Analytical asymptotic structure of the exchange and correlation potentials at a metal surface

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PHYSICS ELSMER

LETTERS

A

Physics Letters A 2 12( 1996)263-269

Analytical asymptotic structure of the exchange and correlation potentials at a metal surface Alexander Solomatin, Viraht Sahni Depamnent of Physics, Brooklyn College of the City University of New York, Brooklyn, NY 11210. USA

Received 26 October 1995;accepted 12 January 1996 Communicatedby J. Flouquet

Abstract We derive the exact analytical asymptotic structure of the Kohn-Sham theory exchange potential and thereby of the correlation potential at a semi-infinite jellium metal surface. The exchange potential is image-like (-A/x), where A depends on the Fermi energy and surface barrier height, and is precisely f for stable jellium.

Hohenberg-Kohn-Sham 111density-functional theory leads, in principle, directly to the density p(r) of a system of interacting electrons in an external potential via spin-orbitals generated by a local “exchange-correlation” potential ~~,(r>. This potential is the functional derivative of a yet unknown “exchange-correlation” energy functional E,,[ p] in which all the many-body effects are incorporated including those of the correlation contribution to the kinetic energy. Thus, knowledge of the exact properties of the potential or of its exchange V,(T) and correlation am’, components is of importance not only in their own right but also for the construction of approximate energy functionals and the potentials derived therefrom. For finite nonuniform electron density systems such as in atoms, molecules and metallic clusters, it is well established [2-41 that the asymptotic structure of the exchange-correlation potential Gus which is - l/r is a consequence of correlations due to the Pauli exclusion principle. However, there remains at present a controversy as to the physical origin of the asymptotic image potential structure (- 1/4x) of vX,(r) at a jellium-metal vacuum [5] interface. Von Barth et al. [2] stated (without proof) that the exchange potential u,(r) at a surface decays exponentially, and that therefore the image potential structure is a Coulomb correlation effect. This view is supported by both Sham [3] and the jelfium-slab-metal calculations of Eguiluz et al. [6] in whose work vX(r) decays as -xe2 asymptotically. On the other hand, the calculations of Harbola and Sahni [7,8] performed for the semi-infinite jeflium-metal surface, show the image potential structure to be due to Pauli correlations. The conclusions of both Eguiluz et al. [6] and Harbola and Sahni [7,8] are, however, arrived at numerically, the calculations in each case being performed for specific values of the bulk-metal Wigner-Seitz radius. Thus, Dobson [9] states: “There is much remaining to be said on this issue, and since both sides of the argument have so far only presented numerical evidence, this question still remains to be settled once and for all by a rigorous analytica analysis.” Elsevier Science

B.V.

PI/SO375-9601(96)00054-O

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In this Letter we resolve the above controversy by determining the exact analytical asymptotic structure of the Kohn-Sham theory exchange potential v,(r) at a semi-infinite jellium-metal surface. Since the asymptotic structure of the exchange-correlation potential vXC(r) is the image potential, the asymptotic structure of the correlation potential v,(t) is then also exucfly known anulyticully. Furthermore, these results are valid for urbirrury bulk-metal density, and for the exact self-consistent orbitals. We show that the asymptotic structure of the v,(r) in the classically forbidden region far from the metal surface decays as +Vz, where Vxs(r) is the Slater [lo] potential due to the Fermi hole charge. We further show that when only Pauli correlations are OPM(r) as defined via the optimized potential method assumed present, the corresponding exchange potential vX [l l] (OPM) also vanishes asymptotically as +V,“(r>. Furthermore, we obtain these structures by deriving the asymptotic structure of the Slater potential to be - (~s( p>/ X, where x is the distance from the metal surface and /3 * a parameter which is the ratio of the surface barrier height W to the Fermi energy lr of the metal. In contrast, for finite systems, the asymptotic structure of V,(T) in the classically forbidden region (which is that of r&)) and of vFPM(r> deta y s as V,“(r), which in turn vanishes as - as/r with us = 1. The corresponding correlation potential v&r) for atoms decays [2,3] as - (u/2r4, where (Y is the polarizability of the positive ion, whereas we now obtain that at the semi-infinite jellium metal surface it decays as -[l - 2o,( /3)1/4x. We begin by determining the asymptotic structure of the Slater potential V:(r). Next we show the asymptotic structure of the Kohn-Sham theory exchange potentials v,(r) and r~~‘~~(r)to be image-potential-like with the structure - +QJ p)/x. Finally, we show that for a slab geometry of jellium metal, the exchange potentials decay asymptotically as l/x2. The Slater potential V:(r) is defined [lo] as

(1)

V,“(r)=/dr/p,(r,i)/lr-r’l,

where the Fermi hole px(rr r’) at r’ for an electron at r is given in terms of the idempotent Dirac density - l k) as px(r, I’) = - 1y(r, r’) 12/2p(r). with y(r, r) = p(t). For matrix y(r, r’) = 2X, IJ~ (r)&(J)0(e, the jellium-metal surface, the Kohn-Sham orbitals are I,@) = m exp[i(k,, - x,,)l&(x), where (kll, r,,) are the momentum and position vectors parallel to the surface and (k, x) the vectors in the perpendicular direction. In dimensionless coordinates (normalized to the Fermi momentum k, = l/ars, (Y-’ = (%r)‘13, rs is the Wigner-Seitz radius), V:(r) may then be written as dk 4,( z)G( k, k’; z)~~‘(z),

