- Email: [email protected]
Exchange corrections to the density-density correlation function at a surface
Surface Science 68 (1977) 368-376 0 North-Holland Publishing Company
EXCHANGE CORRECTIONS
TO THE DENSITY-DENSITY
CORRELATION
FUNCTION AT A SURFACE
Greger LINDELL NORDITA,
Blegdamsevej I 7, DK-2100
Copenhagen
(b, Denmark
The problem of calculating the density-density correlation function and of including the effects of exchange and correlation are discussed. It is shown in a model calculation how a particularly simple surface model can be easily extended to include exchange corrections. The effects of these corrections on the surface plasmon dispersion are found to be small.
1. Introduction
An important concept in the theory of the homogeneous sity-density correlation function
x(r, r’, t - t’) =iO(t
electron gas is the den-
- t’) <[n(r, t), n(r’, t’)]) ,
which contains information about both the static and the dynamical properties of the bulk system. This quantity is also useful for the study of extremely inhomogeneous electron systems such as metal surfaces, and has received attention from many workers [l-5] who have studied the properties of surface plasmons and the surface energy in different surface models. This paper will deal with two aspects of the density-density correlation function at surfaces, how to go beyond the Hartree or mean-field approximation to treat exchange and correlation and how to formulate the Hartree problem such that it can be numerically solved on a computer for any surface model. In section 2 we study the general problem of introducing local field corrections to the mean-field theory. This is done by a generalization of the method of Singwi et al. [6] to the surface case. In section 3 we look at the shape of the exchange hole around an electron close to a surface. In section 4 we illustrate how the very simplest exchange corrections to the Hartree response function, also in approximate form, leads to an almost analytically solvable expression for the surface plasmon dispersion relation. We will use the (semi-) Classical Infinite Barrier Model [7] where the response function for the non-interacting electron gas at a surface is approximated by a 368
G. Lindell /Exchange
369
corrections
direct and a reflected part x0@ - r’, cd) = x&Jr
- r’, cd) + &lk(r*
(2)
- r’, w) .
The vector r* is the mirror image of r in the surface plane.
2. Correlation corrections to the response function The response of a classical gas of interacting particles to an external potential can be studied by linearizing the equation of motion. The equation thus obtained can also be applied to the electron gas as was pointed out by Singwi et al. [6] (forthwith referred to as SSTL). They were able to reduce the quantum mechanical equation of motion for the density-density response function to the classical form by a truncation of higher-order Green’s functions. In this way they were able to calculate corrections to the simplest RPA treatment of the response function in an electron gas. These local field corrections are expressend in terms of the static correlation function g(rl, rz). Before writing down a formula for the induced charge in the SSTL formalism valid at a surface we have to describe our nomenclature briefly. The surface is assumed to be perpendicular to the z-direction and all vectors, both in real and in fourier space, will be divided into a component along the z-axis and a remaining part. Furthermore, this separation will be explicitly shown for the fourier transform of a function. r = R + ze,
,
q=Q+qe,,
f(r) = Qq w$iQz) exp(hz) ./IQ, 4) . The Fourier transform &rr,
d - 1=
X exp[iQ.
(3)
of ,g(r,, r2) - 1 is written as
c WQ, kl, kd f?,kl& (R, -R,)]
exp[i(krzr
- k2z2)] .
Note that two variables are needed for the transformation ordinary bulk result is simply retrieved by putting
WQ, k,> k2)
= &utk(Qt
k,)
hklk2
dt x’(r’, t’) P”(r’,
in the z-direction.
t’) ,
The
(5)
.
whereupon the FWA-assumption that X0, the irreducible nects the effective potential with the induced density p(r, t) = ldr’
(4)
polarization
diagram, con-
(7)
370
G. Lindeil /Exchange
can be used to obtain an integral equation
P(Q, z) =Jda”
corrections
for the induced charge:
x”(Q, z, z”) V=‘(Q, 2”)
+ sda’
p(Q, z’) .I’dzU x’(Q, z, z”) u(Q, z” - z’)
+sh’
p(Q, 2,~“) Cexp[iqz”) Q
X kxq, H(Q - Q’, 4 - 4’, k2) 7’
u(4’) exp [-i(kz
+ 4’)z’f .
