Ab initio structures and polarizabilities of sodium clustersLeeor Kronik, Igor Vasiliev, Manish Jain, and James R. Chelikowsky Citation: The Journal of Chemical Physics 115, 4322 (2001); doi: 10.1063/1.1390524 View online: http://dx.doi.org/10.1063/1.1390524 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/115/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On the theoretical determination of the static dipole polarizability of intermediate size silicon clusters J. Chem. Phys. 117, 11158 (2002); 10.1063/1.1521761 Erratum: “Ab initio structures and polarizabilities of sodium clusters” [J. Chem. Phys. 115, 4322 (2001)] J. Chem. Phys. 115, 8714 (2001); 10.1063/1.1412232 An ab initio study of the monoxides and dioxides of sodium J. Chem. Phys. 109, 4267 (1998); 10.1063/1.477074 Ab initio pseudopotential calculation of the photo-response of metal clusters J. Chem. Phys. 106, 6039 (1997); 10.1063/1.473608 Ab initio calculations of Ru, Pd, and Ag cluster structure with 55, 135, and 140 atoms J. Chem. Phys. 106, 1856 (1997); 10.1063/1.473339

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Ab initio structures and polarizabilities of sodium clustersLeeor Kronik, Igor Vasiliev,a) Manish Jain, and James R. Chelikowskyb)

Department of Chemical Engineering and Materials Science, and Minnesota Supercomputing Institute,University of Minnesota, Minneapolis, Minnesota 55455

~Received 11 October 2000; accepted 14 June 2001!

We present quantitativeab initio calculations for Na cluster structures and polarizabilities, for allcluster sizes up to 20 atoms. Our calculations are performed by combining anab initiocore-corrected pseudopotential and a gradient-corrected density functional within a real spaceapproach. We find the cluster bonding to be very floppy and catalog a host of low-energyquasi-degenerate isomers for all second-decade clusters. The existence of these isomers results in aband of polarizability values for each cluster size even at zero temperature. This eliminates any finerstructure in the polarizability curve. We further show that the experimental polarizability values areconsistently underestimated by calculations at zero temperature. By computing the effects ofstructure expansion and distortion due to a finite temperature we arrive at a quantitative agreementbetween theory and experiment. ©2001 American Institute of Physics.@DOI: 10.1063/1.1390524#

I. INTRODUCTION

The physical and chemical properties of simple metalclusters have attracted considerable theoretical attention, ow-ing to the elementary valence electron structure of theseclusters.1,2 In particular, Na clusters appear to have receivedthe most scrutiny, because of the availability of a relativelylarge body of experimental data.2

The geometry of stable clusters is one of their most in-teresting properties and is often a prerequisite to an accuratecalculation of electrical or optical properties. Many authors,using models of various sophistication levels, have attemptedto determine ground state structures.3–18 Unfortunately, clus-ter structures are generally not amenable to direct experimen-tal determination. Thus, it is customary to test the obtainedstructures indirectly, by computing an experimentally acces-sible quantity that depends on structural properties.

One of the most straightforward experimental techniquesfor studying clusters is the measurement of their static dipolepolarizability, i.e., the change in cluster dipole with an ap-plied static electric field.1,19 Detailed experimental data forthe dependence of the polarizability dependence on the so-dium cluster size were provided by Knightet al. some 15years ago20 ~a second set of experimental data was reportedrecently6!. An early calculation, based on a jellium model,has correctly captured the ‘‘overall’’ nature of the experimen-tal polarizability curve, namely, its generally decreasing na-ture and its dips at ‘‘magic numbers’’ corresponding toclosed shell clusters.21 However, this important calculationdid not provide good quantitative estimates of polarizabilityvalues and failed to predict any finer aspects of the polariz-ability curve. This indicated that structural aspects are impor-tant in calculating polarizability, leading to many polarizabil-

ity calculations based on models which take the ionicstructure into account.4–6,15,17

Despite this great body of work, no attempt at a system-atic ab initio computation of cluster geometry and polariz-ability for all clusters in the range of continuous experimen-tal data ~namely, all clusters up to 20 atoms20! has beenreported. This is primarily due to computational difficulties.Rigorous quantum-chemical techniques, such asconfiguration-interaction calculations,3 are clearly limited tosmall clusters and do not allow for a search of ground statestructures based on molecular dynamics. A major simplifica-tion is afforded by density functional theory~DFT!, which isformally exact, but practically dependent on the quality of anapproximate exchange-correlation functional.4–6 In the ab-sence of further approximations, such calculations have, todate, not been able to exceed nine atom clusters. Other ap-proaches that are more suitable to larger clusters, such asmodel potentials,7 Huckel model calculations,8–10 orlocal,11,12 spherically13,14 or cylindrically15 averaged pseudo-potentials, are too approximate to provide any quantitative fitwith experimental data.

A natural way of retaining the accuracy level of DFTwhile alleviating much of the computational load is the useof ab initio pseudopotentials within the framework of a DFTcalculation.22 While several such studies have indeed beenperformed and have yielded many important and usefulresults,16–18 they fell short of quantitative agreement withexperiment even on the basic benchmark of the sodiumdimer bond length. More significantly, computed polarizabil-ity values17 fell short of an equivalent all-electron computa-tion using the same exchange-correlation functional. As aresult, it has been assumed that a pseudopotentials approachis not sufficiently accurate for reproducing polarizabilityvalues.4 Consequently, recent rigorousab initio studies haveexclusively used the computationally demandingall-electronDFT approach.4–6

Beyond computational difficulties, even the most accu-

a!Current address: Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801.

b!Electronic mail: jrc@msi.umn.edu

JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER 9 1 SEPTEMBER 2001

43220021-9606/2001/115(9)/4322/11/$18.00 © 2001 American Institute of Physics

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rate calculations consistently underestimate the experimentalpolarizability values, typically by as much as 15% to 20%.Recent studies have suggested that this discrepancy is due toa finite temperature effect on the experimental polarizabilityvalues.23–25

Here, we show thatab initio pseudopotentials can beincorporated within DFT calculations of structures and po-larizabilities at no inherent loss of accuracy and with a sig-nificant reduction in computational load. Furthermore, acombination of a gradient-corrected exchange-correlationfunctional and a core-corrected pseudopotential provides fora quantitativeagreement with experiment. We use this ap-proach to provide cluster geometries and a calculation of thepolarizability curve atT50 for all clusters up to 20 atoms.Finally, we show that the remaining difference between thetheoretical and experimental polarizability curves is due tothe effect of a finite temperature on cluster structures. Oncethis effect is incorporated in the calculations, a quantitativeagreement between theory and experiment is obtained.

