For a similar cluster set, previous studies, also using GGA functionals, have foundvalues of -349 kJ·mol-1,38 -346 kJ·mol-1,21 and -336 kJ·mol-1 79 for Ebulk

coh . All of the reportedvalues underestimate the bulk cohesive energy. The presented results show significantlysmaller errors for the best functionals, which can partially be attributed to the greateraccuracy of all-electron calculations compared to pseudopotential methods, as well asto the full optimization of the clusters in all degrees of freedom. Furthermore, inclusionof the Pd19 cluster with its untypically high cohesive energy may be a corrective influ-ence on the extrapolated values, as it somewhat pushes the fitted line towards highercohesive (absolute) energies.

29


Cohesive Energy of Au clusters

The cohesive energy for the set of clusters Au13. . . Au79 was determined for the scalarrelativistic case and fitted to linear functions in the same way as for the Pd clusters.Table 4.6 lists the parameters of the fitted function and the extrapolated bulk cohesiveenergy Ebulk

coh .

Table 4.6.: Cohesive energies Ecoh and linear fit of Ecoh vs. CN av for Aun clusters, from scalar relativisticcalculations.


Au13 -270.3 -184.8 -195.2 -192.6 -230.8 -214.7 -179.3 -225.8 -178.2 -225.3Au19 -300.7 -206.3 -217.6 -214.7 -257.7 -240.3 -200.0 -252.9 -198.8 -252.3Au38 -331.8 -226.3 -239.1 -236.3 -285.0 -271.3 -219.9 -280.8 -218.6 -280.1Au55 -343.7 -235.9 -249.0 -246.1 -296.2 -283.9 -229.3 -292.1 -227.9 -291.4Au79 -357.8 -244.6 -258.4 -255.4 -308.3 -297.2 -237.9 -304.6 -236.4 -303.9

r2 a 0.991 0.987 0.988 0.989 0.991 0.995 0.988 0.991 0.988 0.993kb -29.1 -19.9 -21.0 -20.9 -25.8 -27.8 -19.5 -26.2 -19.4 -26.2bc -112.6 -77.4 -81.5 -79.3 -90.7 -62.6 -73.8 -83.5 -73.8 -83.2Ebulk

cohd -461.8 -315.8 -333.6 -330.3 -400.6 -395.9 -307.6 -398.2 -305.8 -397.4

a Correlation coefficient of linear fit; b Slope; c Axis intercept;d Extrapolated bulk cohesive energy in kJ·mol-1 (Eexp

coh = -366.6 kJ·mol-1 80);

The regression analysis yields good linear fits for the relativistic cohesive energies ofgold clusters as a function of the average coordination number (r2 ≈ 0.99). The slope ofthe fitted function k falls in the range -29.1 kJ·mol-1. . . -19.4 kJ·mol-1 as determined withthe 10 exchange-correlation functionals. The axis intercepts b lie between -112.6 kJ·mol-1

and -62.2 kJ·mol-1, thus spanning a range of 50 kJ·mol-1.Ebulkcoh is predicted in the range -305 kJ·mol-1. . . -462 kJ·mol-1, the best agreement with

the experimental value is found by M06-L, followed by VMTsol, VT{84}sol, and PW91.The first three functionals predict |Ebulk

coh | up to 31 kJ·mol-1 too high, whereas the lattergives a 33 kJ·mol-1 too low extrapolated absolute cohesive energy. These results agreewith those of a previous study on Aun clusters which found Ebulk

coh = -349 kJ·mol-1 38 us-ing a GGA functional — a somewhat better agreement with the experiment than wasfound in this work. As that investigation used a somewhat different set of clusters anddifferent basis sets, the results are not fully comparable in a quantitative way, but theobserved trends and accuracies agree rather well.

Compared to palladium, for gold the agreement with experiment is substantiallyworse; the most accurate functional deviates by 8% (an order of magnitude worse thanthe best relative deviation for Pd). The energies are generally smaller by absolute valuethan observed for Pd, i.e. the curves are shifted upward relative to the bulk value. More-over, the functionals giving the best results are different from the best functionals for Pd.

30


The closest agreement is found with the M06-L functional, which did not exhibit a par-ticularly good performance for palladium; the GGA functionals which were found tobe rather good for Pd (BP, PW91, PBE) find cohesive energies that are more than 30 to50 kJ·mol-1 too low, while those that predicted too strong binding for Pd are now closerto the experimental value (in terms of absolute value), but still overestimate the bindingenergy. Thus it is likely that the marginally better prediction of Ebulk

coh by M06-L, VMTsol,and VT{84}sol is mainly a result of error cancellation; the high (absolute) cohesive ener-gies found previously with these functionals are partially compensated by the generaldecrease of (absolute) cohesive energies observed for gold.

The grouping of functionals with respect to Ebulkcoh found for palladium remains the

same for gold, and the range of values within each group also remains similar. Forgold, however, none of the groups contains the experimental value. Rather, Groups 2and 3 lie around 30 kJ·mol-1 too low and too high, respectively. As found for Pd, the newfunctionals of the VMT- and VT{84} families do not predict significantly better cohesiveenergies, and perform either very similar to other functionals within their group orslightly worse.

Having discussed the results for the scalable behavior of cohesive energies, the nextsection reports on the investigations of ionization potentials and electron affinities ofthe various clusters.

4.1.3. Ionization Potential and Electron Affinity

The ionization potentials (IP ) of the set of relativistic Pdn clusters was determined usingthe ∆SCF method according to

IP = E(Pd+n )− E(Pdn) (4.6)

where E(Pd+n ) is the single-point energy of the ionized Pdn cluster at the previously

optimized geometry, and E(Pdn) is the total energy of the optimized Pdn cluster. Theelectron affinity EA is calculated in an analogous way,

EA = E(Pdn)− E(Pd−n ) (4.7)

using the single-point energy of the cluster with an added electron E(Pd−n ).It should be noted that in the present context IP refers to the vertical ionization potential,

where the geometry of the neutral and ionized species is the same. In contrast to this,the term adiabatic ionization potential is used for the situation where the cation structureis relaxed independently.81 Both of these values are experimentally available; in order toextrapolate from IP to the work function, however, one needs to determine the verticalIP . In analogy to this concept, the energy of the anionic cluster is also determined at the

31


geometry of the neutral cluster.As demonstrated in previous studies,79 it is possible to fit the ionization potential to

linear functions in n−1/3, a quantity proportional to the cluster radius†. The calculatedIP and EA for the clusters are fitted to the following expressions:

IP = k · n−1/3 + ΦIP (4.8)

EA = k · n−1/3 + ΦEA (4.9)

where n is the number of atoms in the cluster and ΦIP and ΦEA denote estimates of thework function (Φ) of the bulk metal. Due to the overlapping valence and conductionbands in metallic systems, adding or removing one electron requires the same amountof energy, so that in the ideal case both estimates are equal.

Scaling of the Ionization Potential

Table 4.7 shows the ionization potentials for the set of clusters calculated using the 10exchange-correlation functionals. The values were fitted as linear functions of n−1/3 andthe fit parameters are given in the lower half of the table. Figure 4.3 shows plots of IPvs. n−1/3 and the lines fitted to the data.

Table 4.7.: Ionization potential (IP , kJ·mol-1) for Pdn (n=13. . . 147) with 10 exchange-correlation function-als, from scalar relativistic calculations and parameters of the linear fits.


Pd13 648.8 623.8 617.4 610.5 617.2 581.3 605.4 601.0 604.9 600.8Pd19 616.6 605.5 598.2 593.0 595.4 570.9 589.3 581.2 588.9 581.0Pd38 633.0 602.3 596.6 588.2 601.3 548.4 582.4 585.8 581.7 585.6Pd55 593.4 572.7 565.9 559.5 567.2 515.4 553.1 551.5 552.5 551.3Pd79 596.3 574.2 567.2 560.7 568.7 –a 554.6 553.4 554.0 553.1Pd147 573.3 549.2 542.1 535.9 544.8 –b 529.9 529.6 529.4 529.3

r2 c 0.776 0.909 0.901 0.918 0.850 0.907 0.926 0.854 0.927 0.854kd 277.1 293.5 294.9 295.2 277.8 377.3 303.2 277.3 304.0 277.5ΦIP

e 527.9 500.8 493.6 486.9 499.9 425.7 477.9 484.7 478.2 484.3∆exp.

f 24.9 -2.3 -9.4 -16.3 -3.1 -77.3 -25.1 -18.3 -24.8 -18.7

a Problems with electronic convergence; b No optimized geometry available;c Correlation coefficient of linear fit; d Slope;e Axis intercept = extrapolated work function, kJ·mol-1 (Φexp = 503 kJ·mol-1 80);g ∆exp. = ΦIP − Φexp

†Previous studies79 also use an average cluster radius R to characterize the scaling of IP and EA. How-ever, as this quantity would depend on the optimized structure given with each functional, for compa-rability it is preferable to use n−1/3. Comparison of the two fitting methods gave very similar resultsin both cases.

32


450

500

550

600

650

0 0.1 0.2 0.3 0.4 0.5

IP,

kJ m

ol-1

n-1/3

Φexp = 503 kJ mol-1

VWNBPPW91PBEPBEsolM06LVMTVMTsolVT84VT84sol

Figure 4.3.: Ionization potential (IP , kJ·mol-1) in Pdn clusters vs. n−1/3, from scalar rela-tivistic calculations with 10 XC functionals. Axis intercept corresponds to workfunction ΦIP . Φexp = 503 kJ·mol-1. Points are connected to guide the eye.