2,
Or, if we would like, an integral equation for the response tunction x that connects the external potential Fxt with the induced charge p. This step was the natural continuation in the bulk case, where an analytical solution could be obtained. In this article we will restrict ourselves to some of the consequences of eq. (8) for the surface plasmon dispersion in a simple model. In order to do this, we will first study the implications of a surface on the Hartree correlation function which will be used as a basis for refinements that include exchange.
3. Calculation of the exchange hole
We have seen how the exchange-correlation hole or g(r,, r2) enters the calculation of induced charge as a function of external potential, and also the plasmon dispersion. In this section we will calculate this depletion of charge around a particular electron as due to exchange effects only. This exchange hole will then be included in the formulae of the previous section as a first approximation. In this way we will essentially have made a treatment of the surface response equvalent to what was done in the bulk case by Hubbard [ 81. I.& us start by writing down the formal relation between the response function, the structure factor, and the probab~ity P(r,, rt) of finding one electron at rl and another at r2:
S(rl,r2)=-fj dwImx(rl,r2,w>, 0
To keep only exchange effects amounts to replace x in this expression by x0. It turns out to be rather cumbersome to calculate g(rt, r2) in this fashion. Only for the case, consistent with the CIBM model, where we negIect all terms except those that depend on z - Z’ or z + z’ is it possible to rearrange the terms in a simple
371
G. Lindell / Exchange corrections
way to give 9Z(kdrl
r*) = 1 g CIBM(r,,
-
r21)
2 (kkArI - r2l)* 6
= (x2,y2,
-22)
9j:(kF(rl
- r;l)
2 (kFlrl
- rz/)*
,
(10)
where r; is the mirror image of r2. A simpler approach is to use the connection particle density matrix for fermions: dr,,r2)
C
=
’
between
p(r,, r2) and the one-
(11)
Jli(rl) tik(r2) .
Ikl
The function g(rl, r2) is then given by g(rl,r2)=
1
-’2 p(rl,Idrl,r2)12 rdp(r2t
where p(rl, r,) is to the evaluation computer. In the bulk and a surface
r2)
(12)
’
the density at rl. The calculaton of p(rl, r2) immediately reduces of a one-dimensional integral and can easily be performed on a regions of flat potential it is in fact possible to isolate an ordinary contribution:
“F
drl,r2)
s
=
dk (k$ - k*)“* @&,) p&a)
Ji(R(k;:
- k2)1’2)
0 =
k$jl(kFIr1 7r2 kFkl
r2l)
- r2l
“F
1
dk (k; - k*)“* cos[k(z + z’) + 2&k] J1(R(k$ - k*)“*)
+n2R s
.
(13)
0
It is assumed in this expression that the electronic wavefunctions are plane-wavelike parallel to the surface and that their z-dependent parts are phase-shifted trigonometric functions inside the surface potential barrier. A possible ansatz for this barrier is an infinite potential step, the Infinite Barrier Model (IBM). The exchange hole assumes a particularly simple form in this model dr,,r2)
=
1
jl(kdrl - hi)_ jdkdr, -r2*1) 2
-2
2
k&,
- r2l
kdr, -- r2l
I
(14)
There are structural similarities between this result and the corresponding expression in the CIBM model as given by eq. (lo), where only the cross product between
G. Lindell /Exchange
372
corrections
0
-5 ztau) Fig. 1. The exchange hole in the CIBM model drawn whereg assumes the values 0.9, 0.8, etc.
for kF = 1 (rs = 1.92).