II. COMPUTATIONAL APPROACH

A. The real-space pseudopotential DFT method

Our computational approach is based on employingabinitio pseudopotentials within density functional theory.26

Specifically, we solve the Kohn–Sham equations in the form:

S 2¹2

21(

RW a

Vps~rW2RW a!1VH~rW !1Vxc@rc~rW !# Dcn~rW !

5Encn~rW !. ~1!

Atomic units are used throughout unless otherwise noted. InEq. ~1!, the true potential of each ion situated atRW a is re-placed by a pseudopotential,Vps(rW2RW a), which accounts forthe core electrons of each Na atom.En and cn(rW), the nthKohn–Sham single-electron eigenvalue and eigen-wave-function, respectively, pertain tovalence electrons only.VH(rW) is the Hartree~electron-electron! potential.r(rW) is thevalence charge density, given by

r~rW !52 (n,occup

ucn~rW !u2, ~2!

where the summation is over all occupied orbitals.rc(rW) isthe partially core-corrected charge density~discussed later!,andVxc@rc(rW)# is the exchange-correlation potential.

The quantitative success of the computation dependscritically on the quality of the specific exchange-correlationfunctional and the pseudopotential used. For the exchange-correlation functional, we tested both the local density ap-proximation ~LDA !, i.e., Vxc@rc(rW)#>Vxc„r

c(rW)…, and thegeneralized gradient approximation~GGA!, i.e., Vxc@rc(rW)#>Vxc„r

c(rW),¹rc(rW)…. We used the LDA functional of Cep-erley and Alder, as parametrized by Perdew and Zunger,27

and the GGA functional of Perdew, Burke, and Ernzerhof.28

For theab initio pseudopotential, we used the Troullier–Martins recipe.29 In this approach, the pseudo-wave-functions differ from the all-electron ones only inside a givencore radius. Inside this core, the pseudo-wave-functions arereplaced by a 12th-order even polynomial that maintains

continuity of the wave-function and its first four derivatives,a zero curvature at the origin, and a conservation of the normof the original wave function.

As the exchange-correlation functional is generally non-linear, use of the valence charge alone may result in error ifthere is a significant overlap between the core and valencecharge densities, requiring a nonlinear correction.30 This cor-rection was demonstrated to be very important for accuratelyreproducing the results of all-electron DFT calculations for avariety of atoms and small molecules,31 and for sodium inparticular.18,30,31 Here we follow the approach of Louieet al.,30 who corrected the charge density used for evaluatingthe exchange-correlation potential by adding afixed ‘‘par-tial’’ core density,rcorr

c ~i.e., a density that isnot subject toiterations for self-consistency!, in the form

rc~rW !5r~rW !1(RW a

rcorrc ~ urW2RW au!, ~3!

where

rcorrc ~ urW2RW au!5H A sin~BurW2RW au!/urW2RW au, r ,r 0 ,

rc~rW2RW a!, r>r 0 ,~4!

rc(rW2RW a) is the core charge density of a single atom, andr 0

is a parameter that is usually the radius around the singleatom at which the core charge density is between one andtwo times larger than the single atom valence charge density.

Two bases are often used for localized systems of thetype studied here. A plane-wave basis requires a very largesuper-cell, most of which is vacuum, in order to minimizespurious cluster-cluster interactions. Use of a Gaussian basisavoids this difficulty, but requires a proper choice of a basisthat would ensure convergence. This is a nontrivial task,even if ultimately successful, especially when a proper ac-count of polarized wave-functions~i.e., subjected to an elec-tric field! is required.5,6

In order to avoid such difficulties, we solve Eq.~1! di-rectly in real space, by using a high-order finite-differenceexpansion for the Laplacian operator and solving for the val-ues of the wave-functions on a three-dimensional grid. Theboundary condition is introduced by placing the grid inside alarge spherical domain and demanding a zero wave-functionoutside this domain. The Kohn–Sham equations are there-fore discretized:32

21

2h2 F (n152N

N

Cn1c~xi1n1h,yj ,zk!1 (

n252N

N

Cn2c~xi ,yj

1n2h,zk!1 (n352N

N

Cn3c~xi ,yj ,zk1n3h!G

1F(a

Vpsa @xi8 ,yj8 ,zk8#1VH~xi ,yj ,zk!

1Vxc~xi ,yj ,zk!Gc~xi ,yj ,zk!5Ec~xi ,yj ,zk!, ~5!

wherexi , yj , zk are the discrete grid coordinates,h is thegrid spacing,Cn are the coefficients of the finite difference

4323J. Chem. Phys., Vol. 115, No. 9, 1 September 2001 Ab initio structures and polarizabilities of sodium clusters

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expansion,N is the order of the expansion~typically 6 toassure convergence32!, and Vps

a @xi8 ,yj8 ,zk8# is the discreteform of the pseudopotential operator corresponding to theath atom. Solving Eq.~5! is tantamount to solving theKohn–Sham equations@Eq. ~1!# without a specific basis~or,formally, on a basis of Dirac delta functions!. Convergence istrivially assured by decreasing the grid spacing,h, and in-creasing the boundary sphere radius,R, until changes in thecomputed quantities are within the desired accuracy level. Astarting point forh can be obtained by considering an iso-lated atom.

For Vpsa , we cast the pseudopotential in itsreal space

Kleinman–Bylander form~written in nondiscrete form fornotational convenience!:32,33

Vpsa ~rW2RW a!c~rW !5Vloc

a ~rW2RW a!c~rW !