The IPs show a monotonous decrease with increasing cluster size (smaller n−1/3),with the exception of the cluster Pd38, for which some functionals (most notably VWN,to a lesser extent the three sol-functionals) predict an increase in IP from Pd19. Com-pared to the fit of Ecoh and dav, the IPs show significantly greater scatter, making thelinear fit therefore much less accurate; r2 ranges from 0.85 to 0.93, with VWN as an out-lier (r2 = 0.78). The slopes k of the fitted lines show some variability, ranging from 257 to304 kJ·mol-1, with one outlier for the M06-L functional, for which two data points couldnot be determined due to insufficient convergence.

The axis intercept of the fitted function corresponds to the IP at infinite cluster size,i.e. an estimated work function of the metal, ΦIP . The extrapolated values range from478 to 500 kJ·mol-1 with one outlier, M06-L (ΦIP = 425.7 kJ·mol-1). The experimentalvalue of Φexp = 503 kJ·mol-1 80 is reproduced to within 25 kJ·mol-1 for most functionals.

As for Ebulkcoh and dbulkav , the extrapolated values of the work function ΦIP cluster in

distinct groups, albeit with slight differences compared to the groupings previouslyobtained. PBE and PBEsol change places and the deviation of the M06-L functional issignificantly more pronounced. However, due to the missing data points in the fit of theM06-L data, ΦIP (M06-L) is considered somewhat unreliable. With regard to IPs, thefunctionals can be categorized in the following four groups:

33


1. VWN is the only functional that significantly overestimates ΦIP .

2. BP, PW91, and PBEsol predict the value of Φexp to within 4 kJ·mol-1.

3. PBE, VMT, VMTsol, VT{84}, and VT{84}sol underestimate Φexp by 16. . . 25 kJ·mol-1.

4. M06-L finds Φexp 77 kJ·mol-1 too small.

Group 2 contains the functionals which yield the most accurate extrapolated workfunctions: BP, PW91, and PBEsol. As noted in the previous section, the BP and PW91functionals are also the most accurate in determining the extrapolated cohesive ener-gies. This agrees with the common trend of GGA functionals being superior for com-puting energies (of metal clusters), which is also substantiated by the other GGA func-tionals (Group 2). While being less accurate than Group 2 the functionals still give goodvalues for ΦIP . In contrast to the results for cohesive energies, PBEsol in this case ex-hibits a better performance than PBE, with the error a factor of 4 lower. This observationis somewhat at odds both with the less accurate energies of PBEsol found above, andwith the behavior of VMTsol and VT{84}sol, which, although closely related to PBEsol,show a much larger error.

The four novel functionals of the VMT- and VT{84} families all predict ΦIP in therange of 477. . . 485 kJ·mol-1: the sol-variants of the functionals occupy the lower limit ofthis range, giving virtually identical values of ∼484 kJ·mol-1, while VMT and VT{84}understimate Φexp more strongly, both yielding ΦIP ≈ 478 kJ·mol-1. As seen previously,the difference between the corresponding variants of the two families is minor, which isnot unexpected, in view of the very similar analytical forms of the exchange functionals;due to the fact that the exchange enhancement factors F VMT

X and F V T84X differ primarily

in the large-gradient limit (i.e. at fast-varying densities), only a small difference is likelyto be observed in systems like transition metal clusters, where the densities are ratherhomogeneous. In contrast to the findings for Ecoh, however, the sol-functionals givemore accurate estimates of the work functions (25% smaller error), which are almost asgood as those from the PBE GGA. Broadly speaking, the VMT- and VT{84} functionalsdo not show an improved performance for the computation of IP and ΦIP , and give asomewhat larger error than other GGA functionals. They are, however, equally accurateas the other functionals in reproducing the linear scaling and qualitatively predict thenoted trends.

The results for the ionization potentials agree with those of previous studies that in-vestigated the scalability of this quantity.79 Using the BP functional, non-relativistic cal-culations, and Pd clusters optimized in the breathing mode, Vent79 finds e.g. IP(Pd147)

= 532 kJ·mol-1 and IP(Pd55) = 550 kJ·mol-1, in good agreement with the GGA results re-ported here. For the smaller clusters Pd13 and Pd19 the values differ more strongly andthe present results do not show the same trend of IP(Pd19) > IP(Pd13). This observation

34


in the previous study may be related to the neglect of relativistic contributions and someother methodological differences. Nevertheless, the extrapolated ΦIP of 482 kJ·mol-1 isin a similar range as the results presented here, which are typically in better agreementwith the experimental work function, while showing the same tendency to underesti-mate Φ.

In addition to extrapolating the linear fit of the ionization potentials to the bulk value,it is possible to approximate the atomic ionization potential IP(Pd) by evaluating thefitted function at n = 1. Using this approach an extrapolated IP(Pd) of 805.0 kJ·mol-1 isobtained with the VWN functional and 794.3 kJ·mol-1 using BP values; the other func-tionals yield values between 760 and 785 kJ·mol-1. These results compare very favorablyto the experimental atomic ionization potential IP exp(Pd) = 804.4 kJ·mol-1,82 with errors<10 kJ·mol-1in the best cases. For comparison, the atomic ionization potential as com-puted with the BP functional is 867.2 kJ·mol-1, thus carrying an error >63 kJ·mol-1. Theextrapolated IP benefits from error cancellation due to the larger number of data pointsin the fit, which is not available for single atomic calculations. As the presented extrap-olation procedure does not require the determination of atomic ground state energies(and thus avoids the computational difficulties associated with them), this constitutes anoteworthy improvement in the calculation of atomic IP values.

Scaling of the Electron Affinity

The electron affinities calculated with the set of exchange-correlation functionals aregiven in Table 4.8, along with the parameters of the fitted lines. Figure 4.4 shows a plotof EA vs. n−1/3 and the linear functions.

Compared to the ionization potential, the trend observed for EA is reversed as theelectron affinity increases with increasing cluster size; the cluster Pd38 again lies some-what outside of that trend. All functionals other than VWN predict a slightly higherEA for Pd38 than Pd55, violating the otherwise monotonous increase. In spite of thisoutlier, the linear fits for this dataset are noticeably improved compared to IP , typicallyr2>0.94, with VWN again exhibiting a slightly worse fit.

The values of the extrapolated work function ΦEA determined with all functionalsare lower than the experimental value, ranging from 462 to 499 kJ·mol-1 (ΦEA(M06-L) =438.5 kJ·mol-1). The spread of ΦEA is thus much smaller among the functionals than forΦIP . A consequence of this narrower range of extrapolated values is the partial disap-pearance of the grouping of functionals. VWN and M06-L remain the upper and lowerbounds to ΦEA, bracketing the remaining 8 functionals in a 25 kJ·mol-1 range, withinwhich the VMT- and VT{84}-families cluster at the lower limit, and BP and PW91 at theupper. The remainder, PBE and PBEsol give intermediate results, between Groups 2 and3. The best agreement with the experimental work function is obtained with the VWN

35


250

300

350

400

450

500

0 0.1 0.2 0.3 0.4 0.5

EA

, kJ m

ol-1

n-1/3

Φexp = 503 kJ mol-1

VWNBPPW91PBEPBEsolM06LVMTVMTsolVT84VT84sol

Figure 4.4.: Electron Affinity (EA, kJ·mol-1) in Pdn clusters vs. n−1/3, scalar relativistic cal-culations with 10 XC functionals. y-intercept corresponds to work functionΦEA. Φexp = 503 kJ·mol-1. Points are connected to guide the eye.

functional, followed by BP and PW91, which already deviate by ∼16 kJ·mol-1. M06-L isagain an outlier and lacks two data points, yielding ΦEA(M06-L) the least accurate andalso least well-defined value.

The similarity of the values ΦEA predicted by the VMT- and VT{84}-functionals ismore pronounced than for ΦIP discussed above. All four functionals predict ΦEA of462. . . 465 kJ·mol-1, underestimating the experimental value by 37. . . 40 kJ·mol-1. In thiscase the differences between the two functionals are too small to discern noticeable be-tween the two variants of the functionals. These functionals do not show an advantageover other GGA functionals for the computation of EA or ΦEA, as already noted for theionization potential. The typical errors are found to be larger by a factor of two thanthose found with, e.g., BP, while the accuracy of the fit remains the same.

36


Table 4.8.: Electron Affinity (EA, kJ·mol-1) for Pdn (n=13. . . 147) with 10 exchange-correlation functionals(scalar relativistic) and linear fit parameters (Φexp = 503 kJ·mol-1 80).


Pd13 311.9 281.3 275.6 269.6 278.2 226.7 263.4 261.6 262.8 261.3Pd19 298.0 293.0 285.2 280.2 278.4 254.6 277.1 263.1 276.7 262.9Pd38 364.1 355.6 349.2 343.2 345.8 303.0 335.9 330.3 335.3 330.1Pd55 366.0 352.7 345.4 339.4 342.1 300.4 334.3 326.3 333.8 326.0Pd79 394.0 377.1 369.8 363.7 368.4 – a 358.5 353.0 358.0 352.7Pd147 409.8 388.5 381.3 375.3 382.5 – b 369.8 369.8 369.2 366.9

r2 c 0.909 0.959 0.954 0.956 0.940 0.951 0.966 0.946 0.966 0.943kd -475.9 -488.4 -484.4 -482.2 -487.4 -491.8 -480.7 -498.3 -480.4 -491.1ΦEA

e 498.7 486.5 478.4 471.9 477.4 438.5 466.0 465.4 465.4 462.6∆exp.

f -4.3 -16.5 -24.6 -31.1 -25.6 -64.6 -37.0 -37.6 -37.6 -40.4

a Problems with electronic convergence; b No optimized geometry available;c Correlation coefficient of linear fit; d Slope;e Axis intercept = extrapolated work function, kJ·mol 1; f ∆exp. = ΦEA− Φexp

Previous work by Vent79 finds EA-values of 375 kJ·mol-1 for Pd147 and 334 kJ·mol-1 forPd55, which agree well with the range of GGA results computed in the present study;the electron affinity values for the smaller clusters agree similarly well. While Ventreports virtually identical values for ΦIP and ΦEA, the present results indicate a generalunderestimation of ΦEA. As this upper bound of 15 kJ·mol-1 is within the margins oferror dictated by the difference in methods and choice of clusters, the agreement is goodalso for the extrapolated values.

