} }< <

LDA Electronic Structure Calculationson Au Cluster13

L. LAMAREEquipe Optique Submicronique, Laboratoire de Physique de l’Universite de Bourgogne, URA 1796,´Faculte Sciences Mirande, B.P. 138, 21004 Dijon, France´

F. MICHEL-CALENDINIEquipe Materiaux pour l’Optique, Laboratoire de Physique de l’Universite de Bourgogne, URA 1796,´ ´Faculte Sciences Mirande, B.P. 138, 21004 Dijon, France´

Received August 29, 1995; revised manuscript received November 22, 1995; accepted November 28,1995

ABSTRACT

The electronic structure of the cubo-octahedral Au cluster was investigated within a13self-consistent molecular-orbital Slater-type-orbital framework. The scalar relativistic

Ž .calculated density of states DOS for the gold cluster under consideration show relativelysimilar features to those obtained experimentally or theoretically earlier calculated byfully relativistic methods. Q 1997 John Wiley & Sons, Inc.

Introduction

he electronic structure, optical, magnetic, andT structural properties of small metallic clus-ters is an area of extensive experimental and theo-retical research because of its relevance to surfacephenomenon, growth morphology, heterogeneouscatalysis, and materials science and technologyw x1]7 . In particular, the structural properties ofsmall bare clusters of gold have been studied by

Ž . w xtransmission electron microscopy TEM 8]11 ,

Ž . w xscanning tunneling microscopy STM 12 , andŽ . whigh-resolution electron microscopy HREM 13,

x14 . These studies have shown that small goldparticles are consistent with both cubo-octahedralŽ . Ž . w xO and icosahedral I geometries 8]14 . Theh h

cubo-octahedral geometry is the structure corre-sponding to the local arrangement of atoms in

Ž .face-centered-cubic fcc crystalline gold.In the present work, we report the electronic

structure of the Au cubo-octahedral bulklike13

cluster within the LCAO]LDA]MO frameworkŽ w x.using the Amsterdam density functional ADF 15

molecular code. The Au]Au distance is taken to be

( )International Journal of Quantum Chemistry, Vol. 61, 635]639 1997Q 1997 John Wiley & Sons, Inc. CCC 0020-7608 / 97 / 040635-05

LAMARE AND MICHEL-CALENDINI

˚Ž .the same as that in crystalline gold 2.88 A , whichis consistent with the relatively small bond-lengthcontraction effect observed for noble-metal parti-cles of around 13 atoms compared to that of bulk

w xmetal 16 . Further, it has been shown from consid-erations of double group theory that Jahn]Tellerdistorsion effects do not occur for the O goldh

w xcluster contrarily to the I gold cluster 17 . More-hover, the choice of the Au-bulk geometry approxi-mation for the O gold cluster allows us to com-hpare directly our results to those of Arratia]Perez

w xet al. 17 . Both scalar relativistic and nonrelativis-tic spin-restricted calculations are carried out.Relativistic treatment is performed using the

w x Ž .quasi-relativistic 18 QR-SR method and the ze-w x Ž .roth-order regular approximate 19 ZORA-SR

method. An analysis of total and partial DOScurves, Mulliken populations, and energy diagramfor the different cases is presented and our resultsare compared to fully relativistic self-consistent-

Ž .field Dirac-scattered-wave SCF]DSW results ob-w xtained by Arratia-Perez et al. 17 . This study indi-

cates that a scalar relativistic treatment is alreadyenough to reproduce the main part of the relativis-tic effects on the features of the gold density of

Žstates splitting of the d-band and s]d hybridiza-.tion in the bonding molecular orbitals .

Theoretical Details

The electronic-structure LDA]LCAO]MO cal-culations were performed using the Amsterdam

Ž w x.Density Functional molecular code ADF 15 . Thisprogram is based on a fragment oriented approach

Ž .and uses Slater-type orbitals STO as basis func-tions. Molecular point group symmetry is invoked

Ž .to construct the symmetry fragment orbitals SFOŽof the molecule from which molecular orbitals are

.built up and to reduce the numerical effort to bedone for calculating the Fock matrix elements byusing an highly optimized cellular partition nu-merical integration scheme. For describing goldvalence atomic states, we used a triple-dzeta qual-ity basis to which is added one 6 p STO polariza-tion function. The basis and fit functions opti-mized coefficients, taken from the ADF databasew x15 , are reported in Table I. The frozen-core ap-proximation, used up to the 5p atomic level, al-lows an efficient treatment of the inner atomicshells. The Coulomb potential is evaluated by way

TABLE IBasis functions and fit functions for gold atom

(in Au cluster The superscript c denotes13)core states .