1(l ,m

Glma f lm

a ~rW2RW a!

3@v la~rW2RW a!2v loc

a ~rW2RW a!#, ~6!

where Vla(rW2RW a) is the pseudopotential component corre-

sponding to thelth angular momentum,Vloca (rW2RW a) is one

specific angular momentum component, chosen as the localone, f lm

a (rW2RW a) is the lth radial atomic pseudo-wave-function, multiplied by thelmth spherical harmonic, and theprojection coefficientsGlm

a are calculated as

Glm5^f lmuVl2Vlocuc&

^f lmuVl2Vlocuf lm&. ~7!

One advantage of casting the pseudopotential in the realspace Kleinman–Bylander form is thatVl2Vloc differs fromzero only inside the pseudopotential core radius. Conse-quently, the nonlocality of the operator is limited to a smallregion around each atom. The only other nonlocal contribu-tion to Eq.~5! is due to the finite-difference approximation ofthe Laplacian operator, but it too extends only to a fewneighbors around each grid point. Consequently, the matrixrepresentation of Eq.~5! is very sparseand highly effectiveiterative diagonalization procedures34 can be employed.Here, the generalized Davidson method was used.35

We note in passing that for calculations of bulk Na prop-erties, aimed at testing the quality of our approximations~seelater in this work!, a standard plane-wave-based calculation22

was employed. It used thesameexchange-correlation func-tionals and pseudopotential constructs as in the real spaceanalyses.

B. Computation of structures and polarizabilities

For structural minimization procedures, we have com-puted the force acting on atoma, FW a , by using the Hellman–Feynman theorem.36 Simply put, this theorem states thatwhen taking the derivative of the total energy with respect toatomic positions to obtain forces, onlyexplicit dependenceon coordinates needs to be taken into account. This providesfor another reason for preferring a real-space grid over aGaussian basis set—force contributions due to the explicitdependence of the basis set on the atomic position~known asPulay forces37! are inherently eliminated in our approach.

Within the formalism presented earlier, the total groundstate energy is given by32,38

Etot5T@r#1(RW a

E r~rW !Vloca ~rW2RW a! dr

1 (lm,RW a ,n

^f lma uVl2Vlocuf lm

a &@Glma,n#21EH@r#

1Exc@rc~$RW a%!#1Ei 2 i~$RW a%!. ~8!

In Eq. ~8!, T@r# is the kinetic energy of electrons. The nexttwo terms correspond to the contribution of the local andnonlocal pseudopotential components, respectively, to theelectron-ion energy, whereGlm

a,n are the projection coeffi-cients of Eq.~7!, with the indexn running over all occupiedelectronic states.EH@r# is the Hartree~electron-electron! en-ergy, Exc@rc($RW a%)# is the exchange-correlation energy, andEi 2 i($RW a%) is the ion–ion interaction energy, where$RW a%denotes the ensemble of all atomic coordinates.

Application of the Hellman—Feynman theorem to Eq.~8! then yields

FW a52E r~rW !]Vloc

a ~rW2RW a!

]RW a

dr

22(lm,n

^f lma uVl2Vlocuf lm

a &]Glm

a,n

]RW a

2]Ei 2 i

]RW a

2E d3r Vxc@rc~rW !#]rcorr

c ~rW2RW a!

]RW a

. ~9!

It is worth noting that the last term in Eq.~9! resultsdirectly from the use of a partial core correction, due to theintroduction of an explicit dependence of the exchange-correlation functional on atomic positions via the atom-centered core charge densities.38 The core-correction forceterm given here is identical to the one provided by Dal Corsoet al. within the LDA approach.38 In the Appendix, we showthat this expression is in fact valid forany exchange-correlation functional. It was therefore used for GGA calcu-lations as well.

For clusters of nine atoms or more, local minima asso-ciated with metastable cluster configurations were avoidedby performing a systematic search of low-energy clusters.This was achieved via simulated annealing computationsbased on Langevin molecular dynamics.39 In this approach,the atomic positions,RW a , evolve according to the Langevinequation:40

MaRa5FW a2gMaRa1WW a , ~10!

whereFW a is the inter-atomic force acting on theath atom,Ma is its nuclear mass,g is a friction coefficient which de-termines the dissipation rate, andWW a are stochastic forcesdefined as random Gaussian variables with a white noisespectrum,

^Waj~ t !&50, ^Wa

j~ t !Wbj~ t8!&52gMakBTdabd~ t2t8!,

j5$x,y,z%, ~11!

4324 J. Chem. Phys., Vol. 115, No. 9, 1 September 2001 Kronik et al.

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and the angular brackets in Eq.~11! denote ensemble or timeaveraging.

In our simulations, we lowered the temperature gradu-ally from 1450 K to 150 K. Clusters corresponding to dis-tinct minima in the energy versus annealing step curve werefully relaxed by moving the atoms along the force lines. Thefinal relaxation was performed with stringent grid spacingand boundary sphere parameters of 0.37 a.u. and up to 21a.u., respectively, assuring convergence of bond lengths tobetter than60.02 a.u. This procedure generally resulted innumerous nonequivalent cluster geometries and high-energyisomers were discarded.

For polarizability calculations, we have used the finitefield method.41 In this approach, a term corresponding to anexternal electric field is added to the Kohn–Sham Hamil-tonian in the form

S 2¹2

21Veff@rW#2eW•rW Dc~rW !5Ec~rW !, ~12!

where the effective potential,Veff@rW#, is the sum of thepseudopotential, Hartree, and exchange-correlation terms ofEq. ~1! andeW is the applied electric field. The dipole momentof the system,mW (eW ), is given by

mW ~eW !5E r~rW !→r~rW,eW !rW dr . ~13!

The components of the polarizability tensor,agd , aredefined as

agd5]mg~eW !

]ed52

]2Etot~eW !

]eg]ed, g,d5$x,y,z%, ~14!

whereEtot is the total energy of the system. The value mea-sured in experiments is the average polarizability,^a&, givenby

^a&5axx1ayy1azz

35

1

3tr~agd!. ~15!