Relations between IP and EA

Because the amount of energy required to remove an electron from a (bulk) metal isequal in magnitude to the energy gained when adding one, the difference between IP

and EA should converge to zero as the cluster size approaches infinity. The residualdifference ∆Φ = ΦIP − ΦEA quantifies the error made when extrapolating the workfunctions from IP and EA. The quantity IP − EA can be fitted as a function of n−1/3,giving ∆Φ as the axis intercept

IP − EA = k · n−1/3 + ∆Φ (4.10)

where in the ideal case ∆Φ→ 0.Table 4.9 gives the parameters of the linear fits for all 10 functionals. The linear depen-

dency is fulfilled very well, with r2 typically greater 0.99, and slopes ranging between765 and 784 kJ·mol-1, except for VWN (k = 753 kJ·mol-1) and M06 L (k = 869 kJ·mol-1). For

37


most functionals ∆Φ is between 12 and 22 kJ·mol-1, with M06-L giving the only negativevalue.

Although ∆Φ is small compared to the magnitude of Φ (<3%), the error in IP and EA

does not entirely cancel out. The data show a systematic overestimation of ∆Φ, whichseems chiefly caused by the underestimation of EA noted above.

The DFT treatment of ionic species, especially of anions, has received a lot of atten-tion.23,83 A major problem in calculating the energies of anions is self-repulsion error,which cannot be adequately compensated by the (mathematically local) XC approxima-tions currently used. For small systems this may lead to occupied states with positiveenergies, i.e. an unbound electron. However, as the size of the system increases this is-sue is mitigated, because the charge is distributed over a larger area.83 For the clusters athand, no occupied states with positive energies are found even for the smallest model.This is an indication that here the self-interaction error is not very pronounced; never-theless, since the self-interaction cannot be entirely corrected by any of the functionalsstudied, some residual error is likely to remain.

Table 4.9.: Linear fit parameters for IP −EA (kJ·mol-1) for Pdn (n=13. . . 147) as a function of n−1/3 with10 exchange-correlation functionals (from scalar relativistic calculations).


r2 a 0.977 0.998 0.996 0.997 0.991 0.999 0.998 0.990 0.998 0.991kb 753.0 781.9 768.6 777.4 765.2 869.1 783.9 775.6 784.4 768.7∆Φ c 29.2 14.2 19.1 15.0 22.5 -12.8 13.0 19.3 12.8 21.8

a Correlation coefficient of linear fit; b Slope;c Axis intercept = extrapolated work function difference, kJ·mol 1;

4.1.4. Conclusions

Having described and interpreted the results for the scalability of the four quantitiesdav, Ecoh, IP , and EA, they can now be compiled into a complete overview of the perfor-mance of the 10 XC functionals studied, the influence of the relativistic treatment, andthe overall quality of the results.

In general all the studied quantities exhibit good scalability with cluster size. Partic-ularly the average bond lengths and cohesive energies of the clusters are almost perfectlinear functions of the average coordination number. While some outliers were foundfor the calculated values of IP and EA, all of the properties can be extrapolated to bulkvalues with fairly high reliability. Due to the larger number of surface atoms, which aresubject to a greater “buckling” and contraction of the structure, the decrease of dav withdecreasing cluster size can readily be rationalized. Accordingly the cohesive energyof the clusters also increases with increasing cluster size, as the coordination number

38


increases and the atoms on average form more bonds. As the values of IP and EA

of the clusters are examined in relation to cluster size, both quantities converge to thebulk work function Φ for infinite clusters, but do so from opposite directions. Whilethe ionization energy of the free atom is greater than Φ, the atomic electron affinity issignificantly smaller. This can be rationalized with the increasing metallic character ofthe clusters, where the cost in energy of removing an electron from the HOMO progres-sively decreases, and the energy gain of adding an electron to the LUMO increases. Thisscalability is also well-reproduced by the clusters in the present study, where both sets,IP and EA, scale linearly with the cluster size as measured by n−1/3.

For the accuracy of both dav and Ecoh it can be seen that the treatment of relativistic ef-fects is necessary to obtain good values in comparison with experiment. The inclusion ofscalar relativistic treatment causes a contraction of bond lengths by approximately 5 pmand an increase of up to 100 kJ·mol-1 in cohesive energy for palladium. These changessubstantially decrease deviations from the experimental values. The magnitude of theserelativistic contributions is not strongly affected by the choice of XC functional.

Based on the results obtained for each of the scalable properties the XC functionalscan be classified into 4 groups of different functional behavior.

Table 4.10.: Four groups of XC functionals according to the combined results forscalable properties.

Group I VWN shortest dav, strong “overbinding”, over-

estimated IP and EA

Group II PBEsol, VMT-

sol, VT{84}sol

accurate dav, Ecoh too high, good IP but

somewhat underestimated EA

Group III BP, PBE, PW91 overestimated dav, best Ecoh, IP , and EA

Group IV VMT, VT{84},M06-L

M06-L strongly elongated bond lengths

and large error for IP and EA; VMT and

VT{84} similar to Group III, but less ac-

curate energies, and longer dav

The classification of functionals and their defining characteristics are listed in Table4.10. Both due to the computed results and the different nature of the functional (theonly LDA functional in the study), VWN forms a group of its own. The VWN func-tional shows the trends typical for LDA methods for all of the parameters, i.e. shortbond lengths and strongly overestimated binding energies. Although popular for thedescription of some extended solids,23 VWN may therefore not be the first choice forthe cluster problem at hand, except perhaps for the calculation of geometries.

Extrapolated results from Group II functionals reproduce experimental bond lengthsvery well and improve the VWN energies drastically, while still falling far short of the

39


accuracy of other functionals for Ecoh. This behavior is expected, as the PBEsol func-tional was designed with the goal of improving lattice constants at the expense of atomicenergies57 and thus less accurate values of Ebulk

coh . IP values from PBEsol, extrapolatedto infinitely large clusters, agree well with the work function of bulk Pd, perhaps sincethis quantity does not depend on atomic energies, thus showing that total cluster ener-gies are computed with good accuracy. The other two functionals in Group II virtuallydo not improve upon the bond lengths and cohesive energies found with PBEsol, whilegiving somewhat less accurate IP and EA.

The 3 GGA functionals of Group III give the most accurate values for Ebulkcoh as well

as for IP and EA. The PBE functional is an improved version of PW91, which itselfis based on BP22 and this trend is reflected in the increasing accuracy of the cohesiveenergy determined with these functionals, although differences within this group aregenerally small. Because of the high accuracy of these functionals for the determinationof cluster energies, they seem to be the method of choice for all energetic aspects of thestudied metal clusters; however, the error of 5 pm for optimized geometries is ratherhigh so that the Group III functionals are not universally optimal, either.

The final group of functionals is composed of the meta-GGA M06-L and the two re-maining GGA functionals in this study, VMT and VT{84}. While the former is an excep-tion to many of the trends observed for the other functionals, the latter two are rathersimilar and also show similar behavior as the Group III GGAs. VMT and VT{84} givesomewhat longer dav than the other GGA functionals and less accurateEcoh, IP , and EA;as they are akin to PBE, using the same parameters in a somewhat different expression,this behavior may be expected. However, the modified functional form does not repre-sent an improvement on the properties computed for the studied clusters. The resultsdetermined with the M06-L functional are of mixed accuracy. Although rather goodvalues for cohesive energies were obtained, the computed dav are unrealistically longand IP and EA values show errors twice as large as the least accurate GGA functionals.For the systems at hand no improvement of accuracy is seen that would lead one torecommend this meta-GGA, particularly in view of the higher computational demand.Some additional unique traits of this functional are also discussed in Section 4.2.

Broadly speaking, this study has not identified a single “optimal” functional which isable to predict all of the quantities studied with sufficiently high accuracy. One seemsto be faced with an inevitable trade-off between good cluster geometries and good ener-gies. For the former, the sol-type functionals PBEsol, VMTsol and VT{84}sol are the bestchoice, while energies are best determined with the well-established GGA functionalsBP, PW91, and PBE. The present study shows that the choice of functional is crucial —as results may differ by well over 100 kJ·mol-1or 10 pm — and can be a major source oferror. The data presented here may thus serve as a guideline which of the functionals tochoose for which application, aware of the strengths and weaknesses of each.

40

Master’s Thesis Magnetism in Small Pd Clusters

4.2. Magnetism in Small Pd Clusters

The previous section demonstrated the scalability of various structural and energeticproperties of clusters, which were found to correlate linearly with the cluster size, asmeasured by n−1/3. As the scaling behavior was studied with various functionals, thedata indicated a trend of decreasing total magnetic moment with increasing cluster sizefor most functionals. However, the M06-L functional did not follow this trend, insteadan increasing spin polarization is predicted. In order to further examine the opposingtrends of the functionals, the clusters Pd19 and Pd38 were chosen as model systems andinvestigated aiming to characterize primarily the structural dependence of the clusters’magnetic moment. The first part of this analysis quantified the bond lengths in the dif-ferent clusters, followed by an investigation of the stability of different structures at dif-ferent numbers of unpaired electrons. This set of calculations used the optimized struc-tures obtained with various GGA functionals and determined the relative stabilities ofclusters at various fixed spin polarizations using the M06-L functional. Conversely,various GGA functionals were used to quantify the stability of the structure optimizedwith the M06-L functional. The final part of the study used cluster structures modifiedto contain one systematically stretched metal-metal bond to determine the stabilities ofa high-spin and low-spin configuration as a function of the bond distance.