The contour
lines are
. A
y(au) O-
((
-L
I
I
-5
0 z(a u
1
Fig. 2. The exchange hole evaluated with wavefunctions kF = 1 and the step height = 1.5 times the Fermi energy.
solving
a step barrier
model
with
G. Lindell /Exchange corrections
313
the two terms to be squared in eq. (14) is missing. These terms would depend on both z and z’ and are physically significant since the exchange hole in the CIBM model can assume values below 0.5 close to the surface. A contour plot of the exchange hole in the CIBM model and a model where the surface is represented by a finite step barrier are shown in figs. 1 and 2. The surface in these plots is located at z = 0 and the CIBM hole resembles the overlap between two holes located symmetrically relative to this surface. In the finite barrier case, the hole will be defined for all values of z and will extend into the vacuum region. We would like to point out that the functions calculated here are almost equivalent to the ones studied by Moore and March [9] by quite another method, except that for our purposes it has been more convenient to concentrate on g(r, , rz) rather than on P(r, , rz) as they have done.
4. Surface pasmon dispersion in the CIBM model with exchange corrections In section 2 we saw that the induced integral equation
~((2, z) = j
dz’ HQ, z’) j
_m
charge of the surface plasmons fulfill the
dz” x”(Q, z, z”, w) F(Q, z”, z’) ,
-03
(15)
where F(Q, z”, z’) contains both the Hartree potential and the local field corrections. In this section we will solve this equation with two approximations. Firstly, the function H will contain only the exchange part of the exchange-correlation hole, and secondly this hole will be given only by the first two terms in the CIBM approximation (eq. (10)). This, in fact, leaves us only with bulk corrections to the Hartree picture, on the other hand it leads to an integral equation of a particularly simple form. In eq. (16) is shown the explicit form of F(Q, 4, o) including a further approximation to the analytical form of G(Q, 4) made by Hubbard (see Singwi et al.) [8,6].
F(Q, 4) = HQ> 4) 11 - G(Q, 411 = u(q) - 1/2(Q* + q* + 9;) . The Fourier representation
x&&Q.
(16)
- in the CIBM model - of x0 (Q, z, z”, w) is
z, z”, w) = r dq{exp[i& --m
- z”)] + exp[iq(z + z”)]} x”,(q, w) ,
(17)
where x! is the Lindhard function. It is tempting to proceed directly by integrating the variable z” but one must be careful here, since the integration range only extends over a half-space so we do not
374
G. LindeN /Exchange
obtain a simple delta-function 0
s
dz” exp[i(q’ - s)z”] = -iP
-03
corrections
of 4 - (7’ but rather 1 Y 6-4
+ w4
(18)
- 4% >
and it is the principal parts term that includes the information relevant to our surface plasmon calculation. A few lines of algebra lead to the result
~z'PCQ,~') 7 & {expPtqz+Qz'N -co _m
P(Q,z) =I& i
X [l- ~(Q,q,w)l +exp[Kqz +(Q2+ &)“*)I [1 - e(Q,q, w)lG(Q,q) + Cevti& - ~'11 + expbh + ~'111 I1- 4Q, 4,~111I
(19)
where e(Q, 4, w) and o(Q, 9, w) are defined as
4Q,rl, a>= 1 + ~(Q,q)x~~Q,q, o), (20)
4Q,qtw)= 1+ 4Q,q)x%Q,q,dG(Q,q).
One now takes the cosine transformation of eq. (19) since this immediately will replace the integra1 over the first two (bulk) terms to simple products. It has to be a cosine rather than a Fourier transform since our functions only are defined for z
(21) where the kernel K(k, K) is of the separable type 2 kl(k,
K)
= c
U&C)
Q(K)
,
i= 1
u,(k) =
$Q, 4, @I- 1 a(Q,q,w) ' -G(Q,k)u,(k),
@z(k)=
and a solution is given by
ajj =
s
dK f+(K)
_m
1 “I(K,=~
2nQ2 +
K=
’
u2~fo = (Q2+ qt)“= uItKj Q '
exists if Det (1 “- A) vanishes since this is a homogenous
Q(K)
.
(221 (22)
equation. A
(23)
G. Lindell /Exchange
1.20
I
I
I
375
corrections
I
1.15 -
W/E, 1.10 -
1.05 -
.90 .O
I
I
I
I
.05
.lO
.15
.20
.25
Q/K, Fig. 3. The surface plasmon exchange COrreCtiOnS(------)
dispersion fOrkF
in the ,Richie-Marusak
model ( -)
and with
= 1.
