Correlated wave function theory of metal surfaces

NF 7

Physica 108B (1981) 871-872 North-Holland Publishing Company

CORRELATED

WAVE

FUNCTION

Chia-Wei

THEORY

OF METAL

Woo, Xin Sun and Tiecheng

SURFACES

Li

Department of Physics, University of California, San Diego La Jolla,

California

92093,

U.S.A.

Electronic density profiles and surface energies are calculated for metal surfaces over a wide range of densities in the jellium approximation. The theory m a k e s use of correlated wave functions, which consist of determinants of model single particle orbitals (taken to be solutions for a variational surface potential) modified by a pair correlation factor. Integral equations relating the wave function and the distribution functions previously derived by one of the authors and his coworkers are employed. Divergences which appear in the integral equations and the energy expression are shown to cancel exactly. Surface energies obtained are close to those extrapolated from experimental data on liquid metals, higher than those of Lang and Kohn using the density functional formalism, and cornparable to those of the same authors after including discrete lattice corrections. W e discuss here the implications of these results, surface excitation spectra, renormalization of adatom wave functions, and effective interactions between adatoms.

The conventional many-body theoretic treatment of the metal surface is by using the density functional formalism invented by Kohn and cow o r k e r s (1). W e h a v e s o u g h t a n a l t e r n a t i v e w h i c h w i l l (i) p r o v i d e u s w i t h s u r f a c e w a v e f u n c t i o n s ; (ii) v a r y t h e w a v e f u n c t i o n r a t h e r t h a n t h e d e n s i t y f u n c t i o n ; (iii) b e f r e e of a d e n s i t y g r a d i e n t e x p a n s i o n ; (iv) t r e a t t h e h o m o g e n e o u s b u l k a n d t h e i n h o m o g e n e o u s s y s t e m in t h e s a m e f r a m e w o r k ; a n d (v) l e n d i t s e l f to s y s t e m a t i c improvement as well as extensions beyond the jellium approximation without resorting to a flrst-order p e r t u r b a t i o n t h e o r y . We h a v e i n t h e main succeeded to find such a theory. This is a report on our progress, and on future applicat i o n s of t h e t h e o r y t o t r e a t i n g s u r f a c e e x c i t a tions and adsorption.

the metal. Our trial wave function for the g r o u n d s t a t e , d~, t a k e s t h e f o r m of a F e e n b e r g J a s t r o w f a c t o r : F =- ~ e x p 1 u ( r . . ) m u l t i p l i e d

i
to a determinant

U

of s i n g l e p a r t i c l e

orbitals:

~DC~(~). W e choose u(r) = -2e2(~Wp)-l[(l-e-br)/r], where t~ ~ 4rrpe2/m denotes the plasmon freF quency, and b is a variational parameter whose value is taken to be that determined for the b u l k (2), s o a s t o g u a r a n t e e t h a t d e e p in t h e m e t a l t h e a s y m p t o t i c b e h a v i o r of t h e w a v e f u n c tion is consistent with that for the bulk. F o r t h e s i n g l e p a r t i c l e o r b i t a l s , we t a k e t h e e i g e n f u n c t i o n s of an e l e c t r o n m o v i n g i n s o m e model field v(r). ['2~

v 2 + v ( ~ ) ] ¢Pa( ~ ) = ~acOa ( ~ ) "

(2)

Our theory begins with the Hamiltonian

H =~

_I~2V2

•

,

2m

i

i~~ +

2 e

..,

_

• j l~i-rj I

i

dR'

~fd •

e 2n+(~) ~ ._.~___

[ri_i~ I

n+(R)n+(R') i 'l

,

,

v(~) w i l l b e p a r a m e t e r i z e d . In f a c t t h i s i s h o w the variational parameters enter our calculat i o n s , v(~) i s n o t , b u t i s u n d o u b t e d l y r e l a t e d t o , the effective surface potential. Ours represents a p a r a m e t r i c a p p r o a c h t o w a r d d e t e r m i n i n g it self- consistently.

(t)

where n+(R) denotes an input static distribut i o n of p o s i t i v e c h a r g e s . In t h e j e l l i u m a p p r o x imation, it is a semi-infinite step function

which ends abruptly on the Gibbs surface, n(~), the electron density profile, and n+(~) both approach the m e a n bulk density p deep inside

0Y/8-4363/81/0000-0000/$02.50 © North-HollandPublishingCompany

The actual calculation proceeds in four steps. F i r s t , we s -@ o l v e t h e e i g e n v a l u e e q u a t i o n-@ (2) f o r a c h o s e n v ( r ) . In t h e p r e s e n t c a s e , v ( r ) i s taken to vary only along the direction z normal to the jellium surface and have a two-parameter h y p e r b o l i c t a n g e n t f o r m . T h e e i g e-@n s o l u t i o n s a r e hypergeometric f u n c t i o n s (3). v ( r ) h a s t o b e positioned carefully relative to the Gibbs surface

871

872 in order to preserve charge neutrality. This is accomplished by a detailed algebraic analysis. We then obtain a non-interacting density funct i o n n 0 ( ~ ) f r o m t h e s e t of t h e s i n g l e p a r t i c l e e i g e n f u n c t i o n s . S e c o n d , we c a l c u l a t e v i a t h e i n t e g r a l e q u a t i o n (4, 5)

~[n(z,l X)/n(zllO)] =f d ~ l f d r z n ( z z [

~')u(r12)g(1,Z)