4.2.1. Introduction

While nickel is a ferromagnetic metal, its 4d homolog palladium is not. However, cal-culations on the density of states (DOS) of bulk Pd have shown it to be “almost” fer-romagnetic,84,85 with a peak of the DOS at the Fermi level.86 It has been suggested,87

that further increasing the DOS at the Fermi level would cause Pd to have a permanentmagnetic moment. Three possibilities have been considered to achieve this: (i) to reducethe coordination number of Pd, (ii) to increase the lattice constant, and (iii) to modifythe symmetry.87 In particular, early calculations already show that the lattice constantof Pd would have to be increased by 5–6% in order to cause a transition to permanentmagnetic behavior in the bulk phase.88

A number of experiments have since been carried out to investigate a potential mag-netic behavior of Pd and the conditions to obtain it. One possibility to increase the lat-tice constant as well as to reduce the coordination number of Pd is to apply it as a thinlayer onto a substrate of greater lattice constant.89 Other studies have prepared Pd clus-ters of various sizes and determined their magnetic behavior. One can expect clustersof increasing size to approach the bulk non-magnetic behavior, which is indeed sup-ported by experimental evidence. A Stern-Gerlach-type experiment on clusters of thesize Pd13 to Pd100 has shown them to be nonmagnetic,90 while a more recent study with

41


photoelectron spectroscopy shows two regimes of magnetic properties: magnetic Pd3

to Pd6 clusters and non-magnetic Pd>15.85 Experiments of a somewhat different naturehave also studied the magnetic properties of Pd nanoparticles prepared in solution89 orembedded in a porous carbon matrix.87 Both papers also report slightly ferromagneticbehaviors for large particles (>20 nm), which, however, may be caused by the inter-action of Pd with the stabilizing ligands and the matrix, rather than size-effects of thenanoscale palladium.

Theoretical investigations on magnetic properties of Pd nanostructures also abound.Several studies predict palladium to be magnetic when prepared as a monolayer witha lattice constant stretched to about that of Ag,91 or bulk Pd with a 5.3% stretched lat-tice constant.88 A first paper on the magnetic properties of Pd13 clusters was publishedin 1993 by Reddy, Khana, and Dunlap,92 in which the authors computed a magneticmoment of 1.56µB for an icosahedral geometry of Pd13, but none for the octahedral (fcc-derived) geometry. In contrast, a subsequent set of calculations with a tight-bindingmodel, found very similar total magnetic moments of ∼7.5µB for both the icosahedraland octahedral isomers of the Pd13 cluster.93 A third study94 on fcc Pd13 found a valueof approximately 6µB as total cluster magnetic moment.

Clusters smaller than Pd13 have also been investigated computationally in a separatestudy, both as neutral clusters and as anions Pdn

-.95 Beginning at triplet (i.e. µ = 1µB)for Pd2, the magnetic moment per atom was found to decrease smoothly as 1/n to µ ≈0.4 for Pd7, while the anions showed more a irregular trend of the magnetic momentwith cluster size. Few studies exist on the magnetism of clusters larger than Pd13. Oneinvestigation of an octahedral Pd19 cluster with LDA calculations96 found a stable para-magnetic state with six unpaired electrons at an energy about 59 kJ·mol-1 lower than theclosed-shell configuration. This magnetic moment also persisted upon varying the lat-tice constant of the cluster to larger and smaller values, and it was found that at about4% increase in lattice constant (relative to the bulk value), the nonmagnetic state wasno longer stable. Pd55 and Pd147 have also been investigated.97 The cubic isomer of Pd55

was found to favor configurations with low total spin, while the icosahedral structurestabilized 26 unpaired electrons. A similar result was also found for Pd147, where theicosahedral structure adopted a 60µB magnetic moment and the cubeoctahedron wasnonmagnetic.97

In summary, it can be stated that both experimental and theoretical evidence find verysmall palladium clusters to be magnetic, which, however, rapidly become non-magneticat nuclearities at or beyond Pd13. Moreover, the structure of the system, that is, both thesymmetry and interatomic distances play an important role for the relative stabilitiesof magnetic and non-magnetic configurations. Elongated metal-metal bonds stabilizehigher magnetic moments, as do the icosahedral isomers of larger clusters (which show

42


higher coordination numbers but larger interatomic distances) ‡.While various Pd clusters have already been investigated, DFT calculations especially

for larger clusters are still somewhat scarce. Previous investigations typically do notevaluate the influence of varying the exchange-correlation approximations, althoughparticularly in such fairly compact transition metal systems the treatment of electroniccorrelation is of central importance. Furthermore, relativistic effects in Pd can be ex-pected to play an important role, which may not be treated accurately enough withstandard pseudopotential methods, especially considering the very subtle energy dif-ferences between configurations with different magnetic moments.

4.2.2. Trends of Magnetic Behavior in Palladium Clusters

Both experimental and theoretical studies of small palladium clusters have shown themto be magnetic in the ground state.85,90,92,96,97 However, as the cluster size increases be-yond a few dozen atoms, the magnetic moment rapidly decreases and approaches thenonmagnetic configuration of the bulk already at small system sizes. Fig. 4.5 showsthe number of unpaired electrons nu in fully optimized Pdn clusters as a function ofthe average coordination number, determined with various exchange-correlation func-tionals§. As evident from the plot, all investigated functionals reproduce the trend ofvanishing magnetic moment with increased cluster size with the exception of the M06-L functional, which predicts an increasing spin. Extrapolating this trend, the M06-Lfunctional may erroneously predict palladium metal to be magnetic.

The value of nu ≈ 6 in Pd13 is reproduced by all exchange-correlation functionalsfor the relativistic calculations and also agrees well with previously published results,namely 7.593 and 6.94 In contrast, in the corresponding non-relativistic calculations allfunctionals yield a magnetic moment of zero. It stands to reason, therefore, that besidescausing a sizeable contraction of the structure, relativistic effects also contribute sub-stantially to the magnetic moment of the cluster. This relativistic nature of the magneticconfiguration may be why the very first study of Pd13, which did not consider relativis-tic behavior, found no unpaired electrons.92 Later investigations used scalar relativisticpseudopotentials95 or semi-empirical fits93 and found the clusters to be magnetic.

Pd19 is the only cluster, where nonrelativistic calculations gave a non-zero magneticmoment, which in fact was typically greater than the one obtained from the relativisticcalculations (nu > 6.5 for all GGA functionals, nu = 7.9 for M06-L, 5.5 for LDA). This may

‡Symmetry unrestricted calculations in many cases find the icosahedral structures to be slightly morestable than the octahedral isomers.97 However, as icosahedral structures are non-crystallographic,their properties cannot be readily generalized to the bulk limit.

§For these calculations the number of unpaired electrons is unconstrained and the code chooses the moststable configuration based on the starting geometry and the corresponding potential energy surface.Line broadening was enabled as in the calculations in the previous section.

43


5

10

15

20

25

30

6 6.5 7 7.5 8 8.5

nu

CNav

VWN

BP

PW91

PBE

M06L

PBEsol

VMT

VT84

VT84sol

VMTsol

Figure 4.5.: Number of unpaired electrons for Pdn clusters of varyingsize

be rationalized with the particular structure of the octahedral cluster: Here the relativis-tic contraction for the apical atoms in the octahedron is particularly pronounced; in theabsence of this, these atoms are bound by substantially elongated bonds in addition tobeing strongly undercoordinated. As mentioned above, these factors contribute to lo-calize electrons and cause a magnetic moment.87 All other clusters have more compactstructures without single isolated corner atoms of CN 4, and seem thus less susceptibleto this behavior.

Already for nu of the Pd19 cluster M06-L diverges from the trend of the other function-als, finding 7.68 unpaired electrons (0.40 per atom), in contrast to the other GGAs (nu ≈5) and LDA (nu = 3.9). For larger clusters the divergence is even more pronounced, asall GGAs and LDA yield a fully quenched spin for Pd38 or Pd55. The curve for M06-Lstrongly goes up, giving a magnetic moment of nu = 33.5 for Pd79 (0.42 per atom), af-ter a noticeable dip at Pd55. While the number of unpaired electrons per atom remainsapproximately constant for M06-L, all other functionals predict this quantity to rapidlyapproach zero, faster than nu itself.

Having established that the magnetic moment of the clusters is primarily influencedby relativistic and structural effects and that the obtained GGA results are plausible andin agreement with previous studies, it remains to be clarified why the M06-L meta-GGAbehaves in an anomalous way, which is the topic of the following sections.

44


4.2.3. Structural Aspects

Table 4.11 shows the average bond lengths dav in optimized Pd clusters as computedwith different functionals. As noted previously, the shortest bond lengths are obtainedwith VWN, followed by PBEsol. The bond lengths determined with the other threeGGA functionals are very similar, and approach the bulk value of 275 pm in the Pd79

cluster. Extrapolation of this trend leads to the generally expanded structures typical ofmost GGAs (see 4.1.1).

The M06-L results are up to 10 pm longer than the VWN bond lengths, and already thethird-smallest cluster, Pd38, has a greater value of dav than bulk palladium. Moreover,the linear increase of bond lengths with increasing cluster size is not reproduced entirelyby the M06-L functional, giving a shorter average bond length for Pd55 than for Pd38.Compared to a typical GGA functional, such as BP, the M06-L bonds are extended byup to 2 pm, although the gap narrows with increasing cluster size. In fact, of all tenfunctionals in this study M06-L produces the longest bond lengths.