Notice that when C(Q, 4) is zero, the criterion reduces to
l-
dK e(Q, _,%
K,
e(Q,
w) K,
o)
Q
1 m
=”
(24)
which is exactly the relation found by Richie and Marusak [lo]. In fig. 3 we show the surface plasmon dispersion relation including exchange corrections compared with the Ritchie-Marusak result. The inclusion of exchange corrections does lower the dispersion curve although not to a significant degree. We believe that a realistic calculation of this quantity with a better approximation to the irreducible polarization diagram should include exchange corrections at least to the order outlined here. Harris [ 1 l] has also treated exchange corrections to surface plasmon dispersion and found corrections of the same magnitude as we have found here.
5. Summary This article has dealt with some aspects of the calculation of response functions at surfaces. The treatment of realistic potentials and of exchange-correlation effects
316
G. Lindell / Exchange corrections
has been outlined. It has been shown that the exchange effects in a simple surface model can be included without significantly increasing analytical or numerical comple~ty~
References [ I] [2] [3] [4] [S] [6] [7] [8] [9] [ 101 [II]
D.M. Newns, Phys. Rev. Bl (1970) 3304. V. Peuckert, 2. fhysik 241 (1971) 191. P.J. Feibelman, Phys. Rev. B9(1974) 5077. J.E. Inglesfield and E. Wikbog, Solid StateCommun. 14 (1974) 661; 16 (1975) 335. D.C. Langreth and J.P. Perdew, Solid State Commun. 17 (1975) 1425. K.S. Singwi, M.P. Tosi, R.H. Land and A. Sjalander, Phys. Rev. 176 (1968) 589. A. Griffin and J. Harris, Can. J. Phys. 54 (1976) 1396. J. Hubbard, Proc. Roy. Sot. (London) A243 (1957) 336. I.D. Moore and N.H. March, Ann. Phys. (NY) 97 (1976) 136. R.H. Ritchie and A.L. Mamsak, Surface Sci. 4 (1966) 214. J. Harris, J. Phys. CS (1972) 1757.
EXCHANGE CORRECTIONS
TO THE DENSITY-DENSITY
CORRELATION
FUNCTION AT A SURFACE
Greger LINDELL NORDITA,
Blegdamsevej I 7, DK-2100
Copenhagen
(b, Denmark
The problem of calculating the density-density correlation function and of including the effects of exchange and correlation are discussed. It is shown in a model calculation how a particularly simple surface model can be easily extended to include exchange corrections. The effects of these corrections on the surface plasmon dispersion are found to be small.
1. Introduction
An important concept in the theory of the homogeneous sity-density correlation function
x(r, r’, t - t’) =iO(t
electron gas is the den-
- t’) <[n(r, t), n(r’, t’)]) ,
which contains information about both the static and the dynamical properties of the bulk system. This quantity is also useful for the study of extremely inhomogeneous electron systems such as metal surfaces, and has received attention from many workers [l-5] who have studied the properties of surface plasmons and the surface energy in different surface models. This paper will deal with two aspects of the density-density correlation function at surfaces, how to go beyond the Hartree or mean-field approximation to treat exchange and correlation and how to formulate the Hartree problem such that it can be numerically solved on a computer for any surface model. In section 2 we study the general problem of introducing local field corrections to the mean-field theory. This is done by a generalization of the method of Singwi et al. [6] to the surface case. In section 3 we look at the shape of the exchange hole around an electron close to a surface. In section 4 we illustrate how the very simplest exchange corrections to the Hartree response function, also in approximate form, leads to an almost analytically solvable expression for the surface plasmon dispersion relation. We will use the (semi-) Classical Infinite Barrier Model [7] where the response function for the non-interacting electron gas at a surface is approximated by a 368
G. Lindell /Exchange
369
corrections
direct and a reflected part x0@ - r’, cd) = x&Jr
- r’, cd) + &lk(r*
(2)
- r’, w) .
The vector r* is the mirror image of r in the surface plane.
