Quantum Monte Carlo characterization of small Cu-doped silicon clusters: CuSi 4 andCuSi 6I. V. Ovcharenko, W. A. Lester Jr., C. Xiao, and F. Hagelberg Citation: The Journal of Chemical Physics 114, 9028 (2001); doi: 10.1063/1.1367375 View online: http://dx.doi.org/10.1063/1.1367375 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/114/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Binding of hydrogen on benzene, coronene, and graphene from quantum Monte Carlo calculations J. Chem. Phys. 134, 134701 (2011); 10.1063/1.3569134 Quantum chemical assessment of the binding energy of CuO+ J. Chem. Phys. 134, 064304 (2011); 10.1063/1.3537797 Benchmark all-electron ab initio quantum Monte Carlo calculations for small molecules J. Chem. Phys. 132, 034111 (2010); 10.1063/1.3288054 Quantum Monte Carlo study of porphyrin transition metal complexes J. Chem. Phys. 129, 085103 (2008); 10.1063/1.2966003 The interaction of oxygen with small gold clusters J. Chem. Phys. 119, 2531 (2003); 10.1063/1.1587115

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Quantum Monte Carlo characterization of small Cu-doped silicon clusters:CuSi4 and CuSi 6

I. V. Ovcharenkoa) and W. A. Lester, Jr.b)

Department of Chemistry, University of California, Berkeley, California 94720-1460

C. Xiaoc) and F. Hagelbergd)

The Computational Center for Molecular Structure and Interactions Department of Physics,ATM and General Sciences Jackson State University, Jackson, Mississippi 39217

~Received 29 November 2000; accepted 2 March 2001!

The relative energies, binding energies, and adsorption energies of three CuSi4 and two CuSi6clusters have been computed in the fixed-node diffusion Monte Carlo~FNDMC!, CASSCF, andB3LYP DFT methods. These results are compared with the earlier Hartree–Fock~HF! and B3LYPDFT investigations of these systems by two of us@C. Xiao and F. Hagelberg, J. Mol. Struct.:THEOCHEM 529, 241 ~2000!#. The very close energy level spacing of the isomers underconsideration confirms the previous work of Xiao and Hagelberg. The FNDMC results show somequalitative discrepancies with B3LYP DFT, and HF findings. They also confirm the appropriatenessof the B3LYP DFT method for the prediction of the most stable CuSi4 isomer, while the CASSCFmethod compares more favorably with FNDMC for adsorption energies than B3LYP DFT.© 2001 American Institute of Physics.@DOI: 10.1063/1.1367375#

I. INTRODUCTION

Considerable interest in pure silicon and metal-dopedsilicon clusters is now found in chemistry as well as in mo-lecular and solid state physics. One of the reasons for thisattention is that the properties of these cluster systems arenotably different from those of the bulk materials.1 In addi-tion, promising applications of materials with novel proper-ties based on metal-doped silicon clusters and metal-dopedfullerenes2 are of significant importance for the semiconduc-tor industry. Numerous experimental3–6 and theoretical7–13

studies have been devoted to the determination of equilib-rium geometries, electronic and bonding structures, as wellas structural transitions of different size pure silicon andalkali–metal-doped silicon clusters. It has been shown, not-withstanding the occurrence of carbon and silicon in thesame column of the periodic table, that there are big differ-ences in the geometries of the clusters of these elements.This point has been clearly demonstrated in spectroscopicfindings for medium size silicon clusters by Rinnen andMandich5 and in theoretical studies of small silicon clustersby Fournieret al.7

Despite the general interest in metal-doped silicon clus-ters, the number of studies devoted to this area is limitedcompared to the analogous metal-doped carbon clusters.Note that the main interest in doped silicon clusters has beenin the alkali–metal systems,3,5,11,14 in contrast to those in-volving transition metals.15,16 The influence of transitionmetals on the geometry and bonding of doped silicon clustersas well as the difference in properties from alkali–metal-

doped silicon clusters warrants investigation. This directionforms the object of the present study.

