The Density Functional Description OP Atomic Clusters

THE DENSITY FUNCTIONAL DESCRIPTION OF ATOMIC CLUSTERS R.O. Jones Institut fUr FestkBrperforschung del' Kernforschungsanlage JUlich, D-5170 JUlich, Federal Republic of Germany The density functional formalism (OFF) Is being used increasingly In calculations of surfaces with and without adsorbates. Within the OFF framework, the local spin density (LSD) approximation gives a very good description of trends In excitation and bonding energies In atoms and molecules. Although there are systematic departures from experiment, the approximation appears to be an appropriate basis for surface calculations.

1. Cluster calculations and surface physics The nature of the bonding between atoms at surfaces is of central importance to surface science and many experimental and theoretical techniques seek direct or indirect information about it. From the theoretical point of v l cw , the most direct information would result from a calculation of the t,(,Vd energy of the system of nuclei and electrons as a function of geometry. The minimum energy ~:ould give the equilibrium structure of the surface and of any adsorbed atoms or molecules. ~lore­ over, the variation with ad s cr-ba t.e position would lead Lnrae d l at.e Iy to activaticn e ne i-g lcu

1'01'

eli r ruc i on and to paths and heats of reactions.

Since these are important quantities for the experimentalist, it is not surprising that numerous approximate methods have been developed for studying total energy variatiolJs. The complicated many-body pr-ob l em cf surface plus ada t oms lI:ould be Greatly simplified if the bonding were sufficiently localized that only near neighbours of the adatoms I~ed be considered. Model calculations for a jellium surface indicate that density changes on chemisorFtion are indeed localized to within a few Kngstr5m (1] and provide support for the cluster approach to cncmt s or-pt.Lou , illustrated in Fig. 1. The reduction of the

probl~c,

to one of molecular dimensions

~eans

that, in

principle, thcor-o ti cn t n.e t ho de of quant um chemistry could be applied. 'l'he pr-c d i et i v « I)()~ill' ,,1' total energy calculat i on" for tile cluster in j.'ig. 1 wouLl t Len p:ll':l1lel those of mol c cul ai- binding c'llc'rgy calculations in gonc r-a l , At present J the trail I t Lona 1 ab i nHio methods of' quantum chemi stry I nappr-opr-Ln to fer the clusters which may be relevant for chcm i s or-p t Lon , Eartree-}'«(']( ca l cu l at i one , wheI'e the wave I'unc ti on of the

SC"C'!IJ

cluster it, apjr-ox i mat c u t,y a :dr.gle SlRtc'r determinant or "configurat i on"

J

r.avo b cc n po r-r or-n.cd I'r i- e);l'!::i ..' orption c l uut er-s such a3 cU:,CO with

:n

Fig. 1: Two adatoms in the nelgnbourhood of a surface. The cluster approach to chemisorption focuses on the adatoms and the near neighbours (encircled) • encouraging results [2]. In general. however, a single configuration wave function cannot be expected to yield a reliable total energy. The Slater determinant which leads to the lowest energy is only one of many with comparable energy and an accurate total energy would require a linear combination of all such determinants - a procedure known as "configuration interaction" (Cl). The rapid increase in the number of relevant configurations with increasing numbers of electrons means. however, that the method is difficult to apply to all but the simplest systems. 2. 'rhe density functional approach The density functional [3] provides an alternative approach to calculating the total energy of a many-electron system. As in the Thomas-Fermi picture. the density n( r ) p lays a central role. since it can be shown [3] that all ground state properties can be expressed as functionals of nCr). For example, the total energy in the presence of an external field. Ve xt• may conviently be written e xt (! ) + (1) E{n] = Ts[n] + I d! n(!} ~(!)] + Exc{n]

[v

i

Ts{nl is the kinetic energy of a system of non-interacting particles with density n. _(!:) is the Coulomb potential and EXC is the exchangecorrelation energy. The ground state density minimizes E[n] and may be found by solving the Hartree-like equations (2)

[ _ ~1

v2

+ Vext(r} + Vxc(_r) _ -

t

n

34

]

~

n

:

0

N

The density nCr) = ~ f n IW nl 2 , where the occupation numbers f n must n=1 be compatible with the symmetry of the state in question, N is the number of electrons and the external field Ve xt arises, in the present context, from the nuclei. The exchange-correlation potential Vxc(r) • 6Ex cln]/6n. If the local spin density (LSD) approximation is used for ExclnJ. the numerical problem reduces to the solution of single-particle equations with a local effective potential. and modern methods of band theory are ideally suited for this purpose. In fact, several groups have adapted band structure programs to layer geometries and are calculating total energy differences or pressures in the surface context I ~]. There is a tendency amongst authors who use the local density functiqnal to emphasize the areas of agreement between calculation and experiment. In the present lecture. I should like to examine the description of atoms and small molecules using the LSD appr-ox imat.Lon , and to point to systematic departures from experiment. It should be emphasized that there is a fundamental difference between the configuration interaction approach to electronic structure and the density functional method. In principle. the former can lead to an exact solution of the Schr5dinger equation for the many-electron system. In contrast, use of the density functional involves an approximation which rmst be tested ill cases wher'e accurate experimental information is ava I Lab J c , In the cont.e xt of surface physics there would be little sense in s t.udy i ng oxygen chemisorption on a metal surface if' neither the oxygen mol('cule nor the metal-metal bond could be described using the method. 3. Lm> description of atoms 'I'ho density functional f'c r-ma Li sm focuses att.ention on the ground s t.a t.e total encl'gy, 11 quantity \~hiell ill dif'ficult to mensur-e , POl' light atoms, successive ionization enc r-gi c s may be summed. but for most gysterns, it ie cn('J'P;Y clif'fel'ences v:hich arc mor-e readily available. In this lectul'e, we shall cmphallizc the latter. although it may be noted that the LSD ilpproxima ti on genera 11y underestiroa tes the magn i tude of' atomic total cl:el'gic:;. As an example of energy d'i I'f'e r-erice s , Pig. 2 shows cx per-i ment a I :lilt! calculated excitation energies f'r-om the 1S (:.;2) r;round state (Of the group I1A atoms to excited states tlith p and d-occupancy 15]. Apart from the dI fferences \'1hich ari ao from the neglect of r-el a t riv i a ti c effects in the ca t c ul a t Lonu , SClLe interesting trends ar-c- apparent. Firstly, the clJ('j'gy i-c-qu i t-od to excite an s-electron to a p 01' d-level is undcl'estimated by the ]'SD ClppT'oximation. Secondly, 35

6

5

::;

4

EXPT

CALC IS

'"l w 3 I

~

2

10

O\..---L----'L-........----'_...L----O'_----'--L-_'----L----'L-........---''---~___' He Be Mg Co Sr Ba ~ He 8e Mg Ca Sr Ba Ra

Fig. 2: Calculated and experimental energies in group IIA atoms. relative to ns 2 ground state. there are striking irregularities in both experiment and theory as the atomic number increases. The pairwise irregularities show an interestine correlation with trends in atomic valence functions. as shown in Fig. } (6). Comparable correlations are also evident in atomic excitation energies and molecular binding energies in other groups (7). An underestimate of the s-d transfer energy is also evident in Fig. 4. where excitation energies of iron series atoms and ions. relative to the d n- 2s 2 atomic state. are shown. Also shown for comparison are the corresponding results for the Hartree-Fock approximation. It is evident that the LSD approximation provides a better description of the systematics across the row. in particular the spin flip energy which causes the breaks between Cr and Mn. The Hartree-Fock method excludes correlations between electrons of opposite spins nnd it is not surprising that energies involving spin flips are poorly reproduced. In spite of the good description of the trends in the energy differences. it is important to note that the s-d transfer energy in the atoms is underestimated by approxim~tcly 1 eV. In first row atoms. the a-p transfer energy is also underestimated by a n Imf Lar- amount. Althoup;h 36

t .

~

t

.

~

Pig. 3: Valence functions for group II A atoms: (a) s-functions for is (ns 2) state; (b) p-functlons for 3p (ns 1np1) state. The dashed curve Is the p function correspondins to the 1s2p configuration of He.