2. Correlation corrections to the response function The response of a classical gas of interacting particles to an external potential can be studied by linearizing the equation of motion. The equation thus obtained can also be applied to the electron gas as was pointed out by Singwi et al. [6] (forthwith referred to as SSTL). They were able to reduce the quantum mechanical equation of motion for the density-density response function to the classical form by a truncation of higher-order Green’s functions. In this way they were able to calculate corrections to the simplest RPA treatment of the response function in an electron gas. These local field corrections are expressend in terms of the static correlation function g(rl, rz). Before writing down a formula for the induced charge in the SSTL formalism valid at a surface we have to describe our nomenclature briefly. The surface is assumed to be perpendicular to the z-direction and all vectors, both in real and in fourier space, will be divided into a component along the z-axis and a remaining part. Furthermore, this separation will be explicitly shown for the fourier transform of a function. r = R + ze,
,
q=Q+qe,,
f(r) = Qq w$iQz) exp(hz) ./IQ, 4) . The Fourier transform &rr,
d - 1=
X exp[iQ.
(3)
of ,g(r,, r2) - 1 is written as
c WQ, kl, kd f?,kl& (R, -R,)]
exp[i(krzr
- k2z2)] .
Note that two variables are needed for the transformation ordinary bulk result is simply retrieved by putting
WQ, k,> k2)
= &utk(Qt
k,)
hklk2
dt x’(r’, t’) P”(r’,
in the z-direction.
t’) ,
The
(5)
.
whereupon the FWA-assumption that X0, the irreducible nects the effective potential with the induced density p(r, t) = ldr’
(4)
polarization
diagram, con-
(7)
370
G. Lindeil /Exchange
can be used to obtain an integral equation
P(Q, z) =Jda”
corrections
for the induced charge:
x”(Q, z, z”) V=‘(Q, 2”)
+ sda’
p(Q, z’) .I’dzU x’(Q, z, z”) u(Q, z” - z’)
+sh’
p(Q, 2,~“) Cexp[iqz”) Q
X kxq, H(Q - Q’, 4 - 4’, k2) 7’
u(4’) exp [-i(kz
+ 4’)z’f .
2,
Or, if we would like, an integral equation for the response tunction x that connects the external potential Fxt with the induced charge p. This step was the natural continuation in the bulk case, where an analytical solution could be obtained. In this article we will restrict ourselves to some of the consequences of eq. (8) for the surface plasmon dispersion in a simple model. In order to do this, we will first study the implications of a surface on the Hartree correlation function which will be used as a basis for refinements that include exchange.
3. Calculation of the exchange hole
We have seen how the exchange-correlation hole or g(r,, r2) enters the calculation of induced charge as a function of external potential, and also the plasmon dispersion. In this section we will calculate this depletion of charge around a particular electron as due to exchange effects only. This exchange hole will then be included in the formulae of the previous section as a first approximation. In this way we will essentially have made a treatment of the surface response equvalent to what was done in the bulk case by Hubbard [ 81. I.& us start by writing down the formal relation between the response function, the structure factor, and the probab~ity P(r,, rt) of finding one electron at rl and another at r2:
S(rl,r2)=-fj dwImx(rl,r2,w>, 0
To keep only exchange effects amounts to replace x in this expression by x0. It turns out to be rather cumbersome to calculate g(rt, r2) in this fashion. Only for the case, consistent with the CIBM model, where we negIect all terms except those that depend on z - Z’ or z + z’ is it possible to rearrange the terms in a simple
371
G. Lindell / Exchange corrections
way to give 9Z(kdrl
r*) = 1 g CIBM(r,,
-
r21)
2 (kkArI - r2l)* 6
= (x2,y2,
-22)
9j:(kF(rl
- r;l)
2 (kFlrl
- rz/)*
,
(10)
where r; is the mirror image of r2. A simpler approach is to use the connection particle density matrix for fermions: dr,,r2)
C
=
’
between
p(r,, r2) and the one-
(11)
Jli(rl) tik(r2) .