Various theoretical methods applied to pure and metal-doped silicon clusters have revealed equilibrium geometriesthat are rather similar in selected respects. Recently, two ofus,16 hereafter referred to as XH, carried out a Hartree–Fock~HF! and Density Function Theory~DFT! study of pure andCu-doped Si4 and Si6 clusters and found an energy differ-ence of only 0.005 eV among CuSi4 isomers and 0.832 eVfor CuSi6 isomers.

Grossman and Mitas1 reported an investigation of puresilicon clusters~up to Si20) with the quantum Monte Carlo~QMC! method.17–19 They found that the fixed-node diffu-sion Monte Carlo~FNDMC! method results in binding ener-gies for small silicon clusters lying within 4% of experiment.Similar good agreement of QMC binding energies with mea-sured values was also shown by Greeff and Lester20 for si-lanes, and by Liet al.21 for the cohesive energy of solidsilicon. From these studies it is clear that if one wishes todetermine the geometry of silicon clusters and precisely cal-culate their bond energies, then an accurate treatment of elec-tron correlation is required.

Small Cu-doped silicon clusters~three CuSi4 and twoCuSi6) were investigated in the present study. The optimizedgeometries for these systems determined by XH16 are usedhere. The relative energies, binding and adsorption energiesfor the isomers under consideration are reported as deter-mined in the FNDMC, B3LYP DFT,22 CASSCF,23 and HFmethods. The localized effective core potential~ECP!approach24 was used in DMC computations for efficiencyand to reduce statistical error.

The paper is organized as follows. Section II summa-rizes the methods and scope of calculations performed, andthe FNDMC method with ECPs. Section III reports and dis-

a!Electronic mail: ovcharen@gumbo.cchem.berkeley.edub!Electronic mail: WALester@lbl.govc!Electronic mail: xiao@twister.jsums.edud!Electronic mail: hagx@twister.jsums.edu

JOURNAL OF CHEMICAL PHYSICS VOLUME 114, NUMBER 20 22 MAY 2001

90280021-9606/2001/114(20)/9028/5/$18.00 © 2001 American Institute of Physics

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cusses results obtained with the various methods. Conclu-sions form the content of Sec. IV.

II. METHODS AND STRUCTURES

A. Structures

The focus here is on three CuSi4 and two CuSi6 clusterspreviously investigated by XH.16 They optimized the geom-etry of these systems with the B3LYP DFT method. Thesegeometries were used in the present study and the structuresare presented in Figs. 1 and 2. The cluster labels are those ofRef. 16 with modified atom numbering.

Isomers 4a and 4b have a rhomboid Si4 base distin-guished by whether the Cu atom is located over the Si4 base~face adsorbed Cu! or exterior of the base~edge adsorbedCu!. The 4c isomer corresponds to a significantly distorted

Si4 base with the adsorbed Cu atom over the Si~4!–Si~5!bridge. Isomer 6a can be viewed as a substitutional geom-etry, arising from the Si7 (D5h) ground state structurethrough replacement of one equatorial Si atom by Cu, whileisomer 6b has an adsorbed Cu atom above a distorted base ofSi atoms. Isomers 4a and 4b were reported by XH to benearly isoenergetic and HF, CASSCF, and DFT methodswere found to differ in the specification of the lowest energyconformation. In contrast to CuSi4 , the lowest energy ar-rangement for the CuSi6 clusters was isomer 6a from allmethods.

B. HF, CASSCF, and DFT methods

Our earlier HF and B3LYP DFT computations were car-ried out with the GEN basis set, i.e., a 6-31G* Popleet al.25

basis set for Si and the Wachter’s basis set26 augmented by dand f polarization functions for Cu. For the present B3LYPDFT and CASSCF methods, we used a 6-311G** Pople ba-sis set25 for Si together with the 6-311G** Wachters–Haybasis set26,27 for Cu. This choice of basis set is qualitativelyclose to the GEN basis set and provides a measure of thebasis set dependence of the DFT results.

The configuration space used in the CASSCF calcula-tions involved at least three virtual orbitals and two doublyoccupied orbitals. The number of active electrons and orbit-als in which they are distributed was chosen~active elec-trons, orbitals! as ~4,6!, ~5,6!, and~6,7! for the singlet, dou-blet, and triplet states, respectively.