101

Co

Sc:

Ti

V

Ct

..,

Fe

Co

Hi

Cu

Pig. _: EnercJ differences in iron series ataas and Ions: (a) HP calculations; (b) LSD values. Experimental values are dashed. See Ref. 15l. 37

the.. deviations from experiment are systematic, they indicate that the LSD approximation will lead to errors in binding energies in cases where the relative s-, p- and d-contributions to the charge density change. An example would be the cohesive energy of carbon or silicon. where the ground state of the atom is s2 p2 and the bond in the solid has a greater p-character. 4. LSD description of small molecules The LSD approximation has been applied to a large number of diatomic molecules (6-81, as well as to small polyatomic molecules such as H20 and NH (9]. Spectroscopic constants such as dissociation ener3 gies. vibration frequencies, bond angles and equilibrium internuclear separations are generally given quite satisfactorily. Por heteronuclear molecule•• such as CO, the variation of dipole moment with internuclear separation is reproduced remarkably well. Por small molecules with s or sp-valence electrons. the overall agreement with "experiment is better than obtained by Hartree-Fock calculations and comparable to the accuracy ot CI calculations where they exist. One ot the most interesting series ot calculations has proved to be the group IIA diatomic molecules Be2 to Ba2 (61. where the ground state has an equal occupancy of bonding and antibonding orbitals and is repulsive in the Hartree-Fock approximation. In Pig. 5. LSD calculations 4

",

41\

\

'A ,~. .

\

t i

';80.

/

•

•

~

".MgCoSrBaAa Fig. 5: Calculated values ot dimer binding energies (full curve. lett scale) and bulk cohesive energies (broken curve. right scale) calculated in Ret. (10]. Experimental values are given as crosses where available. See Ret. (6]. 38

the cohesive energies and dimer binding energies tor group IIA elements are compared with experimental values where available. The zigzag behaviour shown by the calculated dimer binding energies is quite different from the expectations of those who view these molecules as being bound by long-range van der Waals' forces. Which scale with the atomic polarizabilities. Recent CI calculations indicate that Be2 is more strongly bound than Mg2 and it would be interesting to have experimental confirmation. The LSD functional calculations indicate that the bonding is a result of sp- and sd-hybridization due to overlapping charge densities on the two atoms. In Pig. 2. we have shown that both s-p and s-d transfer energies are underestimated by the LSD approximation and the systematic overestimate in the bindins energy is to be expected. Experience with small molecules suggests that the LSD approximation should be capable of describing trends in bonding in the context shown in Pig. 1. One of the few calculations to be performed so far is the Cu cluster calculation by Baerends and co-workers [11]. who 5CO found that the method overestimated the strength ot the bonding between the CO molecule and the substrate. This is perhaps an indication that five atoms is too few to provide an adequate representation of a surface. If this proves to be the case. then an efficient scheme for embedding the cluster in an appropriate medium may be essential [12].

~f

5. Modifications of the LSD approximation The systematic deviations from experiment obtained using the LSD approximation suggests that modifications may exist which reproduce atomic excitation energies. for example. very well. A useful starting point for such a discussion is the exact expression for the exchangecorrelation energy [13.14] 1

Exc[n] = ~ ~

f d_r n(_r) J dr' n -

xc(

,. ) r. r -r

Ir-r'l

Where the exchange-correlation hole 2

xc e nCr. r'-r) • nCr') J o

dA [g(r.r';A)-l]

is expressed in terms of the exact pair-correlation function g(r.r';A). The formula (3) is obtained by integrating over the coupling constant A in the presence of an external potential such that the density is independent of A. The expression depends only on the spherical average of the xc-hole and not on the precise shape and a sum-rule shows that the xc-hole contains one electron. Por approximations Which. like the local 39