Only the diagonal elements of the polarizability tensor needto be calculated, and the rotational invariance of the traceoperation assures that the result is coordinate independent, asappropriate.

The diagonal tensor elements,agg , are found by calcu-lating eitherE(eW ) or mg(eW ), at eW50 and eW56Deg , andevaluating either derivative in Eq.~14! using a finite-difference expression. In agreement with previous work,41

we found that for polarizability calculations, a larger bound-ary sphere~of 25 to 27 a.u.! was needed to assure conver-gence. However, a somewhat larger grid spacing of 0.4 a.u.could be used. The polarizability values presented below areaccurate to better than60.1 Å3/atom.

Experimental polarizability values at a finite temperatureinclude an additional contribution from dipole rotation in theexternal electric field. In the high-temperature or low-fieldlimit, the effective measured polarizability,aeff , becomes41

aeff5^a&1m2

3kT. ~16!

All high-temperature polarizability values given later in thiswork were corrected according to Eq.~16!.

III. CLUSTER STRUCTURES

We start by testing the suitability and accuracy of ourcomputational approach. To that end, we computed threequantities for which accurate experimental data are available:the Na dimer bond length, and the Na bulk lattice parameterand modulus. A comparison between the experimental val-ues, our computed values, and previous LDA and GGA val-ues reported in the literature, is given in Table I.42–46

Analyzing the LDA results of the Na dimer we find, inagreement with Ro¨thlisberger and Andreoni,18 that the core-corrected pseudopotential results agree with the all-electronones to within;1% at most. However, in the absence ofcore-correction the pseudopotential results underestimate theall-electron ones by;2.5% to 3%. Interestingly, the resultsof Martins et al.16 and Moulletet al.17 fall short of the all-electron ones by;4%, indicating a problem with the specific

TABLE I. GGA and LDA calculations of several experimentally known Na quantities: the dimer bond length,the bulk lattice parameter, and the bulk modulus. ‘‘all-e’’—all electron, ‘‘c-ps’’—core corrected pseudopoten-tial, ‘‘n-ps’’—non-core-corrected pseudopotential.

Quantity Experiment

GGA LDA

all-e c-ps all-e c-ps n-ps

Na2 bondlength ~a.u.!

5.82a 5.84,b 5.85c 5.83d 5.71,b 5.70c 5.66,d 5.68e 5.55,d 5.56e

5.47,f 5.48g

latticeparameter~a.u.!

7.98h¯ 7.90,d 7.92i 7.71j 7.66d,i

7.63,k 7.72l7.54,d 7.53i

7.59l

bulkmodulus~GPa!

6.8h¯ 7.5,d 7.7i 9.0j 9.1d,i

9.6,k 9.5l8.7d,i

aReference 42. gReference 16.bReference 4. hReference 43.cReference 5. iReference 44.dThis work. jReference 45.eReference 18. kReference 46.fReference 17. lReference 30.

4325J. Chem. Phys., Vol. 115, No. 9, 1 September 2001 Ab initio structures and polarizabilities of sodium clusters

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choice of pseudopotential, or its implementation, in thesecalculations. We find the same trends in the computation ofthe bulk lattice parameter, in agreement with the results ofRefs. 30, 44, and 46.

The essential problem with a LDA approach is that eventhe all-electron calculations differ significantly from the ex-perimental values—the dimer bond length and the bulk lat-tice constant are underestimated by;2% and by;3.4%,respectively, and the bulk modulus is overestimated by;32.4%. This discrepancy is significantly reduced by usingGGA. In our GGA calculations, the dimer bond length agreeswith experiment to an insignificant;0.1%, the bulk latticeconstant is underestimated by only;1%, and the bulkmodulus is overestimated by;10%. Our dimer results agreeto better than;0.5% with those of all previous all-electroncalculations.4–6 For the bulk we are not aware of any all-electron GGA calculations. However, our results are inagreement with the pseudopotential results of Kanget al.44

The results of Table I establish the combined use of agradient-corrected exchange-correlation functional and acore-corrected norm-conserving pseudopotential for aquan-titative analysis of Na structures. It is the accuracy of theformer and the simplicity of the latter~especially when

implemented in real space! that facilitates this study. To thebest of our knowledge, this is the first comprehensive andquantitative study of all Na clusters up to 20 atoms.

Ground state structures found for Na3 to Na8 clusters areshown in Fig. 1. Some additional low-lying isomers of Na3

and Na6 are given in Fig. 2, together with their calculatedenergy difference from the ground state. The geometry ofthese structures is in excellent agreement with that found byprevious all-electron calculations.4,5 To summarize theseknown structures briefly, we found an obtuse triangle forNa3 , a planar rhombic structure for Na4 , a planar trapezoidalshape for Na5 , a ‘‘close to planar’’ pentagonal pyramid forNa6 , a bi-capped pentagonal pyramid for Na7 , and aD2d

structure for Na8 . For Na3 , where the equilateral triangle isJahn–Teller unstable, the potential energy surface is veryflat. The obtuse distortion is more energetically stable thanthe linear one and the acute one by only;15 and;50 meV,respectively. Na6 marks the transition from planar structuresto three-dimensional ones and the planar isomer is higher inenergy than the three-dimensional one by only;35 meV. Wefind the agreement with all-electron calculations for Na5 tobe particularly noteworthy, as previous pseudopotential

FIG. 1. Ground state geometries for Na3 to Na8 clusters. Letters denotedimensions detailed in Table II.

FIG. 2. Low-energy isomers for Na3 and Na6 clusters. Numbers denote totalenergy differences~per entire cluster! in meV.

TABLE II. GGA and LDA calculations of bond lengths for Na3 to Na8 ground state clusters. All lengths givenin a.u. Bond labels refer to Fig. 1. ‘‘all-e’’—all electron, ‘‘c-ps’’—core corrected pseudopotential, ‘‘n-ps’’—non-core-corrected pseudopotential.