C. Fixed node DMC

The fixed node DMC method is a stochastic solution ofthe importance-sampled imaginary-time Schro¨dinger equa-tion:

2] f ~x,t!

]t52D“

2f ~x,t!1~EL~x!2ET! f ~x,t!

1D“•~ f ~x,t!FQ~x!!, ~1!

where a density functionf (x,t) is a product of a trial func-tion CT(x) and the time dependent solution to the Schro¨-dinger equationf~x,t!. The latter function converges to theexact ground state solutionF0(x) at large imaginary timet.This density function form includes importance samplingand leads to the termFQ(x) in ~1!, called the quantum force,given by

FQ~x!52“CT~x!

CT~x!. ~2!

The diffusion constantD is equal to 0.5 in atomic units. Thelocal energyEL(x) is the energy of the trial function at thechosen coordinatex:

EL~x!5HCT~x!

CT~x!. ~3!

Here H is the Hamiltonian for the system, whileET is anenergy offset chosen to be close to the exact ground stateenergy.

FIG. 1. Schematic structure, atom labeling, and relative energies of CuSi4

isomers.

FIG. 2. Schematic structure, atom labeling, and relative energies of CuSi6

isomers.

9029J. Chem. Phys., Vol. 114, No. 20, 22 May 2001 Copper doped silicon clusters

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The trial functionCT(x) was chosen as a product of aSlater determinant and a Schmidt–Moskowitz 10-parametercorrelation function28

CcSM5eU. ~4!

Here U is a function of relative positions of electrons andnuclei and contains electron–electron, electron–nuclear, andelectron–electron–nuclear correlation factors. The correla-tion functionU contains 10 coefficients that are optimized tominimize statistical error and to recover a significant portionof the correlation energy.

The usual optimization procedure in QMC involvesminimizing the variances2 of the local energy:

s~EL$x%!251

N (i 51

N

~EL~xi !2ET!2, ~5!

where the summation is over theN characteristic sampleelectronic configurations or walkers. Instead of variance op-timization, we minimized the local energy functional:29

f ~EL$x%!51

N (i 51

N

uEL~xi !2ETu. ~6!

The reasons for this choice are discussed below.

D. ECP approach

The most significant hurdle to the performance of QMCcomputations arises from the stochastic nature of the method.The statistical error in calculated energy values depends onthe length of the calculation or the CPU time. The longer thecomputation of the system the more precise the results ob-tained. A procedure, which drastically reduces the computa-tional effort, is the use of effective core potentials~ECPs! forcore electrons and explicit treatment of only the valenceelectrons. Hammondet al.24 and Ceperley30 pointed out thatthe computational time for QMC calculations is proportionalto Za whereZ is the atomic number anda is of order 5.5 to6.5. With the use of ECPs, the computational time depen-dence is;Zeff

3.4, whereZeff is the effective atomic numberresulting from the use of an ECP and is given byZ–Ncore,

24

whereNcore is the number of core electrons.The ECP approximation leads to the replacement of the

core–valence repulsion and the core–valence orthogonalityconditions by the pseudopotential,31

UECP5(A

(i 51

Nval S Ul max11A ~r iA!

1(l 50

l max

(m52 l

l

uYlm~V iA!&UlA~r iA!^Ylm(V iA!u D ,

~7!

where the indexA sums over only those atoms having apseudopotential. For atomA, V iA is the solid angle of elec-tron i from A, l max is the largest angular momentum amongthe core electrons, andUl

A is a radial pseudopotential foratom A, which depends only on the electron–nuclear dis-tancer iA and the angular momentuml. The nonlocal charac-ter of Eq. ~7! leads to the inability to apply it directly in

QMC and to a localization approximation obtained by allow-ing UECPto act on the valence wave functionCval as impliedin Eq. ~3!.

The ECPs developed by Stevens, Basch, and Krauss~SBK! have been shown to produce good agreement betweenQMC energies and experiment.32 We used these ECPs forthe description of the core electrons for the atoms in thepresent study. We note that recently the method of softECPs, which excludes repulsiver 22 and attractive Coulombsingularities, has been extended to QMC33 and applied34 tosecond- and third-row atoms of the periodic table. Soft ECPshave been shown to reduce the variance and to enable longertime steps in DMC calculations. These ECPs do not pres-ently exist for Cu, but are expected to become availableshortly.