Ikl
The function g(rl, r2) is then given by g(rl,r2)=
1
-’2 p(rl,Idrl,r2)12 rdp(r2t
where p(rl, r,) is to the evaluation computer. In the bulk and a surface
r2)
(12)
’
the density at rl. The calculaton of p(rl, r2) immediately reduces of a one-dimensional integral and can easily be performed on a regions of flat potential it is in fact possible to isolate an ordinary contribution:
“F
drl,r2)
s
=
dk (k$ - k*)“* @&,) p&a)
Ji(R(k;:
- k2)1’2)
0 =
k$jl(kFIr1 7r2 kFkl
r2l)
- r2l
“F
1
dk (k; - k*)“* cos[k(z + z’) + 2&k] J1(R(k$ - k*)“*)
+n2R s
.
(13)
0
It is assumed in this expression that the electronic wavefunctions are plane-wavelike parallel to the surface and that their z-dependent parts are phase-shifted trigonometric functions inside the surface potential barrier. A possible ansatz for this barrier is an infinite potential step, the Infinite Barrier Model (IBM). The exchange hole assumes a particularly simple form in this model dr,,r2)
=
1
jl(kdrl - hi)_ jdkdr, -r2*1) 2
-2
2
k&,
- r2l
kdr, -- r2l
I
(14)
There are structural similarities between this result and the corresponding expression in the CIBM model as given by eq. (lo), where only the cross product between
G. Lindell /Exchange
372
corrections
0
-5 ztau) Fig. 1. The exchange hole in the CIBM model drawn whereg assumes the values 0.9, 0.8, etc.
for kF = 1 (rs = 1.92).
The contour
lines are
. A
y(au) O-
((
-L
I
I
-5
0 z(a u
1
Fig. 2. The exchange hole evaluated with wavefunctions kF = 1 and the step height = 1.5 times the Fermi energy.
solving
a step barrier
model
with
G. Lindell /Exchange corrections
313
the two terms to be squared in eq. (14) is missing. These terms would depend on both z and z’ and are physically significant since the exchange hole in the CIBM model can assume values below 0.5 close to the surface. A contour plot of the exchange hole in the CIBM model and a model where the surface is represented by a finite step barrier are shown in figs. 1 and 2. The surface in these plots is located at z = 0 and the CIBM hole resembles the overlap between two holes located symmetrically relative to this surface. In the finite barrier case, the hole will be defined for all values of z and will extend into the vacuum region. We would like to point out that the functions calculated here are almost equivalent to the ones studied by Moore and March [9] by quite another method, except that for our purposes it has been more convenient to concentrate on g(r, , rz) rather than on P(r, , rz) as they have done.
4. Surface pasmon dispersion in the CIBM model with exchange corrections In section 2 we saw that the induced integral equation
~((2, z) = j
dz’ HQ, z’) j
_m
charge of the surface plasmons fulfill the
dz” x”(Q, z, z”, w) F(Q, z”, z’) ,
-03
(15)
where F(Q, z”, z’) contains both the Hartree potential and the local field corrections. In this section we will solve this equation with two approximations. Firstly, the function H will contain only the exchange part of the exchange-correlation hole, and secondly this hole will be given only by the first two terms in the CIBM approximation (eq. (10)). This, in fact, leaves us only with bulk corrections to the Hartree picture, on the other hand it leads to an integral equation of a particularly simple form. In eq. (16) is shown the explicit form of F(Q, 4, o) including a further approximation to the analytical form of G(Q, 4) made by Hubbard (see Singwi et al.) [8,6].
F(Q, 4) = HQ> 4) 11 - G(Q, 411 = u(q) - 1/2(Q* + q* + 9;) . The Fourier representation
x&&Q.