In accordance with other studies, we observe a preference of high-spin configurationsfor structures with expanded metal-metal distances. This can be rationalized by a sharp-ening of the electronic density of states, which is progressively more localized at atomicnuclei. The long bonds predicted by the M06-L functional are thus a first argument forthe peculiar behavior of the spin. This reasoning also accounts for the dip in clustermagnetic moment at Pd55, which coincides with a smaller dav in that cluster. Since Pd55

is a cuboctahedron with, compared e.g. to Pd79, few exposed corner atoms, the structuremay thus be predicted especially compact; Pd79 contains 24 corner atoms of CN 6, whilePd55 contains only 12 corner atoms of CN 5. Furthermore, a previous study also reportsa drop in magnetic moment for Pd55,97 citing the cuboctahedral structure as a reason.

As expanded structures likely are the reason for the high spin polarization, it remainsto be clarified whether the functionals predict different relative stabilities with differentstructures. For this purpose, the optimized structures and the functionals were mutuallyinterchanged, as described in the following section.

Table 4.11.: Average Bond Lengths (dav, pm) in Pd13 . . . Pd79clusters, scalar relativistic calculations with severalfunctionals

VWN BP PW91 PBE PBEsol M06-L

Pd13 262.5 270.1 270.0 270.4 265.6 272.0

Pd19 264.2 271.6 271.4 271.6 267.0 273.4

Pd38 265.8 273.2 273.1 273.3 268.6 275.2

Pd55 266.2 273.7 273.6 273.8 269.0 275.0

Pd79 267.5 274.9 274.8 275.0 270.3 275.9

45


4.2.4. Stability of Fixed Numbers of Unpaired Electrons

Calculations with fixed numbers of unpaired electrons are a popular method to studythe relative stability of different magnetic moments84 and have been used in some of thestudies of magnetism in Pd clusters mentioned above.94,96 Such calculations are basedon fixing nu in a system, and systematically determining the total energies at a numberof different values of nu. Thus a curve of stability vs. magnetic moment is obtained,which allows one to asses what magnetic moments are energetically preferred. Withthe aim to clarify both the influence of the functional and of the cluster structure on themagnetic moment preference, we selected the Pd19 and Pd38 clusters as model systemsand carried out a set of fixed-spin calculations, while varying the functional and thecluster geometry.

Pd19

Fig. 4.6(a) shows the stability of the Pd19 cluster at fixed nu = 0. . . 9, as a total energydifference relative to the closed-shell case, E(nu = 0). The calculation was carried outwith two different functionals, BP and M06-L, each at two geometries, relaxed with BPand M06-L. All points are single-point calculations and the geometries are those relaxedat variable spin (i.e. the structures obtained from the optimizations carried out previ-ously). The notation “Functional//Geometry” is used to indicate which XC functionalis applied at which optimized geometry.

The first striking feature of the graph is the different stability of each functional at itsnative geometry (i.e. BP//BP and M06-L//M06-L): The curve for BP has a compara-tively shallow minimum at nu = 4, stabilizing this low-spin configuration by 26.3 kJ·mol-1,while the M06-L functional yields the high-spin, nu = 8 configuration as 63.0 kJ·mol-1

more stable than the nonmagnetic state. This observation confirms the opposite spinpreferences of the two functionals also without the influence of line broadening; thepeculiar high-spin behavior of M06-L is clearly evident from the graph, and the con-trasting preferences of the two functionals are apparent.

When the two geometries are exchanged, i.e. the M06-L geometry is treated with BP(“BP//M06-L”) and the BP geometry with M06-L (“M06-L//BP”), the trend is almostreversed (see arrows in Fig. 4.6(a)). The minimum for BP//M06-L shifts away from thelow-spin configuration and now adopts high-spin, at roughly the same relative stabil-ity. Conversely, for M06-L//BP, a sudden secondary minimum at the low-spin stateappears, just slightly higher in energy (by 9.4 kJ·mol-1) than the high-spin configuration.

From these observations it can be concluded that the structure of the cluster has in-deed a major influence on the stability of the spin. The exchange of minima with theexchange of the two structures for BP suggests that the magnetic moment of the clusteris rather sensitive to the structure at which the calculation is carried out. The aver-age metal-metal distances in the two structures differ by approximately 2 pm — less

46


than 0.75% — yet the magnetic moments are qualitatively different. While, in termsof relative energies, the differences between the two spin states are subtle, the effect isnevertheless present.

A perhaps even more striking observation is the appearance of the double minimumin the M06-L//BP curve. Apparently, apart from the structural change, another effectoccurs as well, which is unique to the behavior of M06-L. The low-spin preference in-duced by the structure switch seems to be counteracted by an intrinsic tendency of M06-L to favor high-spin configurations. In this combination of structure and functional, itappears that the system “cannot decide” whether to adopt a high or low spin. Thus oneis led to conclude that cluster structure is not the only cause of the magnetic moment,but that another functional-related influence is also present.

In order to increase the understanding of this double-minimum phenomenon, thestudy of complementary functional-structure fixed-spin calculations was extended toinclude also the VWN and PBEsol functionals and the corresponding relaxed geome-tries. Fig. 4.7 shows the analogous plots for the three functionals side-by-side. Therelative magnitude of the double-minimum is highlighted. As the structures go from avery compact (VWN, dav = 264.2 pm), through a more expanded (PBEsol, dav = 267.0 pm)to the most expanded (BP, dav = 271.6 pm) structure, the high-spin configuration is pro-gressively stabilized, while the low-spin stability remains roughly at the same level. Thestability curves of VWN and PBEsol qualitatively follow that of BP, showing the sameexchange of stability upon exchange of structure.

In line with the previous conclusions, the increase in average metal-metal distanceprovides an explanation for the progressive increase of high-spin stability. The ongoingpresence of the low-spin minimum (which, however, disappears in M06-L//M06-L)seems to be due to the fact that bond distances are still about 2 pm shorter than in theM06-L-relaxed geometry. The next section therefore investigates the influence of indi-vidual bond lengths on this double-minimum behavior.

Pd38

Fig. 4.6(b) shows the analogous fixed-spin-moment calculations for the cluster Pd38.Once again a low-spin and a high-spin configuration have been determined roughlyat twice the magnetic moments, the former at nu = 8, the latter at nu = 18, for a clustertwice the size of the one previously examined. Compared to Pd19 the stabilization en-ergies of the states are in a similar range, which implies that the per-atom stabilizationsare roughly a factor of two lower. The BP functional once again shows a very shallowminimum at nu = 8, in this case for both the BP-relaxed and M06-L-relaxed geometries.M06-L shows a pronounced stabilization of the high-spin configurations at both geome-tries, but once again yields a slight secondary minimum at nu = 8. For this cluster thepronounced exchange of minima upon exchange of the geometries is absent; however,

47


-60

-40

-20

0

0 2 4 6 8

Ere

l, kJ

mol

-1

nu

BP//BP

BP//M06L

M06L//BP

M06L//M06L

(a) Pd19

-80

-40

0

40

80

0 2 6 10 14 18

Ere

l, kJ

mol

-1

nu

BP//BP

BP//M06L

M06L//BP

M06L//M06L

(b) Pd38

Figure 4.6.: Relative stability of Pdn as a function of number of unpaired electrons nu, at BP- and M06-L-optimized structures computed with BP and M06-L

the M06-L//BP curve noticeably reduces the depth of the high-spin minimum, andBP//M06-L is somewhat less sensitive towards high spin polarizations.

4.2.5. Structural Differences Quantified

The average bond length in the clusters, as considered initially, is a rather general quan-tification of the features of a cluster and may not sufficiently capture the subtle differ-ences between structures. For a more differentiated assessment of structural features,this section explores the bond-length distributions in Pd19 and Pd38 and considers theinfluence of single bonds on the magnetic moment of the cluster.

-60

-40

-20

0

0 2 4 6 8

VWN//VWN

VWN//M06L

M06L//VWN

M06L//M06L

0 2 4 6 8

PBEsol//PBEsol

PBEsol//M06L

M06L//PBEsolM06L//M06L

0 2 4 6 8

BP//BP

BP//M06L

M06L//BP

M06L//M06L

nu

Ere

l, kJ

mol

-1

Figure 4.7.: Relative stability of Pd19 as a function of nu, three functional-structure pairs. Highlighted:energy difference between high-spin and low-spin minima for M06-L at different structures.y-axis: energy difference to nu = 0 state (kJ·mol-1)

48


Bond Length Distributions

Contrary to the bulk metal, optimized clusters are characterized by a non-uniform dis-tribution of bond lengths,38 where surface atoms are distinguished by shorter bondsthan atoms within the cluster. Clusters optimized with varying functionals may dif-fer not only with respect to their average bond lengths, but also show different bondlength distributions. These distributions were evaluated for the Pd19 and Pd38 clusters,as optimized with BP and M06-L.

Pd19

When constrained to octahedral symmetry, the cluster Pd19 only has two distinct Pd-Pd distances, one between the central atom and its 12 nearest neighbors (d1) and onebetween an apical atom at the corner of the octahedron and its 4 nearest neighbors (d2).Table 4.12 shows the distances d1 and d2 in the clusters optimized with BP and M06-L. Comparing the two distances it is apparent that d1 is rather similar among the twofunctionals, differing by about 1.2 pm, while d2 deviates much more strongly, by 4.9 pm.The structures given by the two functionals differ almost exclusively in the positions ofthe apical Pd atoms, where M06-L gives a very open structure with the corner atoms farremoved from the rest of the cluster.