Bond

Reference 5—all-e Reference 4—all-e This work—c-ps Reference 17—n-ps

GGA LDA GGA LDA GGA LDA LDA

Na3 a 6.17 5.99 6.13 5.96 6.11 5.96 5.68b 8.62 7.99 8.50 8.09 8.49 8.11 7.15

Na4 a 6.69 6.50 ¯ 6.48 6.58 6.46 6.18b 5.86 5.72 ¯ 5.68 5.84 5.68 5.42

Na5 a 6.45 6.30 ¯ 6.23 6.43 6.22 6.04b 6.54 6.31 ¯ 6.20 6.46 6.25 6.03c 6.70 6.48 ¯ 6.43 6.60 6.50 6.01d 6.75 6.48 ¯ 6.49 6.67 6.52 6.36

Na6 a 6.84 6.58 ¯ 6.58 6.76 6.57 6.27b 6.39 6.22 ¯ 6.19 6.33 6.18 5.89

Na7 a 6.65 6.45 ¯ ¯ 6.61 6.43 6.10b 6.71 6.49 ¯ ¯ 6.64 6.47 6.02

Na8 a 6.99 6.73 ¯ ¯ 6.92 6.75 6.76b 6.36 6.47 ¯ ¯ 6.29 6.14 5.86c 6.68 6.95 ¯ ¯ 6.62 6.43 ¯

d 6.98 6.71 ¯ ¯ 6.92 6.76 ¯

e 6.37 6.18 ¯ ¯ 6.30 6.15 ¯

4326 J. Chem. Phys., Vol. 115, No. 9, 1 September 2001 Kronik et al.

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calculations16,17 have indicated a more symmetric geometry,where the right and left triangles are almost equilateral.

The characteristic bond lengths of the Na3 to Na8 struc-tures obtained in this work, together with bond lengths ob-tained in previous DFT calculations, are given in Table II. Asdemonstrated earlier for the Na dimer and bulk, Table IIshows that bond lengths obtained using GGA are larger thanthose obtained using LDA. Use of a non-core-correctedpseudopotential17 significantly underestimates the all-electron bond lengths. It also disagrees with the computedisomer energy differences.

For clusters computed using all-electron DFT by bothGuanet al.4 and Calaminiciet al.5 ~up to Na6 for LDA andNa3 for GGA!, the two computations agree semi-quantitatively but differ by about 1% to 1.5% in the bondlength. Our bond lengths~as well as energy differences be-tween isomers! appear to be in better quantitative agreementwith those of Guanet al.4 The residual differences are smalland may easily result from the use of different exchange-correlation functionals and different basis sets.

For cluster sizes beyond Na8 , many quasi-degenerateisomers exist.5,17,18 Typical geometries of quasi-degenerateisomers obtained from molecular dynamics, for all clustersfrom Na9 to Na20, are given in Fig. 3. Relative energy dif-ferences between clusters of equal size are also given. Some

structures, differing only by a slight distortion or a minorchange in the position of capping atoms, were omitted forclarity. Our results agree generally with an earlier calculationby Rothlisberger and Andreoni that employed a similarapproach.18 However, our work supersedes that earlier calcu-lation in two aspects:~a! By using GGA, it facilitates a morerealistic comparison with experiments~see the discussion ofpolarizabilities to follow!. ~b! By employing a computation-ally efficient real-space approach~and no doubt by enjoyingthe advances of computer hardware since this earlier work!,we probeall the second decade of clusters, and not just a fewspecific clusters—Na13, Na18, and Na20.

As with any molecular dynamics computation, thesearch for quasi-degenerate isomers is by no means exhaus-tive. Nevertheless, the consistency of the results obtained forall clusters suggests that the existence of an isomer that ismuch lower in energy than those reported here is quite un-likely. For example, Calaminiciet al.5 reported aC2v bi-capped pentagonal pyramid as the ground state one for Na9

~center isomer of Na9 in Fig. 3!. While our calculations didreveal an isomer of lower symmetry and energy~left isomerof Na9 in Fig. 3!, it was lower in total energy by only 27meV.

In agreement with prior work,8,9,18 we find that many ofthe obtained low-energy clusters feature ‘‘motifs’’ of pen-

FIG. 3. Quasi-degenerate low-energy isomers for Na9 to Na20 . Numbers denote GGA-calculated energy differences per atom, in meV, with respect toleft-most isomer in row.

4327J. Chem. Phys., Vol. 115, No. 9, 1 September 2001 Ab initio structures and polarizabilities of sodium clusters

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tagonal symmetry—a type of symmetry that must clearly dieout for very large clusters as it is absent in the bulk—but isstill prevalent even for Na20. Many clusters can be thoughtof as composed of two or more smaller cluster shapes. Ob-servation of ‘‘motifs’’ that are incompatible with bulk bond-ing for clusters of this size, or of a combination of smallercluster shapes, is not unique to sodium clusters and has beenobserved in computations of e.g., silicon clusters.47 In manycases~see, e.g., Na11, Na12, or Na15!, ‘‘motifs’’ resemblinghexagonal symmetry~though not formally possessing it!were also observed. We note that Ro¨thlisberger andAndreoni18 hypothesized that closed-shell clusters may pos-sess fewer quasi-degenerate low-energy isomers than open-shell ones. Our comprehensive study of all second-decadeclusters does not seem to support this hypothesis, as no trendof quasi-degenerate isomer abundance with shell structurewas discerned.

The obtained clusters exhibit very ‘‘floppy’’ bonding.Many ~though not all! low-energy isomers lack any overallsymmetry and are distorted, i.e., show a variation of bondlength. These traits are clearly due to the weak, and weaklydirectional, nature of the Na–Na bond.