Orbital Basis functions Fit functions

c1s 43.45 86.90c2s 32.55 75.30c3s 17.35 61.30c4s 12.20 49.16c5s 6.25 39.35

6s 0.95 31.581.602.75

7s 25.478s 20.24

14.889s 12.12

8.9010s 7.29

5.434.05

11s 3.332.521.90

c2p 36.23c3p 19.05 53.52c4p 10.39 36.37c5p 4.82 25.12

6p 1.73 17.627p 12.548p 9.059p 6.60

10p 4.873.28

c3d 23.00c4d 11.00 44.80

5d 1.55 28.432.855.05

6d 18.477d 12.248d 8.269d 5.66

10d 3.942.50

c4 f 8.245 f 36.206 f 22.728 f 9.569 f 6.376 g 22.118 g 5.789g 3.10

VOL. 61, NO. 4636

ELECTRONIC STRUCTURE OF AU CLUSTER13

of an accurate fitting procedure of the charge den-sity with so-called fit functions which are Slater-type exponential functions centered on atoms.The exchange and correlation energies are evalu-

w x Ž .ated with the Vosko]Wilk]Nusair 20 VWNformulas.

Relativistic Calculations

The relativistic molecular calculations are car-ried out using two different self-consistent scalarvalence-only approaches incorporating mass-veloc-ity and Darwin corrections: the quasi-relativisticŽ . Ž . w xscalar l method Q-SR 18 and the zeroth-orderŽ . Ž .scalar regular approximate method ZORA-SRw x19 .

The QR-SR method is based on the diagonaliza-Ž .tion of the first-order scalar relativistic Pauli

Hamiltonian in the space of zeroth-order solutionsŽ .i.e., nonrelativistic . This approach is well adaptedto the relativistic effects originating from high ki-netic energy valence electrons in a flat potential,and core-avoiding high-angular momentum.

The ZORA-SR formalism results in the use ofthe zeroth-order regularized approximate form ob-

Ž .tained by expanding the scalar Dirac equation inŽ 2 . Ž . 2Er 2c y V instead of E y V r2c as done in

the Pauli approximation. This method avoids thePauli Hamiltonian divergence problems in theneighborhood of atomic nuclei and is variationallystable. The regular approximation therefore gives,in general, better results than does the Pauli ap-proximation for low-energy electrons that move in

Ža strong Coulomb potential such as the core-.penetrating valence s and p electrons . So, the

method yields, in principle, considerable improve-ment for the valence energy levels over Pauli re-sults but requires adapted basis sets to extract itsadvantage; such basis sets are not yet available inthe ADF package. In the calculations, we consid-ered two levels of accuracy for the molecular po-tential V : frozen and full. By ‘‘frozen,’’ we meanthat V represents the potential created by thenuclei and the frozen-core potentials and needs tobe calculated once, whereas for ‘‘full’’ approxima-tion, the V term in the regular Hamiltonian in-cludes the valence Coulomb potential as well andso must be evaluated self-consistently.

For both relativistic approaches, the core poten-tials and densities of the atomic frozen-cores which

are responsible for indirect relativistic effects dueto relativistic changes of core potentials and densi-ties are generated from a fully relativisticDirac]Fock]Slater calculation. The basic atomicfragments used to build up the molecule wasdetermined using the same approximations as forcluster calculations. Moreover, we employed the

w xVWN 20 exchange-correlation functional simi-larly to the nonrelativistic case.

Results and Discussion

Diagrams of the valence energy levels are shownin Figure 1. Both NR and SR treatments for theAu cluster predict 2T as its ground states aris-13 2 g

ing from the electronic configuration t 5 and give2 g

the 5e molecular orbital as the lowest unoccupiedgŽ .molecular orbital LUMO . This is in agreement

w xwith earlier theoretical results 17 . The calculatedFermi energies E are equal to y4.606, y5.426,F

y5.228, and y5.232 eV for NR, QR-SR, frozen-ZORA-SR, and full-ZORA-SR, respectively. Thevalue of the occupied bandwidth is reduced whenpassing from the NR case to the SR case: 7.227 eVŽ . Ž . Ž .NR , 7.035 eV QR-SR , and 6.890 eV ZORA-SR .

FIGURE 1. Nonrelativistic and relativistic valenceenergy-levels of cubo-octahedral gold cluster.

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 637

LAMARE AND MICHEL-CALENDINI

TABLE II(Mulliken valence charges for the difference calculations a: nonrelativistic; b: QR-SR;

)c: frozen-ZORA-SR; d: full-ZORA-SR .

calculation a b c d

Peripherical atomsOrbital

S 1.1510 0.3831 0.6297 0.6347P 0.1202 0.3310 0.3419 0.3423D 9.5762 9.6208 9.6135 9.6110

Atomic gross charge y0.1419 0.1943 0.1157 0.1137

Central atomOrbital

S y0.0434 1.8370 1.3873 1.3837P y0.5116 1.0453 0.7049 0.6980D 9.5503 10.1470 9.9969 9.9873

Atomic gross charge 1.7030 y2.3316 y1.3879 y1.3664

FIGURE 2. Calculated total and local valence densities of states of the cubo-octahedral gold cluster.

VOL. 61, NO. 4638

ELECTRONIC STRUCTURE OF AU CLUSTER13

Here, it is worthwhile to note that the frozen andthe full options in the ZORA Hamiltonian led toquite similar results.