E. Software

Ab initio and GGA DFT calculations were performedwith the GAUSSIAN 94 quantum chemical code.35 All theQMC calculations~including the correlation function optimi-zation step! were done with the QMagiC program.36 The HFtrial functions for QMC calculations were generated usingthe GAMESS37 program. The SBK ECPs and correspondingbasis sets were obtained from the Extensible ComputationalChemistry Environment Basis Set Database.38

III. RESULTS AND DISCUSSION

All energies, relative, binding, and adsorption, for thethree CuSi4 and two CuSi6 clusters were obtained using theab initio CASSCF, B3LYP DFT, and FNDMC methods. Theuse of the FNDMC method was validated by the significantcorrelation energy recovered for the present systems. TheFNDMC method has been shown to recover up to 100% ofthe correlation energy for selected systems.24 Therefore, theFNDMC results provide a useful benchmark for the othermethods. The use of two different, but closely related, basissets for the B3LYP case provides a useful measure of basisset dependence of these results; see Tables I–III. The relativeinsensitivity of the FNDMC method to choice of basis set forthe trial function representation17 is worth mentioning.

A key requirement for an accurate FNDMC calculationis optimized correlation function coefficients. We exploredminimization of the variance, Eq.~5!, and of the local energyfunctional, Eq. ~6!, for optimization in the fixed samplevariational Monte Carlo~VMC! method. Based on the expe-rience of previous studies, 10 000 walkers were used andprovided a good description of the configurational space.

TABLE I. Relative energy~eV! of CuSi4 and CuSi6 clusters.

CuSi4 CuSi6

4a 4b 4c 6a 6b

HF/GEN ~Ref. 16! 0.0 0.086 0.342 0.0 3.106B3LYP/GEN ~Ref. 16! 0.0 20.005 0.095 0.0 0.832B3LYP/6-311G** 0.0 20.020 0.076 0.0 1.013CASSCF/6-311G** 0.0 0.072 0.303 0.0 0.164DMC 0.0 20.13~8! 0.34~12! 0.0 0.53~15!

9030 J. Chem. Phys., Vol. 114, No. 20, 22 May 2001 Ovcharenko et al.

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Variance optimization for all the clusters was found to beunreliable. It was found that the variance minimization for afixed sample encountered difficulty owing to the limitedsample size that is computationally feasible for these sys-tems. It was found necessary to carry out a sequence of VMCand optimization steps~‘‘shaking’’ ! to converge the optimi-zation procedure. The presence of ‘‘bad’’ walkers on thedistribution wings was found and discovered to be correlatedwith problems of variance minimization. Local energy func-tional minimization lowers the influence of ‘‘bad’’ walkers,and is found to be straightforward and able to recover ap-proximately from 50% to 80% of the correlation energy ob-tained by the FNDMC method for the present systems. Theuse of a weighting factor during optimization had differentinfluence for different systems. For all the clusters investi-gated the weighting factor imbalanced the optimization pro-cedure. The possibility of this weighting factor feature hasbeen pointed out by Barnettet al.39

The time step for stability of FNDMC calculations wasfound to be much smaller for the Cu-containing system com-pared to the pure silicon clusters. The following time stepswere found to produce stable FNDMC runs with high accep-tance ratios~more than 99.6%!: 0.0005 a.u. for Cu atom,CuSi4 and CuSi6 clusters; 0.005 a.u. for Si atom, Si4 and Si6clusters.

Calculated relative energies for the clusters under con-sideration are presented in Table I. The basis set dependenceof the B3LYP DFT method is comparatively small and doesnot have a qualitative influence on the results. The maindifference in this case occurs with the energy splitting of the4a and 4b isomers, which differ by up to a factor of 4 for thetwo basis sets, but the magnitude of this effect is signifi-cantly smaller than the precision of the B3LYP DFT method.The HF and CASSCF methods provide consistent descrip-tions of the CuSi4 clusters, but for the CuSi6 clusters thesemethods differ by more than an order of magnitude. Com-parison of the HF, CASSCF, and B3LYP DFT values with

FNDMC results shows no preference for one method overanother for relative energies. The most stable isomer ofCuSi4 is found to be 4b by B3LYP DFT, which is in accordwith FNDMC, and contrary to HF and CASSCF. The 4c and6b isomers are in qualitative agreement for all the methodsas the highest-lying forms.