An interesting consequence of this ‘‘floppy’’ nature isthat computations using model potentials may fail even inthe qualitative prediction of correct low-energy clustershapes. A case in point is Na13, where Huckel calculations10

~as well as LDA calculations not based on moleculardynamics16! predicted the highly symmetric cubo-octahedronas the correct ground state. However, both our results andthose of Ref. 18 suggest that the cubo-octahedron is actuallyhigher in total energy by;0.5–0.6 eV than the isomersshown in Fig. 3. A similar situation is found in the recentwork of Kummel et al., based on cylindrically averagedpseudopotentials,15 even though the latter approximation issignificantly more sophisticated than Hu¨ckel based calcula-tions. One of the low-energy structures predicted in thatwork for Na12 is the pentagonal bi-pyramid. We tested thisstructure explicitly and found that it is higher in total energyby ;0.6–0.7 eV than the isomers shown in Fig. 3. We con-clude that for such ‘‘floppy’’ clusters the use of model poten-tials, which assume certain symmetries, should be viewedwith caution.

IV. CLUSTER POLARIZABILITIES

As before, we start by testing the suitability and accu-racy of our computational approach. To that end, we compareour LDA and GGA results for clusters up to nine atoms withthe all-electron LDA and GGA calculations of Calaminiciet al.,5 the all-electron GGA calculations of Rayaneet al.,6

the pseudopotential LDA calculations of Moulletet al.,17 andthe experimental data of Knightet al.20,48 The results areshown in Fig. 4. To facilitate the comparison with previouscalculations, theC2v isomer of Na9 ~center isomer of Na9 inFig. 3! was used for this computation.

Using Fig. 4, it is readily apparent that ourpseudopotential-based results are inexcellentagreement withthe all-electron results of Ref. 5, for both LDA and GGA.Again, this confirms that our use of pseudopotentials resultsin no inherent loss in accuracy. We find the contributions of

the core electrons to the polarizability, often invoked to ex-plain the discrepancy between all-electron and pseudopoten-tial calculations,4 to be negligible. Simple calculations showthat ~for three-dimensional geometries! the cluster polariz-ability should roughly scale as the volume of the cluster.1

Therefore, we find that the reason that the LDA pseudopo-tential calculations of Ref. 17 underestimated the all-electronLDA values actually lies in their significant underestimate ofbond lengths~see Table II!. Indeed, this underestimate ofpolarizabilities is absent for the single Na atom.17 This com-putational demonstration of the insignificance of core polar-ization is consistent with the recent experimental findings ofRayaneet al.6 These authors found that the polarizabilities ofLi and Na atoms are very similar and that the polarizabilitiesof Li clusters are smaller than that of Na clusters of equalsize by a factor generally commensurate with the shorterLi–Li bond length.

As in the computation of structural parameters, bothLDA and GGA are in qualitative agreement with experiment,but GGA allows for a significantly improved quantitativeagreement. This improved agreement is not solely due to theextended bond lengths reported in Tables I and II, because itis apparent even for the single atom. Instead, it highlights theimportance of gradient corrections in correctly capturing thenature of exchange-correlation in Na atoms. We note that asopposed to bond lengths, for polarizabilities the agreementbetween theory and experiment is still not satisfactory—anissue discussed further below.

It is interesting to note that for the single atom, our po-larizability values are about 0.7 Å3 higher than those ob-tained in Ref. 5 for both LDA and GGA calculations. Forlarger clusters, this effect is offset by that of the somewhatlonger bond lengths found in Ref. 5, resulting in a very closeagreement between the polarizability values in the two cal-culations. The polarizability values obtained in Ref. 6 appear

FIG. 4. A comparison of various theoretical calculations for the polarizabil-ity of Na clusters up to 9 atoms: diamonds—experimental data, after Ref.20; down triangles—all-electron~‘‘all-e’’ ! GGA calculations, after Ref. 6;empty circles and squares—all-electron GGA and LDA calculations, respec-tively, after Ref. 5; filled circles and squares—core-corrected pseudopoten-tial ~‘‘c-ps’’ ! calculations, respectively, this work; up triangles—non-core-corrected pseudopotential~‘‘n-ps’’ ! LDA calculations, after Ref. 17.

4328 J. Chem. Phys., Vol. 115, No. 9, 1 September 2001 Kronik et al.

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to be several percent higher than our own and of those givenin Ref. 5. Perhaps this is due to the different gradient-corrected functional used by these authors, as the bondlengths computed with this functional49 are somewhat largerthan those found in this work and in Ref. 5.

The above discussion establishes the validity of thepseudopotential approach and the superiority of GGA toLDA for calculating polarizabilities. Therefore, we have pro-ceeded to compute the polarizabilities of all quasi-degenerateisomers found~energy differences less thankT!, a selectionof which was given in Fig. 3, using GGA-based pseudopo-tentials only. The resultingT50 polarizability curve, com-pared with the experimental ones, is given in Fig. 5~incircles!. For ease of comparison, we have divided this figureto two parts. The first shows only the older experimental dataof Knight et al.,20 including error bars. The second showsboth experimental data sets6,20 and omits error bars.

Both general trends of the experimental curve—an over-all decreasing polarizability with increasing cluster size andsignificant dips at ‘‘magic’’ numbers—are reproduced in the

theoretical curve of Fig. 5. However, as mentioned earlier,these rudimentary trends were reproduced even by a jelliumcalculation.21 The two novel findings of ourab initio T50curve are as follows.~a! A band of polarizability values,rather than a single value, is found for the second decade ofclusters, so that the ‘‘fine structure’’ of the polarizabilitycurve for the second decade of atoms isnot reproduced theo-retically. ~b! The theoretical results still underestimate theexperimental data consistently~throughout the entire sizerange studied! and significantly~by sometimes as much as20%!, just like the early jellium calculations, despite the useof a first principlesapproach. We explain both findings, start-ing with the latter.

At first glance, one may argue that even though struc-tural properties have been quantitatively reproduced usingGGA, apparently this gradient correction is still not sufficientfor polarizability computations and more elaborate densityfunctionals need to be used. However, we do not support thisview. This is because a close inspection of Figs. 4 and 5reveals that the theoretical result for the single atom is inexcellent agreement with experiment. This indicates thatGGA does capture the essential characteristics of the polar-ized wave function—making an inadequate density func-tional unlikely.