density approxiaation, aatiatJ thia sum rule, there ia a ayatematic cancellation of errora (14J. Non-local aodifications to the LSD functional can be constructed [15J wbich, in addition to aatisfying the above aum rule, lead to exact re.ults in other caaea. One such modification (16J suaranteea that EXC includea a term which cancela exactly the electron Coulomb aelf-interaction (self-interaction corrected or SIC functionalJ. Atomic total energiea calculated using both SIC and ai~le non-local functionals are much closer to experiment than the LSD valuea. However, present indicationa are that enerl7 difterencea, on which we bave placed particular e~baaia. are not iaproved significantly (17). This remaina an interesting and challenging area of reaearch. 6. Concluding rearka The description of excitation and bonding energies of atoma and aaall moleculea provided by the LSD approxiation is remarkably good particularly in view of ita relative numerical aimplicity. Ita application to aore complicated geometriea appears to have very good proapecta. It ia eaaential to note. however. that deviationa trom experimentally meaaured enerl7 differences are inevitable in caaea where bonding leada to a change in the balance between a. p and d contributiona to the cbarge density. Although reaults for a apecific ayatem are likely to depart fro. experiment. the proapecta for deacribing ~ in bonding . in tbe aurface context appear to be very good. Areaa ot particular intereat will be the size of cluster neceaaary to describe the bonding to a aurface. the embedding of a cluster in an appropriate medium and -edificationa of the LSD approximation. Referencea [lJ

(2) (}J (II)

The atoa-jelliua model of chemiaorption baa been atudied by Lang N.D. and Villi... A.R.: Phya. Rev. 8 ~ (1978). 616 for Li. Si and Ci adsorbates. Bacus P.S.: presented at International Topical Conference on Vibration. at Surfacea. Naaur. Belgium, September 1980 Hohenberg P. and Kohn V.: Phya. Rev. 1}6 (1964), 8864; Kohn V. and Sbam L.J.: Phya. Rev. !!Q (1965). All}} Recent eXUlPles of lQer calculationa of aurface propertiea are Jepaen 0 •• Madsen J. and Anderaen O.K.: J. Magn. Magn. Mat. 12-18 (1980).867 (apin densit7 in thin Ni(100) filma); Vang C.S. and Preeaan A.J.: Pb7a. Rev. B 21 (1980), 11585 (aurface atatea. surtace

[5] [6] [7] [8]

[9] [10] [11] [12]

[13] [14] [15] [16] [17]

magnetism and electron spin polarization in nine-layer Ni(100) films). Gunnarsson O. and Jones R.O.: J. Chern. Phys. 72 (1980), 5357 Jones R.O.: J. Chern. Phys. 11 (1979), 1300 Harris J. and Jones R.O.: J. Chern. Phys. 68 (1978), 1190 (group IA dimers)j Phys. Rev. A 19 (1979), 1813 (group IVA dimers) A survey of the results obtained for diatomic molecules using LSD and Xa functionals is given by Baerends E.J. and Ros P.: Int. J. Quantum Chern. S ~ (1978), 169. See also Dunlap B.I., Connolly J.W.D. and Sabin J.R.: J. Chern. Phys. 11 (1979), 4993 Kitaura K., Satoko C. and Morokuma K.: Chern. Phys. Letters £2 (1979), 206 Moruzzi V.L., Janak J.F. and Williams A.R.: 'Calculated Electronic Properties of Metals', Pergamon, Oxford (1978) Baerends E.J.: presented at conference on the Calculation of Atomic Positions at Solid Surfaces, Gif-sur-Yvette, France, May 1980 Examples of approaches to the embedding problem are Grimley T.B. and Pisani C.: J. Phys. C I (1974), 2831; Hyman E.A.: Phys. Rev. B 11 (1975), 3739; Gunnarsson O. and Hjelmberg H.: Phys. Scr. 11 (1975), 97 Harris J. and Jones R.O.: J. Phys. C ~ (1974), 1170 Gunnarsson O. and Lundqvist B.I.: Phys. Rev. B 12 (1976), 4274 Gunnarsson O. and Jones R.O.: Phys. Scr. ~ (1980), 394 Zunger A., Perdew, J.P. and Oliver G.L.: Solid State Commun. ~ (1980), 933 Gunnarsson O. and Jones R.O.: to be published

41