The binding energy per atomEb and the Cu atom ad-sorption energyEad, describing the relative stability of theclusters under consideration, are defined, respectively, as

Eb~CuSin!5@nE~Si!1E~Cu!2E~CuSin!#/~n11!,~8!

Ead~CuSin)5E~Sin!1E~Cu!2E~CuSin!.

The calculated values ofEb for CuSi4 and CuSi6 clusters arepresented in Table II and the computed values ofEad aregiven in Table III.

One sees that the FNDMC binding energies for all theclusters lie between the B3LYP DFT and HF results. A com-parison of the relative values shows that DFT theory overes-timates the binding energy, while the HF theory underesti-mates it. Qualitatively B3LYP DFT binding energies arefound to be closer to the FNDMC ones than HF or CASSCF,but they still are outside of the statistical error bars of theFNDMC results. The important feature found in all the cal-culations is the close value of the binding energy for all theisomers of the particular cluster and the larger binding en-ergy per atom for CuSi6 clusters relative to CuSi4 clusters.

Analysis of Cu atom adsorption energy shows the con-tradictory behavior of the methods applied relative toFNDMC/ECP. Qualitative agreement is found comparingCASSCF and FNDMC methods. They give the same signand similar scaling of the adsorption energy for the CuSi4

and CuSi6 clusters. We note that XH also found a negativeadsorption energy for the 6b isomer of the CuSi6 cluster inthe HF method. The Cu atom adsorption by the Si4 and Si6clusters is found to be endothermic.

IV. CONCLUSIONS

The computation of relative energies, binding energies,and adsorption energies for three CuSi4 isomers and twoCuSi6 isomers were carried out in the B3LYP DFT,CASSCF, and FNDMC/ECP methods. The FNDMC methodrevealed the 4b isomer to be the lowest in energy for CuSi4

cluster followed by the 4a and 4c isomers. The 6a isomerwas found to be the most stable form of CuSi6 . These resultsare in qualitative agreement with B3LYP DFT calculations.This method is found to overestimate binding energies, andthe HF and CASSCF methods to yield underestimates. The

TABLE II. Binding energies~eV/atom! of CuSi4 and CuSi6 clusters.

CuSi4 CuSi6

4a 4b 4c 6a 6b

HF/GEN ~Ref. 16! 1.383 1.366 1.314 1.583 1.139B3LYP/GEN ~Ref. 16! 2.474 2.475 2.455 2.802 2.683B3LYP/6-311G** 2.590 2.594 2.574 2.906 2.762CASSCF/6-311G** 0.567 0.553 0.506 0.632 0.608DMC 1.82~4! 1.85~4! 1.76~5! 2.22~4! 2.15~4!

TABLE III. Adsorption energy~eV! of CuSi4 and CuSi6 clusters.

CuSi4 CuSi6

4a 4b 4c 6a 6b

HF/GEN ~Ref. 16! 0.593 0.508 0.251 0.518 22.589B3LYP/GEN ~Ref. 16! 1.315 1.320 1.221 1.364 0.533B3LYP/6-311G** 2.114 2.134 2.037 2.396 1.382CASSCF/6-311G** 20.471 20.544 20.775 21.781 21.945DMC 21.31~9! 21.18~9! 21.65~12! 22.81~13! 23.34~12!

9031J. Chem. Phys., Vol. 114, No. 20, 22 May 2001 Copper doped silicon clusters

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negative adsorption energy, corresponding to endothermicCu atom adsorption, obtained with the FNDMC method forall the clusters was reproduced only with the CASSCFmethod, while the B3LYP DFT and HF methods resulted inexothermic adsorption.

ACKNOWLEDGMENTS

This study was supported by the National Science Foun-dation through the CREST Program~HRD-9805465!. Thecalculations were performed at San Diego SupercomputerCenter under a grant from the National Partnership for Ad-vanced Computational Infrastructure~NPACI! for which weare grateful.

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