A more reasonable explanation has to do with the role afinite temperature plays in determining cluster structures. Be-cause the experiment is thought to be conducted at manyhundreds of degrees~K! ~the exact temperature of the clusterbeing very difficult to measure directly!,1 one may expect asignificant deviation of experimental values from thosefound from a ground state computation. Several studies haveindicated that, as is to be expected, a finite temperature dis-torts the cluster shape in general, and in particular extendsthe average bond length.8,18,23–25As explained earlier, thisincrease in bond length increases the average cluster volumeand therefore increases the polarizability. The importance ofthis ‘‘structural’’ temperature effect is underscored by thecomparison between theT50 theory and the experiment inFig. 5. Even though a perfect agreement with experiment isobtained for Na1 and only a small discrepancy is found forNa2 , a significant gap between theory and experiment devel-ops from Na3 on, i.e., as soon as the geometry of the clusterstarts playing a significant role in the polarizability. As thecluster gets larger, the impact of an increase in average bondlength on cluster volume~and hence polarizability! shouldincrease. Indeed, the discrepancy between theory and experi-ment in Fig. 5 gradually increases, on average~percentage-wise!, from about 8% to almost 20%, further supporting thisexplanation. An additional hint in favor of this structural ex-planation is that for the Na trimer, both our calculations andthose of Ref. 5 show that the polarizability of the obtusetrimer underestimates the experimental value by over 10%.However, the polarizability of the linear trimer underesti-mates the experiment only 2% to 3%. At a reasonable finitetemperature the linear geometry would be ‘‘sampled’’ andcontribute to the measured value. This effect would explainmuch of the discrepancy between theory and experiment.

Ab initio LDA calculations by Ku¨mmel et al.23 haveshown that this simple structural effect can indeed account

FIG. 5. Polarizabilities of Na clusters; theory versus experiment.~a! Com-parison with Ref. 20, including experimental error bars.~b! Comparisonwith Refs. 20 and 6. Circles—theory atT50 K ~dashed lines are a guide tothe eye!; stars—theory atT5750 K; filled and empty diamonds—experimental data of Refs. 20 and 6, respectively.

4329J. Chem. Phys., Vol. 115, No. 9, 1 September 2001 Ab initio structures and polarizabilities of sodium clusters

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quantitatively for the discrepancy between theory and experi-ment for Na8 and Na10, which are close in number of atoms.Blundell et al.24 have used a simpler extended Thomas–Fermi model for confirming this prediction for~within therange of clusters studied in the present work! the closed shellstructures of Na8 and Na20. Still, one could argue that asboth LDA and Thomas–Fermi approaches underestimate thepolarizability of even a single atom, the structural effectmerely compensates for the initial error, with the temperaturebeing used as ade factofitting parameter. An unequivocalconfirmation of the temperature-modified structure assump-tion requires that an approach yielding the correct singleatom polarizability~namely, GGA! be used. It further re-quires thatone temperaturequantitatively account for thedifference between theory and experiment over a wide rangeof cluster sizes and shell structures.

For attempting such a quantitative fit, we have per-formed Langevin molecular dynamics39 at a fixed tempera-ture of 750 K ~i.e., at the upper limit of the experimentalrange! for all cluster sizes studied. For each cluster size, wesampled ten select cluster structures, obtained at regulartime-step intervals from a fully thermalized molecular dy-namics run. For each of these we ‘‘froze’’ the nuclear coor-dinates and computed the average polarizability using thesame procedure described above. The rigor of such a ‘‘quasi-Born-Oppenheimer’’ approach has been debated before,50

but it is clearly a very useful approximation, used to testwhether the effect of temperature on structural properties isindeed the dominant one here.

The high-temperature polarizabilities thus obtained@in-cluding the dipole correction of Eq.~16!# are also shown inFig. 5. The stars denote the average of the polarizability val-ues obtained and the ‘‘theoretical error bars’’ denote the stan-dard deviation. Figure 5~b! shows that our high-temperaturetheoretical values are in consistentquantitative agreementwith the experimental data of Knightet al.20 ~to within theexperimental error and the theoretical sampling error!.Therefore, we confirm that the most simple picture of tem-perature effects suffices to account for the discrepancy be-tween theory and experiment. The fact that the theoreticalpolarizability values may either slightly underestimate orslightly overestimate the experimental values~as opposed tothe ‘‘customary’’ consistent underestimate! also suggests thatour calculations did not fail to take into account any majormechanism contributing to the polarizability. Thus, the high-temperature calculations given here agree with, significantlygeneralize, and strongly support those of Refs. 23–25.

We wish to stress that we do not claim that the Langevintemperature of 750 K must strictly correspond to the actualexperimental temperature. Such a claim clearly requires atest of numerous higher and lower temperatures and a de-tailed analysis of the sensitivity of the polarizability curve tothe exact temperature used in the simulations. Regardless,the dominant role of temperature-induced structural modifi-cations in Na cluster polarizabilities is clear-cut. Interest-ingly, Rayaneet al.6 reported a good agreement betweentheory and experiment for some of the smaller clusters evenat zero temperature. In light of Fig. 5, this again leads us to

believe that their zero temperature GGA polarizability val-ues, shown in Fig. 4, are too large.

We now turn to addressing the appearance of fine struc-ture, and especially characteristic even–odd oscillations, inthe experimental data, but not in the theoretical results. Odd–even oscillations have also been observed experimentally in,e.g., ionization-potential curves of Na atoms, and explainedsuccessfully even with a relatively simple triaxially de-formed jellium droplet model.51 Our calculations, however,indicate aband of polarizability values, masking any sub-structure in the polarizability curve, at both high and lowtemperature. Moreover, we find a smooth, gradual change ofpolarizability values throughout the second decade of atoms,not only at a high temperature, but even atT50! It is alsoworthwhile to note that Ku¨mmelet al.15 also reported differ-ent polarizability values for different isomers of even-numbered clusters. However, they attributed significance tothe polarizability trends of thelowest energyisomer foundfor each cluster size and used these trends to compare thetwo sets of experimental data. In our opinion, this depen-dence cannot be compared to experiment because a host ofquasi-degenerate isomers will be sampled in the experiment.We expect that any fine structure will be ‘‘smeared out’’ bythe effective averaging over different cluster shapes.