ŽThe Mulliken valence populations analysis see.Table II show a very different behavior for orbital

and atomic gross charges distribution in nonrela-tivistic and scalar relativistic cases. The relativeorbital charge changes occur mainly for s and porbitals and the sign of the corresponding chargecontribution is changed from negative to positivefor the peripherical atoms in a gold cluster. Theatomic gross charge repartition is affected simi-larly: The central gold atom is thus positivelycharged by an amount of 1.71 eV in the nonrela-tivistic case and obtain a negative charge whichvaries from y2.33 to about y1.38 eV when pass-ing from quasi-relativistic to ZORA results.

The calculated nonrelativistic and relativisticDOS curves are presented in Figure 2. The discretelevels have been broadened with Gaussians of a0.6 eV width parameter. The NR-DOS curve showsessentially two main peaks well separated. Thesmall peak, which includes the E level, corre-Fsponds to the occupied s]p band. The higher in-tensity peak, slightly distorted around its top, isdue to the d band. The NR-DOS curve obtained by

w xArratia-Perez et al. 17 presents a third lower-energy peak, attributed to the discrete nature ofthe central gold atom; such a structure does notappear in our NR results. The DOS curves of

Ž . Ž .Figure 2 b ] d show that the d band peak splitsunder relativistic effects. But this splitting remainslow in intensity since the spin]orbit coupling re-sponsible for the main part of it has been neglectedin our calculations. The ZORA results are almostunchanged relatively to the quasi-relativistic ones.This is essentially due to the identical basis setsused in nonrelativistic and relativistic calculations.

Conclusion

This work emphasizes the major role played byrelativistic effects in heavy elements such as gold.The present LDA]LCAO]STO relativistic elec-tronic structures for the Au cubo-octahedral clus-13

ter compare rather well with the fully relativisticw xall-electron DSW calculations 17 and available

w xexperimental data 21 .

References

1. Proceedings on a Symposium on Small Particles an Inorganicw Ž .xClusters, Berlin, 1984 Surf. Sci. 156, 3 1985 .

Ž .2. J. Koutecky and P. Fantucci, Chem. Rev. 86, 539 1986 .3. W. Weltner, Jr. and R. J. Van Zee, Annu. Rev. Phys. Chem.

Ž .35, 291 1984 .4. W. Weltner, Jr. and R. J. Van Zee, in Comparisons of Ab Initio

Quantum Chemistry with Experiment for Small Molecules, R. J.Ž .Bartlett, Ed. Reidel, Dordrecht, 1985 .

Ž .5. J. R. Morse, Chem. Rev. 86, 1049 1987 .6. S. Iijima, in Microclusters, S. Sugano, V. Nishina, and

Ž .S. Ohnishi, Eds. Springer-Verlag, Berlin, 1987 .Ž .7. J. A. Howard and B. Mile, Acc. Chem. Res. 20, 173 1986 .

Ž .8. S. Ogawa and J. Ino, J. Cryst. Growth 13, 48 1972 .9. C. Solliard, Ph. Buffat, and F. Faes, J. Cryst. Growth 32, 123

Ž .1976 .10. C. Y. Yang, K. Heineman, M. J. Yacaman, and H. Poppa,

Ž .Thin Solid Films 58, 163 1979 .Ž .11. L. D. Marks and D. J. Smith, J. Cryst. Growth 54, 425 1981 .

12. D. W. Abraham, K. Sattler, E. Ganz, H. Mamin, R. E.Ž .Thomson, and J. Clarke, Appl. Phys. Lett. 49, 853 1986 .

Ž .13. S. Iijima and T. Ichihashi, Jpn. J. Appl. Phys. 24, L24 1985 ;Ž .Ibid. 24, L 434 1984 .

Ž .14. S. Iijima and T. Ichihashi, Phys. Rev. 56, 616 1986 .Ž15. ADF release 1.1.4 Dept. of Theoretical Chemistry, Vrije

.Universiteit, Amsterdam . E. J. Baerends, D. E. Ellis, andŽ .P. Ros, Chem. Phys. 2, 41 1973 ; G. te Velde and E. J.

Ž .Baerends, J. Comp. Phys. 99, 84 1992 .16. P. A. Montano, G. K. Shenoy, E. E. Alp, W. Schulze, and J.

Ž .Urban, Phys. Rev. Lett. 56, 2076 1986 .17. R. Arratia-Perez, A. F. Ramos, and G. L. Malli, Phys. Rev. B

Ž .39, 3005 1989 .Ž18. P. M. Boerrigter, Thesis Vrije Universiteit, The Nether-.lands, Amsterdam, 1987 ; T. Ziegler, V. Tschinke, E. J.

Baerends, J. G. Snijders, and W. Ravenek, J. Phys. Chem. 93,Ž .3050 1989 .

19. E. van Lenthe, E. J. Baerends and J. G. Snijders, J. Chem.Ž .Phys. 99, 4597 1993 ; R. van Leeuwen, E. van Lenthe, E. J.

Baerends, and J. G. Snijders, J. Chem. Phys. 101, 1272Ž .1994 .

20. S. J. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200Ž .1980 .

21. G. K. Wertheim, S. B. Di Cenzo, and S. E. Yougquist, Phys.Ž .Rev. Lett. 51, 2310 1983 .

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 639