Polarization induced surface dipole moments

Volume 57A, number 2

PHYSICS LETTERS

31 May 1976

POLARIZATION INDUCED SURFACE DIPOLE MOMENTS E. ZAREMBA* Department of Physics, University of California San Diego, La Jolla, California 92093, USA Received 4 February 1976 The work of Antoniewicz on the polarization induced surface dipole moment of a metal-atom system is generalized to include the retarded response of the metallic electrons.

The change in work function arising from the adsorption of atoms on metal surfaces is due to the change in the net surface dipole moment of the metal-atom system. Antoniewicz [1] has recently pointed out that part of this change can be ascribed to the polarization (or van der Waals) forces between the atom and the metal. In treating this interaction it was assumed that the metallic electron density responded immediately to the instantaneous dipole moment of the atom which corresponds to the assumption of the atom interacting with its electrostatic image in the metal. Using this approximate form of the interaction, the induced atomic dipole moment was obtained in a variational manner. In this note we present a generalization of this previous calculation which takes into account the retarded response of the metallic electrons and thereby treats the atom and the metal on an equal footing. Furthermore, as is important in problems of this nature, a reference plane is defined with respect to which the position of the atom near the metal surface is to be reckoned. We imagine that the metal-atom separation is sufficiently large for overlap between the atom and metal wavefunctions to be negligible. In this situation, exchange effects can be neglected and the interaction in the non-relativistic limit is simply the Coulomb interaction V=fdrfdr’p~~(r)v(r_r’)pm(r’).

(1)

Here, pt(r) is the total charge density of the ith subsystem (i = a,m) and u(r) = if r is the Coulomb potential. (Atomic units are used throughout, el = m = 11 = 1.) The electronic density of the ith subsystem can be obtained conveniently starting from an expression found in ref. [2]. To second order in V the change in the electronic density ~n’(r) = fli(T) nb(r) between the noninteracting and interacting systems is found to be —

(~n1(r))=

—

f ~fdr1

dr4 u(r1 —r3) u(r2

—

r4)Di(r1, r2, r; i~)x’(r3 r4 i~) ,

(i’~f) .

(2)

Here xkr, r’; i~)and Di(u, u’, r;i~)are the continuations to complex frequencies of the Fourier transforms x~(r,r’;w)i

f d(t~t’)e ?(t_t’)(T[~nl(r,t)5n1(r’,t’)})

and

*

National Research Council of Canada postdoctoral fellow.

156

,

(3)

Volume 57A, number 2

Ii(u,u’,r;w)

PHYSISC LETTERS

fdt

31 May 1976

fdt’e_4~.~(t_t’)
(4)

T is the time ordering operator and the angular brackets denote ground state expectation values. In obtaining (2) we have made use of the fact that the transforms defined in (3) and (4) have poles only in the second and fourth quadrantsof the complex-co plane. The expression for the perturbed density in (2) is equally valid for both the atomic and metallic subsystems. Its evaluation requires knowledge of the density-density response function of one subsystem and the triplet correlation function defmed in (4) of the other. This latter quantity has the spectral representation

rdwrdw’rfi(~-°’’°) f2(c~,w’)

D’(u,u ,r;i~)=i

—i

—

J 2irJ 2ir

1

(5)

+

I, Lw’(w—i~) (co—i~)(co’—i~)J

where the spectral functions are given by ~

‘

(6)

and -f2(co, co’)

fdt fcit’ e_1Wte1~~)t([ni(r), fl~(u,t)J, fli(U’, r’)}

.

In the following we restrict our attention to the dipole moment induced on a hydrogen atom (1 = a) near the surface of a simple metal represented by jellium. Because of the translational invariance along the surface, only the z-component of the atomic dipole moment is finite: _fdrz(~na(r)).

(8)

Introducing the two-dimensional Fourier transformof the Coulomb potential in (2), we obtain from (8) the expression

is),

4f(q,i~)g(q, =

(9)

i~ (2)~ e2QZJ ~

where f(q, i~)= fci.rf dr’ e_iKtej~~*T~Xm(r, r’;i~) and g(q,i~)= J’du

fdulf drei~~e_iIC*.u’zDa(u, u’, r;

i~)-

(11)

In (9), L2 is the surface area of the metal, q is a two-dimensional surface wavevector and Z is the position of the atomic nucleus relative to the surface. We have also introduced the complex wavevector x q + iq2. By expandingf(q, i~)and g(q, i~)in powers of q, an expansion of the dipole moment in inverse powers of Z is generated. The expansion off(q, i~)takes the form [3] f(q,i~)=jL 2 ~w+~

r 2

11+2q L

ZC+~2ZB1

CO2

1+0(q3),

~‘

w2

÷~2

(12)

j

157

Volume 57A, number 2

PHYSICS LETTERS

31 May 1976

where w~,is the surface plasmon frequency, ZB is the position of the edge of the positive jellium background and ZC is the centroid of the induced surface charge as defined in ref. [4]. Because of the inversion symmetry of the ground state atomic wavefunction, we note that the leading term in the expansion of g(q, i~)is of order q3,with corrections of order q5. In evaluating D”(u, u’, r; i~)we make the approximation of replacing all excitation energies in its spectral representation by a common value, E. If this is done we find that Da(u,u’,r;i~)~—2 3E2+ ~2 ‘ (~fl~2(u)~Sna(u)5na(u’)t5n~z(r)). (E2+~2)2 Substituting this result together with (12) into (9) and performing the indicated integrations we find pZ

9

(.0

4Z~E

=—-=~

(.0

-t-3Ef2r

~ — ~ (w~,+E)2 L

4Z Z

1

(13)

(14)

J ,

with 2

Zo—ZB=— 14w~~ +9w E+3E

(Z

—z

(15)

).

22w~2~+5w

5~E+3E2 C

B

The quantity Z0 can be used [3,4] to define a reference plane from which the position of the atom is to be measured when evaluating the induced dipole moment. If the limit w5~,-÷°° is taken and E is chosen to be the ionization energy of hydrogen, i.e. E = 4 (au), the expression in (14) reduces to the result of Antoniewicz, apart from the correction of order Z 5. For the simple metals, is smaller than E and the resulting value of p~is typically reduced by a factor of two From the above considerations it is clear that the variational calculation [1] of the atomic wavefunction, is equivalent to the approximation leading to (13). However, a more accurate determination of the atomic dipole moment can be obtained by explicitly evaluating the spectral functions given in (6) and (7). This aspect is currently under investigation for the example of the rare gas atoms and should indicate whether an expression such as (13) with an appropriately chosen average energy is reliable. As mentioned previously, (2) also provides an expression for the induced density in the metal. This quantity is more difficult to evaluate but its contribution to the net surface dipole moment may in fact be less important than that of the atom. This is suggested by considering the linear response of a metal to a static dipole moment oriented perpendicularly to the surface. The average of the induced surface density over any plane parallel to the surface is zero and so does not contribute to the surface dipole moment. —

References [1] [21 [3] [4]

158

P.R. Antoniewicz, Phys. Rev. Lett. 32 (1974) 1424. A.L. Fetter and J.D. Walecka, Quantum theory of many-particle systems (McGraw Hill, New York, 1971) p.83, eq. (8.1). E. Zaremba and W. Kohn, Phys. Rev. B13 (1976) 2270. N.D. Lang and W. Kohn, Phys. Rev. B7 (1973) 3541.