A detailed comparison of the two polarizability experi-ments is shown in Fig. 5~b!. The two experimental sets differgreatly in the magnitude of the odd–even oscillations~theyare much more pronounced in the newer data set!.6 However,they differqualitativelyas well–they disagree on thesignofthe odd–even transition~namely,aeven.aodd or aodd.aeven!going from, e.g., Na9 to Na10 or Na17 to Na18. This indicatesthat in this case the fine structure may have as much to dowith experimental conditions as with theoretical consider-ations. Interestingly, theoretical calculations for GaAs clus-ters also did not reproduce the odd–even polarizability oscil-lations observed in experiment.41

Kummel et al.15 recently analyzed the potential pitfallsof the experiment. They correctly pointed out that we usuallyassume that the experiment measures the unbiasedpolarizabilities—averaged over all present cluster shapes andorientations, but that this is not necessarily realized in ex-periment. Cluster polarizabilities are typically measured us-ing a static, transverse inhomogeneous electric field.1,6,20 Adetailed discussion of experimental techniques is clearly out-side the scope of the present text. Nevertheless, there aremany aspects of the cluster selection and deflection processthat could favor clusters of a given geometry and/or cause apreferential alignment of clusters. Such deviations from the‘‘true’’ average polarizability could easily result in odd–evenoscillations. We therefore believe that the partial disagree-ment between the two experimental sets of data, as well asthe absence of between them and theory, merits further ex-perimental attention.

Our theoretical results agree more closely with the olderexperimental data set20 than with the new experimental dataset.6 This is mostly because the fine structure is much smallerin the old data and can generally be considered to be of theorder of the experimental accuracy, whereas in the newerdata it is much more pronounced and significant.

4330 J. Chem. Phys., Vol. 115, No. 9, 1 September 2001 Kronik et al.

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We also note that in ionization potential curves, high-temperature effects tend to reduce the odd–evenoscillations.51 Therefore, the difference between the two ex-periments may have been attributable to a lower effectivetemperature in the newer one. However, the lack of suchoscillations even at theT50 theoretical curve points againstsuch an interpretation.

Finally, we note that while the simplistic inclusion ofhigh-temperature effects presented here can surely be refinedfurther, it already quantitatively corrects the consistent un-derestimate of polarizabilities that characterizes zero tem-perature models~irrespective of odd–even oscillations, orlack thereof!. Given the stated experimental error, and thepossible systematic errors associated with the unresolved dis-agreements between the different experiments, such refine-ment seems superfluous at the present time.

V. CONCLUSIONS

In conclusion, we have critically analyzed the successesand failures of previous DFT calculations to construct aquantitative, yet computationally simple, approach for calcu-lations of Na cluster properties. This approach combines agradient-corrected density functional and a core-correctedpseudopotential within a real space approach. We used thismethod for determining structures and polarizabilities for allNa clusters up to 20 atoms. We found the bonding in Naclusters to be very floppy and catalogued a multitude ofquasi-degenerate isomers. We constructed the zero tempera-ture polarizability curve and found that its general experi-mental features, especially its ‘‘dip’’ at magic numbers cor-responding to closed electronic shells, are reproduced.However, the multitude of isomers results in a band of po-larizability values which masks any fine structure of the po-larizability curve. Furthermore, we found that the zero tem-perature polarizability curve consistently underestimates theexperimental values. However, once the extension and dis-tortion of cluster structures at a high temperature are takeninto account, quantitative agreement between theory and ex-periment is obtained.

ACKNOWLEDGMENTS

We acknowledge support for this work by the NationalScience Foundation, the U.S. Department of Energy~DE-FG02-89ER45391!, and the Minnesota Supercomputing In-stitute. LK wishes to thank the Fulbright foundation for apost-doctoral scholarship.

APPENDIX: PROOF OF THE CORE-CORRECTIONFORCE TERM FOR A GENERAL EXCHANGE-CORRELATION FUNCTIONAL

The purpose of this appendix is to prove that the forceterm due to nonlinear core-correction, given in Eq.~9!, isvalid for any exchange-correlation density functional. Allsymbols not defined explicitly here have already been de-fined in Sec. II.

In general, the exchange-correlation energy,Exc@r#, isgiven by

Exc@r#[E d3r F xc@r#, ~A1!

where

Fxc@r#5FxcS r,]r

]r a,

]2r

]r a]r b,....D . ~A2!

With the aid of Eq.~A1!, the exchange-correlation po-tential,Vxc[dExc /dr(rW), can be expressed as

Vxc5E d3r 8dFxc@r~rW !#

dr~rW !

5E d3r 8]”Fxc@r~rW8!#

]”r~rW8!

dr~rW8!

dr~rW !

5E d3r 8]”Fxc@r~rW8!#

]r~rW8!d~rW2rW8!

5]”Fxc@r~rW !#

]”r~rW !, ~A3!

where]”Fxc /]”r denotes a functional derivative, in which onethat takes the dependence ofFxc on the derivatives ofr intoaccount.52

Using Eq.~A1! again, we express the ‘‘core-correctionforce,’’ FW a

corr, as

FW acorr[2

]Exc

]RW a

52E d3r 8]

]RW a

Fxc@rc~rW8!#

52E d3r 8]”Fxc@rc~rW8!#

]”rc~rW !

]rc~rW8!

]RW a

, ~A4!

whererc(rW8) is given by Eq.~3!. However, the only contri-bution to rc(rW8) that depends explicitly onRW a is the core-correction charge density,rcorr

c (rW2RW a). Taking this and Eq.~A3! into account, we get

FW acorr52E d3r Vxc@rc~rW !#

]rcorrc ~rW2RW a!

]RW a

. ~A5!

Equation ~A5! is identical to the core-correction term inEq. ~9!. QED.

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4332 J. Chem. Phys., Vol. 115, No. 9, 1 September 2001 Kronik et al.

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