0

1 f X ,r-9 -9 + 2J0 d~ Jdr 2dr 3 n(z21 ~')n(z3[ ~')u(r23)

x[g(1, Z, 3) - g(Z, 3)] ,

(3)

where n0(T) =- n(z 10); the density profile n(~) n(z[l). The two- and three-particle distribution functions n(z I) n(z2) g(l, 2) and n(zl) n(zz)n(z3)g(l, Z, 3) are governed by hlgher-order coupled integral equations llke Eq. (5). In the third step we must determine g(l, Z) and g(l, 2, 3). Preliminary calculations use the bulk radial distribution function gB(rl210 ) for g(l, 2) and a convolution approximation for g(l, Z, 3). Finally the energy expectation value E ---(%bISlt~>/(~ I~> is obtained and the bulk energy subtracted analytically to yield the surface energy. Since all variational parameters enter only the surface energy part, the latter can be minimized by itself to conclude the calculation. The only theoretical difficulty encountered is t h e a p p e a r a n c e of d i v e r g e n c e s in t h e i n t e g r a l e q u a t i o n (3) a n d t h e s u r f a c e e n e r g y e x p r e s s i o n . I n e a c h c a s e , p r o p e r c o n s i d e r a t i o n of t h e c l e a v ing p r o c e s s , i . e . , o n e t h a t p r e s e r v e s t h e s y m m e t r y b e t w e e n t h e t w o h a l v e s of b u l k m e t a l , l e a d s t o e x a c t c a n c e l l a t i o n of t h e s e d i v e r g e n c e s . Our cMculated density profiles resemble those of L a n g a n d K o h n (6) e x c e p t f o r s m a l l r s . T h e s u r f a c e e n e r g i e s o b t a i n e d (7) f a l l o n t h e d a t a e x t r a p o l a t e d f r o m s u r f a c e t e n s i o n s of l i q u i d m e t a l s . T h e y l i e m u c h a b o v e t h o s e of L a n g a n d E o h n . If t h e c o r r e c t i o n s d u e t o d i s c r e t e n e s s of the lattice are indeed as big as that given by a first-order p e r t u r b a t i o n t h e o r y (6), o u r r e s u l t s will be too high, and much improvement must -9 be brought about by refining our variational v(r). O n t h e o t h e r h a n d , r e c e n t i n d i c a t i o n s (8) a r e that the corrections may well be rather minute if t h e e l e c t r o n s a r e p e r m i t t e d to r e s p o n d to a -9 change in n+(r). While a first-order perturbation calculation does not allow the electrons to d o s o , v a r i a t i o n a l c a l c u l a t i o n s s u c h a s o u r s do. There is hope, then, for determining the source of d i s c r e p a n c y b e t w e e n p r e s e n t w o r k a n d R e f . 6. W e h a v e put a l a y e r e d s t r u c t u r e i n t o n+(~) a n d carried out test calculations.

S u r f a c e e x c i t a t i o n s c o n s i s t of b o t h s i n g l e p a r t i c l e a n d c o l l e c t i v e b r a n c h e s . We h a v e a s s e m bled trial wave functions for both. Single quasiparticles are described by taking one (or a finite number of)¢Pa(~) above the Fermi s e a , and c o l l e c t i v e e x c i t a t i o n is g i v e n i n t h e l o w e s t o r d e r b y applying a density fluctuation operator on ~ . Estimates on the excitation spectra have been made. T h e a v a i l a b i l i t y of w a v e f u n c t i o n s f o r t h e m e t a l surface permits us to evaluate the matrix elements between an unexcited adatom and an adatom coupled to a surface excitation. These matrix elements are vertices which can be eliminated by canonical transformations to yield both t h e e f f e c t i v e m a s s of a r e n o r m a l i z e d a d a t o m and the effective interaction between two adatoms. The formalism and preliminary results for these properties which are central to the c o n s i d e r a t i o n of a d s o r p t i o n p h e n o m e n a w i l l b e presented in the Conference but be published elsewhere. This work was supported in part by the U.S. N a t i o n a l S c i e n c e F o u n d a t i o n t h r o u g h G r a n t No. DMR80-08816, and by Fudan University ( S h a n g h a i , C h i n a ) a n d t h e I n s t i t u t e of P h y s i c s ( B e l j i n g , C h i n a ) w h i c h s u p p o r t e d X. S u n a n d T. L i d u r i n g t h e i r r e s e a r c h l e a v e i n t h e U . S . s i n c e 1979.

REFERENCES

[11

H o h e n b e r g , P. a n d K o h n , W . , P h y s . R e v . 136 ( 1 9 6 4 ) B 8 6 4 ; I
[Z] Chakravarty, S. and Woo, C.-W., Phys. Rev. BI3 (1976) 4815. [3]

Senbetu, L. and Woo, C.-W., B18 (1978) 3251.

Phys.

[4]

Rajan, V.T. and Woo, B 1 8 (1978) 4 0 4 8 .

[5]

Lin-Liu, Y.R., Chakravarty, S. a n d W o o , C . - W . , P h y s . R e v . C18 (1978) 516.

[6]

L a n g , N . D . a n d K o h n , W. , P h y s . R e v . B1 (1970), 4 5 5 5 ; L a n g , N . D . , S o l i d S t a t e P h y s .

C.-W.,

Rev.

Phys. Rev.

Z8 (1973) 225. [7]

[8]

Sun, X . , L i , T. a n d W o o , C . - W . , to b e published in a Springer-Verlag monograph. Monnier,

R. and Perdew,

A.P.S. 26 (1981) 427.

J.

P.,

Bull.