(16)
- in the CIBM model - of x0 (Q, z, z”, w) is
z, z”, w) = r dq{exp[i& --m
- z”)] + exp[iq(z + z”)]} x”,(q, w) ,
(17)
where x! is the Lindhard function. It is tempting to proceed directly by integrating the variable z” but one must be careful here, since the integration range only extends over a half-space so we do not
374
G. LindeN /Exchange
obtain a simple delta-function 0
s
dz” exp[i(q’ - s)z”] = -iP
-03
corrections
of 4 - (7’ but rather 1 Y 6-4
+ w4
(18)
- 4% >
and it is the principal parts term that includes the information relevant to our surface plasmon calculation. A few lines of algebra lead to the result
~z'PCQ,~') 7 & {expPtqz+Qz'N -co _m
P(Q,z) =I& i
X [l- ~(Q,q,w)l +exp[Kqz +(Q2+ &)“*)I [1 - e(Q,q, w)lG(Q,q) + Cevti& - ~'11 + expbh + ~'111 I1- 4Q, 4,~111I
(19)
where e(Q, 4, w) and o(Q, 9, w) are defined as
4Q,rl, a>= 1 + ~(Q,q)x~~Q,q, o), (20)
4Q,qtw)= 1+ 4Q,q)x%Q,q,dG(Q,q).
One now takes the cosine transformation of eq. (19) since this immediately will replace the integra1 over the first two (bulk) terms to simple products. It has to be a cosine rather than a Fourier transform since our functions only are defined for z
(21) where the kernel K(k, K) is of the separable type 2 kl(k,
K)
= c
U&C)
Q(K)
,
i= 1
u,(k) =
$Q, 4, @I- 1 a(Q,q,w) ' -G(Q,k)u,(k),
@z(k)=
and a solution is given by
ajj =
s
dK f+(K)
_m
1 “I(K,=~
2nQ2 +
K=
’
u2~fo = (Q2+ qt)“= uItKj Q '
exists if Det (1 “- A) vanishes since this is a homogenous
Q(K)
.
(221 (22)
equation. A
(23)
G. Lindell /Exchange
1.20
I
I
I
375
corrections
I
1.15 -
W/E, 1.10 -
1.05 -
.90 .O
I
I
I
I
.05
.lO
.15
.20
.25
Q/K, Fig. 3. The surface plasmon exchange COrreCtiOnS(------)
dispersion fOrkF
in the ,Richie-Marusak
model ( -)
and with
= 1.
Notice that when C(Q, 4) is zero, the criterion reduces to
l-
dK e(Q, _,%
K,
e(Q,
w) K,
o)
Q
1 m
=”
(24)
which is exactly the relation found by Richie and Marusak [lo]. In fig. 3 we show the surface plasmon dispersion relation including exchange corrections compared with the Ritchie-Marusak result. The inclusion of exchange corrections does lower the dispersion curve although not to a significant degree. We believe that a realistic calculation of this quantity with a better approximation to the irreducible polarization diagram should include exchange corrections at least to the order outlined here. Harris [ 1 l] has also treated exchange corrections to surface plasmon dispersion and found corrections of the same magnitude as we have found here.
5. Summary This article has dealt with some aspects of the calculation of response functions at surfaces. The treatment of realistic potentials and of exchange-correlation effects
316
G. Lindell / Exchange corrections
has been outlined. It has been shown that the exchange effects in a simple surface model can be included without significantly increasing analytical or numerical comple~ty~
References [ I] [2] [3] [4] [S] [6] [7] [8] [9] [ 101 [II]
D.M. Newns, Phys. Rev. Bl (1970) 3304. V. Peuckert, 2. fhysik 241 (1971) 191. P.J. Feibelman, Phys. Rev. B9(1974) 5077. J.E. Inglesfield and E. Wikbog, Solid StateCommun. 14 (1974) 661; 16 (1975) 335. D.C. Langreth and J.P. Perdew, Solid State Commun. 17 (1975) 1425. K.S. Singwi, M.P. Tosi, R.H. Land and A. Sjalander, Phys. Rev. 176 (1968) 589. A. Griffin and J. Harris, Can. J. Phys. 54 (1976) 1396. J. Hubbard, Proc. Roy. Sot. (London) A243 (1957) 336. I.D. Moore and N.H. March, Ann. Phys. (NY) 97 (1976) 136. R.H. Ritchie and A.L. Mamsak, Surface Sci. 4 (1966) 214. J. Harris, J. Phys. CS (1972) 1757.