Table 4.12.: Distinct bondlengths (pm)in Pd19 in theM06-L andBP-optimizedstructures

Structure d1 d2

BP 273.3 262.5

M06-L 274.5 267.4

With respect to the differences in magnetic moment, it stands to reason that most ofthe difference between the two functionals is caused by the large difference in metal-metal distance for these apical atoms. This allows for the isolation of spin density at thecorners of the octahedron, giving rise to the predicted high-spin configuration.

Pd38

The Pd38 cluster contains a total of six distinct metal-metal bonds, as shown in Table4.13. For this cluster the differences arising from the two functionals are not as apparentas for Pd19. In this case the bonds differ by as little as 0.3 pm and as much as 3.2 pm. d5,the distance most different between the two functionals is the distance between atomson the (100) face of the cluster, again the most strongly undercoordinated atoms, as was

49


Figure 4.8.: Straining a single Pd position in Pd19 (left) and Pd38 (right)

already the case for Pd19. Furthermore, the other distances that differ most between thefunctionals, are also related to the positions of the (100) atoms.

Compared to the smaller cluster, differences between functionals are less pronouncedfor Pd38. This may be why the behavior of the functionals upon exchange of the struc-tures (vide supra), is less pronounced in the Pd38 case. Moreover, atoms on the surface ofthe cluster Pd38 are less undercoordinated than in Pd19, where the smallest coordinationnumber is 4, compared to 6 in the larger cluster. The surface atoms are thus bound morestrongly and cannot get as far away from the cluster. Thus the reduced dissimilarityof bonds in the cluster seems to account for the absence of a high-spin minimum forBP//M06-L. This aspect is not captured by the average bond length alone.

Table 4.13.: Distinct bond lengths (pm) in Pd38 in the M06-Land BP-optimized structures

Structure d1 d2 d3 d4 d5 d6

BP 278.8 273.1 273.6 272.4 271.1 271.3

M06-L 279.1 276.1 274.5 274.5 274.3 273.6

Effect of Stretched Bonds

The above results suggest that a single structural aspect, namely a particular elongatedPd-Pd bond, may play an important role in determining the calculated magnetic mo-ment of these two clusters. Therefore, a set of modified geometries of Pd19 and Pd38 wasprepared based on the M06-L-optimized structures, where one atomic distance was sys-tematically varied. Fig. 4.8 shows the positions of the atoms which were modified in aseries of steps compressing and stretching the initial structure. Along with the indicatedatoms, the set of 6 and 24 symmetry-equivalent atoms moves accordingly.

50


0

30

60

90

BPEfit, kJ mol-1

0

30

60

90

TPSS

-30

0

30

60

370 375 380 385

M06-L

ds, pm

(a) Pd19

0

40

80

120BP

Efit, kJ mol-1

0

40

80

120TPSS

-40

0

40

80

425 430 435 440

M06-L

ds, pm

(b) Pd38

Figure 4.9.: Stability of high-spin (light-colored lines) and low-spin (dark lines) configurations as a functionof ds (pm). Vertical bar indicates ds at optimized geometry. Energy Efit is re-scaled to setminimum of low-spin parabola at energy zero.

The total energies of the multiplicities 4 and 8 (Pd19), and 8 and 18 (Pd38) were de-termined in a series of fixed-spin calculations as a function of the stretched distanceds, measured from the center of the cluster. Besides the BP and M06-L functionals, thecalculations were also carried out with another meta-GGA, TPSS, in order to see if theobserved high-spin preference is a common trend among meta-GGA functionals.

The trends of the energies are reproduced very well by parabolic curves determinedin a least-squares fit to the measured energies. The resulting energies were then normal-ized by setting the minimum of the low-spin parabola to zero, thus making it possibleto compare relative energy differences Efit computed with the various functionals andstructures on a common energy scale.

Pd19

Fig. 4.9(a) shows the relative energies of the high- and low-spin configurations of Pd19

as a function of the distances of the apical atoms. Common among all the functionals

51


is the observation that the high-spin minimum is always located at a stretched distancethat is longer than that of the low-spin configuration. This agrees with the by nowestablished trend of extended structures favoring higher numbers of unpaired electrons.Furthermore it is noteworthy, that the curvature of the parabolas is very similar for thehigh- and low-spin configurations for each functional; therefore one observes a linearbehavior of the relative stabilization of nu = 8 to nu = 4 as a function of stretched distanceds.

For the functionals BP and TPSS the positions of the minima are very similar, indi-cating the optimum apical distance to stabilize nu = 4 is at about 367 pm and 375 pm tostabilize nu = 8. The relative energies for BP are very similar for both nu = 4 and nu =8, but the high-spin configuration is slightly higher in energy for ds<372 pm. The sameobservation holds for TPSS, although low-spin is more strongly favored than for BP andoccurs already for ds<374 pm.

The curve obtained with the M06-L functional is qualitatively and quantitatively dif-ferent from that of the other two functionals. Firstly, both the high-spin and low-spinminima occur at significantly higher values of ds, and M06-L predicts a low-spin ds atthe position of the high-spin minimum of BP and TPSS. The difference in distance be-tween the two minima, however, is significantly shorter than that found with the otherfunctionals. Secondly, the most striking difference between the functionals concerns therelative stability of the high-spin and low-spin configurations. M06-L reverses the or-dering of the curves and predicts the high-spin configuration to be more stable over awide range of distances; nu = 4 becomes more stable only at very compressed geome-tries.

This dataset provides further insight into the nature of the double minimum observedabove. Since M06-L predicts both minima in a very narrow range of ds values, theminimum for nu = 4 can be expected to appear at the slightly compressed structures,which were used for the plots of relative stability (Fig. 4.7), and to only be absent in thefully extended M06-L optimized structure.

Pd38

A very similar picture emerges from the investigation of Pd38. All distance-energycurves are well represented by parabolic fits in ds. The high-spin minima also occur atlonger metal-metal distances than the low-spin minima, although the two are typicallycloser together. The TPSS functional in this case positions the minima at the shortestdistances, followed by BP. Both TPSS and BP yield a more stable low-spin configurationat all values of ds.

The M06-L functional also shows a behavior analogous to the one for the system Pd19.The minimum at low magnetization occurs at approximately the same ds as the high-spin minimum with BP. Also in this case M06-L shows a preference for the high-spin

52


configuration at all stretched lengths, in contrast to the other functionals.The relative energy differences may also explain why the curve of BP//M06-L (Fig.

4.6(b)) does not show a minimum at the high-spin configuration, as would be expectedby analogy from the Pd19 system. Compared to the energy-distance curve for Pd19, forPd38 the high-spin and low-spin curves never intersect, i.e. at no stretching of the geom-etry is the high spin ever energetically favored. The same reason also accounts for thefact that the M06-L//BP curve has only a very shallow dip at the low-spin multiplicityinstead of a pronounced double minimum as in Pd19.

4.2.6. Conclusions

The present study on the magnetic features of Pd clusters has brought to light a numberof significant insights and conclusions, which shall now be summarized.

The magnetic moment of the clusters is chiefly a relativistic effect. The number of unpairedelectrons in Pd13, while around nu = 6 for scalar relativistic calculations, drops to zerowhen relativistic effects are disregarded. This result agrees with previous studies, whichonly detected magnetic Pd13 when relativistic effects were taken into account.92,93,95 Onereason for this is likely the shifting of electronic d-states upon the inclusion of scalarrelativistic treatment.

The magnetic moment of the clusters strongly depends on the cluster geometry. The datashow a strong correlation between the bond lengths in a cluster and its magnetic mo-ment. Expanded structures, especially those with strongly undercoordinated atomssuch as octahedral Pd19, favor electronic configurations with a high magnetic moment.This effect can be rationalized with the stronger localization of electrons at the metalatoms due to the larger interatomic distances. This sharpens the density of states andstabilizes unpaired electrons. This general structural argument accounts for the increasein magnetic moment in the structures of Pd13 . . . Pd79 optimized with M06-L, as they ex-hibit untypically large Pd-Pd bond lengths.

Further support for the strong influence of the cluster structure on its magnetic mo-ment comes from fixed-spin calculations on the Pd19 and Pd38 clusters; Here the moststable spin state of a cluster changes, as the structure is exchanged for one originally op-timized with another functional. In all calculations the M06-L functional shows a pref-erence for the high-spin configuration, most pronounced at the native M06-L-optimizedgeometry, and shows a secondary minimum of varying depth depending on the struc-ture contraction. While the main influence on the magnetic moment is the cluster struc-ture, M06-L seems to be influenced by a secondary effect intrinsic to that functional.

The magnetic moment of the clusters not only depends on the cluster geometry but also on theexchange-correlation functional. One can trace the primary structural influence on the spinback to one particular atomic coordinate in each cluster by analyzing the bond length

53


distribution. Using a set of modified structures where one (unique) atom is succes-sively moved to give stretched geometries, parabolic energy profiles for high-spin andlow-spin configurations are obtained. This dataset indicates that the high-spin pref-erence is not a feature of meta-GGA functionals in general but occurs only in M06-L;this functional shows a qualitatively different behavior and a pronounced preference ofhigh-spin configurations. The data also provide a rationalization for the presence andabsence of the double minimum for M06-L in the fixed-spin-moment calculations.

Two principal influences seem to govern the magnetism of the studied Pd clusters —that of the structure and that of the exchange-correlation functional. The M06-L func-tional has a peculiar tendency towards high-spin configurations, which potentially ex-trapolates to erroneously magnetic palladium metal. While its cause remains uncertain,the present study has unambiguously confirmed and quantified this behavior.

54

5Summary

Metal nanoclusters are an interesting and versatile class of materials and find applica-tions in many fields of nanoscience, especially heterogenous catalysis.7,13,42 At clustersizes larger than a threshold of about 100 atoms properties of clusters become scalable,i.e. proportional to the cluster size, and can thus be extrapolated to bulk quantities.The most popular ab initio-method for the treatment of metal clusters is Density Func-tional Theory (DFT),21 which enables quantum-chemical calculations on clusters of sev-eral hundred atoms.36 However, the main challenge in current DFT methods remainsthe accurate treatment of the exchange-correlation energy, for which no exact expres-sion is known but numerous approximate exchange-correlation (XC) functionals havebeen proposed.23,24 XC functionals are typically designed and validated either for smallmolecules, or, less frequently, for extended solids; thus, their applicability to metal clus-ters, which lie at the interface between these regimes, is not immediately apparent. Thepresent thesis studied the scalability of structural, energetic, and magnetic cluster prop-erties, using a set of ten XC functionals, among them four which have only recently beenpublished.28,29 Besides exploring the scalable behavior of clusters as such, the main fo-cus of this work was a comparative evaluation of these functionals, aiming to assesstheir applicability and performance for Pd and Au clusters.

A set of six Mn clusters, where n = 13, 19, 38, 55, 79, 147 and M = Pd and Au, wasstudied, focusing on the average intermetallic distance dav, the cohesive energy Ecoh,and for Pd the ionization Potential IP and electron Affinity EA. These quantities scalewith the cluster size. They can be fitted with linear functions of the average coordina-tion number of the cluster or n−1/3, a quantity proportional to the cluster radius. Onecan assess the accuracy of the employed functionals by extrapolating these quantities tobulk values and comparing them to experimental values. Three functionals optimizedfor lattice constants of extended solids, PBEsol,57 VMTsol,28 and VT{84}sol,29 predictthe most accurate bulk bond lengths, within 1 pm of the experimental values 275 pmand 288 pm for Pd and Au, respectively. The most accurate cohesive energies are ob-tained using the GGA functionals BP,53,54 PW91,55 and PBE,56 within 10 kJ·mol-1 for Pd(exp. -376 kJ·mol-1), while the M06-L functional50 gives somewhat better agreement

55

Master’s Thesis

with the experiment for Au (-366 kJ·mol-1), albeit with an error of almost 30 kJ·mol-1.While the linear fit of IP and EA with n−1/3 is not as good as for the other quantities,both can be extrapolated to estimate the bulk work function Φ. The BP functional givesthe best agreement for the extrapolated IP , underestimating the experimental value of503 kJ·mol-1 by about 2 kJ·mol-1. All of the functionals underestimate the electron affini-ties, so that the most accurate extrapolated value of Φ is obtained with the VWN52 LDAfunctional. A residual difference of 12. . . 22 kJ·mol-1 remains between the extrapolatedEA and IP , chiefly attributable to inaccuracies of the EA determination. Based on the re-sults for the various quantities, one can categorize the XC functionals into four groups:VWN (Group I); PBEsol, VMTsol, VT{84}sol (Group II); BP, PW91, PBE (Group III);VMT, VT{84}, M06-L (Group IV).

While all of the functionals predict the scalable behavior of the clusters and the cor-responding size-dependent trends, none of them can be considered “optimal” for thecomputation of all of the quantities. The results indicate a fundamental trade-off be-tween accurate geometries and accurate energies, which seem to be mutually exclusivewithin the set functionals. The four novel GGA functionals of the VMT- and VT{84}families are also subject to this fact, and do not improve on the results by the older GGAmethods. The use of a meta-GGA functional, M06-L, does not improve results, either;indeed, it yields unrealistically long bond lengths, and energies one order of mangitudeworse than those found with the best GGA functionals. From the observations made inthis study it is evident that the choice of XC approximation has a profound influence onthe obtained results (geometries differing by 8 pm and energies by up to 160 kJ·mol-1),so that a judicious selection of functional according to the desired property is crucial.

The ability to account for or neglect scalar relativistic effects makes it possible to quan-tify the relativistic contribution to the computed quantities. Interatomic distances con-tract by about 5 pm and cohesive energies increase by up to 100 kJ·mol-1 upon inclusionof relativistic effects. It is only with relativistic treatment that the binding within thecluster is adequately represented, making possible a good agreement with experiment.For Au clusters the GGA cohesive energies are underestimated by 30 kJ·mol-1 and moredespite the scalar relativistic treatment.

The second part of this thesis explored the magnetic properties of Pd clusters. Thenumber of unpaired electrons nu decreases and quickly vanishes, as expected, with in-creased cluster size using the LDA or GGA functionals; the M06-L functional, however,predicts a strongly increasing magnetic moment of the clusters, yielding nu >33 forPd79. When determining the relative stability of Pd19 and Pd38 at various fixed valuesof nu, M06-L was shown to stabilize a high-spin configuration of nu = 8 (Pd19) and nu

= 18 (Pd38) by about 60 kJ·mol-1 relative to the closed-shell configuration. In contrast,the BP functional, chosen as a representative of a typical GGA functional, stabilizeslow-spin configurations of nu = 4 and nu = 8. Upon exchanging the optimized struc-

56

Master’s Thesis

tures obtained with these functionals, the stability of the magnetic configurations is alsoswitched. When applying M06-L to a series of modified cluster structures, where asingle metal-metal distance was artificically stretched and compressed over a range of25 pm, the functional showed a strong preference for the high-spin configuration at vir-tually all geometries. This led to the conclusion that it is not only the expanded clusterstructures found with the M06-L functional that favor the high magnetic moments ofthe clusters; there appears to be an additional intrinsic tendency of the functional toprefer high numbers of unpaired electrons.

In summary, the efforts presented in this thesis provide new knowledge about theapplicability of various exchange-correlation functionals to metal clusters and their be-havior, allowing for more insight into which functional is best used for the computationof particular quantities. Furthermore, the study described and quantified the unusualmagnetic behavior of clusters predicted by the M06-L functional.

57

ABasis Sets

The following all-electron orbital and auxiliary basis sets were used for palladium andgold throughout the present study.

PalladiumReference 71Contraction (18s, 13p, 9d)→ [7s, 6p, 4d]Fit basis (17s, 6r2, 5p, 5d)

Exponents of the Orbital Basis Functions

s p d

α1 0.13500000·10-1 0.904000000·10-1 0.97000000·10–1

α2 0.40067417·10-1 0.214300000 0.26602317

α3 0.11247078 0.508279790 0.77945515

α4 0.48448268 0.131341920·101 0.19901245·101

α5 0.11164752·101 0.303040990·101 0.47897946·101

α6 0.22499606·101 0.680629510·101 0.11168119·102

α7 0.54118888·101 0.146853680·102 0.27050913·102

α8 0.10579348·102 0.322783020·102 0.71709317·102

α9 0.28719757·102 0.710869430·102 0.24247077·103

α10 0.64119284·102 0.167499520·103

α11 0.16514908·103 0.439606140·103

α12 0.37464803·103 0.136983110·104

α13 0.90594369·103 0.584159920·104

α14 0.23981500·104

α15 0.71555559·104

α16 0.25013151·105

α17 0.10935260·106

α18 0.72204907·106

58

Master’s Thesis

GoldReference 72Contraction (21s, 17p, 11d, 7f)→ [8s, 7p, 5d, 3f]Fit basis (20s, 8r2, 5p, 5d)

Exponents of the Orbital Basis Functions

s p d f

α1 0.1000000000·10-1 0.2000000000·10-1 0.5600000000·10–1 0.1700000000

α2 0.2200000000·10-1 0.4600000000·10-1 0.1400000000 0.4200000000

α3 0.5600000000·10-1 0.1000000000 0.3500000000 0.1100000000·101

α4 0.1400000000 0.2100000000 0.8800000000 0.3338171959·101

α5 0.3400000000 0.4550000000 0.2200000000·101 0.9912050247·101

α6 0.8400000000 0.1000000000·101 0.5546597004·101 0.2775035858·102

α7 0.2100000000·101 0.2250000000·101 0.1400382996·102 0.8682437134·102

α8 0.5153470039·101 0.4956506729·101 0.3686191177·102

α9 0.1335270023·102 0.1079928970·102 0.9053779602·102

α10 0.2624551010·102 0.2613319016·102 0.2432032013·103

α11 0.5848051071·102 0.5526807022·102 0.8167215576·103

α12 0.1136140976·103 0.1278000031·103

α13 0.2334391022·103 0.2854565125·103

α14 0.6212769775·103 0.6777907715·103

α15 0.1429497925·104 0.1773843994·104

α16 0.3491379883·104 0.5459124023·104

α17 0.9358392578·104 0.2307280078·105

α18 0.2827494141·105

α19 0.9992878906·105

α20 0.4465295000·106

α21 0.3095417000·107

59

Master’s Thesis

Exponents of the Fit Basis Functions: Pdr2 s p d

α1 0.42860000·100 0.801348340·10-1 0.100000 0.20000α2 0.26268384·101 0.224941560 0.250000 0.50000α3 0.13612590·102 0.968965360 0.625000 0.12500·101

α4 0.64556604·102 0.223295040·101 0.156250·101 0.31250·101

α5 0.33499904·103 0.449992120·101 0.390625·101 0.78125·101

α6 0.27396622·104 0.108237776·102

α7 0.211586960·102

α8 0.574395140·102

α9 0.128238568·103

α10 0.330298160·103

α11 0.749296060·103

α12 0.181188738·104

α13 0.479630000·104

α14 0.143111118·105

α15 0.500263020·105

α16 0.218705200·106

α17 0.144409814·107

Exponents of the Fit Basis Functions: Aur2 s p d

α1 0.92000000000·10-1 0.44000000000·10–1 0.100000 0.20000α2 0.42000000000 0.11200000000 0.250000 0.50000α3 0.20000000000·101 0.28000000000 0.625000 0.12500·101

α4 0.99130134580·101 0.68000000000 0.156250·101 0.31250·101

α5 0.52266380320·102 0.16800000000·101 0.390625·101 0.78125·101

α6 0.25560000620·103 0.42000000000·101

α7 0.13555815430·104 0.10306940078·102

α8 0.10918248046·105 0.26705400460·102

α9 0.52491020200·102

α10 0.11696102142·103

α11 0.22722819520·103

α12 0.46687820440·103

α13 0.12425539550·104

α14 0.28589958500·104

α15 0.69827597660·104

α16 0.18716785156·105

α17 0.56549882820·105

α18 0.19985757812·106

α19 0.89305900000·106

α20 0.61908340000·107

60

BValidation of the VMT[sol]

Implementation

In order to test and validate the implementation of the VMT and VMTsol functionals inParaGAUSS, a series of calculations were done with small molecules, aiming to repro-duce the published values,28 obtained with the original implementation of the function-als in the deMon2k code.65 The optimized bond lengths, total energies and atomizationenergies of a set of 10 molecules (H2, OH, H2O, CO, N2, O2, NO, NH3, CH4, HCN) — asubset of the originally published test data for this functional — were determined whileaiming to reproduce the parameters of the reference implementation as accurately aspossible.

The DZ-ANO orbital basis set98 and the GEN-A2*99 charge fit basis were convertedfrom deMon2k to the ParaGAUSS format, preserving all contractions and exponents. AsdeMon2k uses Hermite Gaussian functions, and ParaGAUSS uses spherical harmonicGaussians to fit the density, a one-to-one correspondence of Coulomb fitting functionsis not possible for d-type functions (six Hermite Gaussian d-functions vs. five sphericalharmonics). Furthermore, ParaGAUSS uses an additional r2 dependent function in thefit basis, which is not present in deMon2k. The charge fit exponents for H, C, and Owere used as-is from the GEN-A2* basis set, and for these elements the exponent ofthe r2-function was set to 1020, thus effectively removing this function from the basis.For nitrogen, the r2 exponents were instead set to the same values as those of the d-functions, so as to increase the flexibility of the basis and “simulate” the sixth d-typefunction present in deMon2k.

For the determination of atomic energies, the symmetry of the atoms was restrictedto C2v and the atomic orbitals were set to a fixed occupation according to Hund’s rule(which was also used by the authors of the reference study). The homonuclear linearmolecules were optimized with D4h symmetry constraints, and the heteronuclear lin-ear molecules with C2v. NH3 and CH4 were optimized with C3v and Td symmetries,respectively. The integration grid was chosen to contain 71 radial shells with 171 an-gular points, resulting in a symmetry-adapted grid size of approximately 1350 points

61

Master’s Thesis Validation of the VMT[sol] Implementation

for H2. Test calculations for H2 with a larger grid (∼4000 points) yielded differences of1.9·10-3 kJ·mol-1 for the total energy and 1.5·10-3 pm for the bond length. The error intro-duced by the grid thus appears to be a very minor contribution; the grid size used in thereference calculations changes dynamically (“adaptive” grid) and is not specified fur-ther in the pertaining publication. The other computational parameters were the sameas used in the cluster calculations (Chapter 3), but no line broadening was used.

Tables B.1 through B.3 show the optimized bond lengths, atomization energies andtotal energies of the optimized molecules as determined with ParaGAUSS, as well asthe relative deviation from the published values.28 For comparison, the results with thePBE and PBEsol functionals are also included. With all four functionals we find aver-age absolute deviations (AAD) from the reference values <0.13 pm for the optimizedbond lengths, <9.3 kJ·mol-1 for the atomization energies and <4.2 kJ·mol-1 for the totalenergies.

Both the optimized bond lengths and total energies of the molecules are in very goodagreement (AAD within “chemical accuracy”) with the reference implementation. Thedeviations of atomization energies are likely due to differences in the energies of thesingle atoms (for which no reference values were published). The one exception to theoverall good agreement is the H2 molecule, where the deviation from the deMon2k at-omization energy (around 23 kJ·mol-1) is twice as large as the others, and the total energydeviates by a factor of four. The cause of this discrepancy is somewhat elusive, but thismolecule may for the purposes of the validation of the functional implementation beconsidered an isolated outlier. The deviations are of approximately equal magnitudefor all four functionals, which additionally validates our implementation of VMT andVMTsol relative to the other (well-established) PBE and PBEsol methods. As the de-viations from the reference for both geometries and total energies show no systematictrend to over- or underestimated values, the differences in results can be predominantlyattributed to the general differences between the two codes, with the specific functionalimplementations equivalent.

We thus, within the error margins dictated by the use of two significantly differentDFT codes, consider the implementation of the VMT and VMTsol functionals success-fully validated.

62


Table B.1.: Optimized bond lengths (pm) for 10 molecules with the PBE[sol] and VMT[sol] functionals:values determined with ParaGAUSS (upper half of table) and deviation from deMon2k referenceimplementation (lower half)

H2 OH H2O CO N2 O2 NO NH3 CH4 HCNa

PBE 74.93 98.43 96.99 114.12 110.55 122.24 115.90 102.33 109.62 107.64 116.27VMT 74.80 98.44 97.01 114.27 110.67 122.46 116.07 102.37 109.67 107.65 116.40PBEsol 75.71 98.57 97.10 113.93 110.41 121.54 115.53 102.50 109.86 108.00 116.17VMTsol 75.76 98.59 97.12 114.07 110.53 121.53 115.60 102.57 109.98 108.12 116.32

PBE -0.08 -0.21 -0.22 0.03 0.02 0.07 0.08 -0.34 -0.05 0.20 0.04VMT -0.14 -0.25 -0.21 0.03 0.01 0.07 0.07 -0.33 -0.02 0.20 0.04PBEsol -0.07 -0.29 -0.23 0.00 0.00 0.07 0.05 -0.37 -0.11 0.10 0.01VMTsol -0.18 -0.30 -0.23 0.00 0.00 0.07 0.03 -0.36 -0.09 0.12 0.02

a left column: C–H bond, right column: C–N bond;

Table B.2.: Atomization energies (kJ·mol-1) for 10 molecules with the PBE[sol] and VMT[sol] func-tionals: values determined with ParaGAUSS (upper half of table) and deviation fromdeMon2k reference implementation (lower half)

H2 OH H2O CO N2 O2 NO NH3 CH4 HCN

PBE 442.6 453.0 973.7 1103.1 1007.2 579.0 701.9 1261.5 1754.9 1349.7VMT 444.1 447.3 961.6 1085.8 990.4 562.2 685.2 1247.9 1740.1 1329.7PBEsol 450.6 471.8 1019.3 1146.7 1035.0 629.5 739.9 1308.1 1813.3 1395.2VMTsol 440.1 461.4 999.5 1133.1 1017.2 619.4 725.8 1280.3 1783.1 1373.0

PBE -20.07 -9.71 -11.48 -10.45 -3.03 -11.93 -8.94 -4.04 -6.41 -6.74VMT -20.24 -9.92 -11.80 -11.12 -3.68 -12.48 -9.51 -4.56 -6.76 -7.40PBEsol -21.52 -9.53 -11.19 -10.14 -2.33 -12.00 -8.68 -2.88 -4.28 -5.43VMTsol -23.07 -9.69 -11.37 -10.55 -2.52 -12.34 -8.95 -3.27 -4.72 -5.80

63


H2

OH

H2O

CO

N2

O2

NO

NH

3C

H4

HC

N

PBE

-1.1

683

-75.

6879

-76.

3861

-113

.234

4-1

09.4

543

-150

.251

6-1

29.8

182

-56.

5155

-40.

4667

-93.

3480

VM

T-1

.173

0-7

5.71

76-7

6.41

54-1

13.2

813

-109

.501

3-1

50.3

048

-129

.868

3-5

6.54

30-4

0.49

29-9

3.39

29PB

Esol

-1.1

489

-75.

4290

-76.

1261

-112

.824

2-1

09.0

472

-149

.761

0-1

29.3

689

-56.

2906

-40.

2720

-92.

9733

VM

Tsol

-1.1

430

-75.

3921

-76.

0847

-112

.767

2-1

08.9

917

-149

.693

2-1

29.3

073

-56.

2529

-40.

2369

-92.

9198

PBE

-20.

14-3

.33

-5.1

5-0

.77

2.41

0.91

0.19

-1.4

3-3

.32

-0.8

1V

MT

-20.

30-3

.84

-5.7

3-1

.84

1.26

-0.2

4-0

.90

-2.1

9-3

.70

-1.7

9PB

Esol

-21.

94-2

.54

-4.4

00.

023.

012.

381.

17-0

.82

-2.1

5-0

.01

VM

Tsol

-23.

50-3

.16

-5.0

5-1

.19

1.67

1.18

-0.0

7-1

.83

-3.0

1-1

.32

Tabl

eB

.3.:

Tota

lene

rgie

sat

rela

xed

geom

trie

s(H

artre

es)

for

10m

olec

ules

with

the

PB

E[s

ol]a

ndV

MT[

sol]

func

tiona

ls:

valu

esde

term

ined

with

Par

aGAU

SS

(upp

erha

lfof

tabl

e)an

dde

viat

ion

from

deM

on2k

refe

renc

eim

plem

enta

tion

(low

erha

lf,in

kJ·m

ol-1

)

64

References

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Scalable Properties of Metal Clusters · 2.1. Transition Metal Clusters – Experiment, Theory, and Applications Metal clusters occupy a unique position between molecules and extended

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