(2)

where G(kv

k’;

J(

z)

q,

z)

=

= ~~‘2/(n’-A’2”“dqJ(q, 0

2jm

-02

z)

dz’ e-9~Z-z’~~k( z’)&(

S,(q) = A2 tan- I[ ( A2 - xj)“‘/x*]

+ jA+A’dq~(q, A- .v

z’),

- XA( AZ - x;)“2,

z)[sA(q)

+s,,(q)],

(3) (4)

(5)

f (A2 - A’2)/q], A = (1 - k2)“2, A’= (1 - k’2)‘/2, and p,(z) is the density A’, x,,, = i[q normalized to the bulk value p = ki/3r2. To make the derivation of the asymptotic structure of the Slater potential more accessible, we initially perform our calculations for the orbitals of a model effective potential. We then prove that the result is model

SJq) = S,(q) I ,4-r

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265

independent and valid for the fully-self-consistent case. The orbitals of the finite-linear-potential model [12] assumed are +,( z) = sin[ kz + S(k)] +D,

+ [B, Ai( &) + Ck Bi( lk)] [ e( z) - e( z - zb)]

6( -z)

exp(-KI,z)8(z--zb),

(6)

E is the energy, W the where k = m, x = {mE) & = u;“~ - &,, lo = k2zii3, zF = +kiz,,/W, barrier height, and Ai( &) and Bi( &> the Airy functions. The phase factor S(k) and the coefficients B,, C, and D, are determined by the requirement of the continuity of the wavefunction and its logarithmic derivative at z=Oand z=zt,. We first determine J( q, z> of Eq. (4) for z > 0 employing the orbitals assumed. Then recognizing that in the asymptotic large z region, the effective value of q - l/z, we expand the resulting expression to obtain q cos

4

J( q, z) = eeqZ -cos q’+k:

k:

2D,D,t + _e-(K+K’)zb Kk

+

s,

a_+

+2tb

dz’

[ B k Ai’( 5’k)

+ C, Bi( &‘)I [ B,! Ai( Q) + C,r Bi( G)]

Kk*

4qDkDkr

-cK+K*jZ 2e

-( Kk +

(7)

’

Kk’)

where k, = k’ T k, and 8, = S(k’) T S(k). In deriving Eq. (7) we have also used the fact that k - k’ for large z so that the effective value of k_- l/z. Next consider the contribution of J(q, z> of Eq. (7) to the first integral in G(k, k’; z) of Eq. (3). The last term of Eq. (7) is exponentially small in the vacuum re ion and does not ( A - A ) zl)/z. Now contribute. The contribution of the second set of terms has a prefactor (1 - expI - +F (A2 - A’2)1/2z ry k!i2z = (k_z)‘/2Z’/2 Z+ 1 for large z. Thus, the contribution of the second term is of 0(1/z). The contribution of the first term of Eq. (7) is readily seen to be ue-”

cos s_

mdu /

0

u2 + a2

(8)

+ a sin S_ - du /0

where a = k-z. Next we note that the second term of G(k, k’; z> of Eq. (3) does not contribute to the leading order. This is because the integral is concentrated about its lower limit A - A’_ k_ - l/z, and q - l/z. In the limit q -P A - A’, X, + A, X,, + -A’, SO that S,,, * 0. Thus, G(k, k’; z) is given by Bq. (8). We next consider the integral over k in Bq. (2) and rewrite it as (l/z) /t da. Again, since for large z, k - k’, Kk z = Kke z + cu where c = k’/Kkt so that f&k(2) - 4kt( Z) eXp< - Cd. Substituting this +k( Z) into Eq. (2) and Using the fact that cos S_ m 1 and sin 6_ m 0 for k, k’ - 1, the Slater potential is then

v
(9)

The term in the first large parentheses is the normalized density p,(z) so on solving the integral in the large brackets we obtain vx”(x) = -as(p); x-+m where p2 = W/+.

p2-1 c%(P)=-

P2

1_ i

ln(P’--1)

G

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Letters A 212 11996) 263-269

To prove that Eq. (10) for Vxs is model independent we divide the z axis into three parts: z 4 - d, - d G z G d, z a d where d is an effective width of the surface region. In the first region which corresponds to the crystal bulk, the potential vX,(z) is constant. Consequently the orbitals there are of the form &( z) = sin[ kz + S(k)], where 6(k) are the appropriate self-consistently determined phase shifts. The contribution to J(q, z) from this region is then the same as before modified by d. However, due to screening, the surface region where the Bardeen-Friedel oscillations [5] exist is small in comparison to the asymptotic electron position: d r%:z for z + =L Furthermore, q = l/z, so that expansion about q then gives the same leading term for J(q, z) as in Eq. (7). This is the term which leads to the coefficient as( p). Next, it is evident that although the self-consistent orbitals in the surface region are slightly different from those assumed above, the contribution of this region to G(k, k’; z) is of 0(1/z) and to V;‘< z> of o(l/z2>. Finally, in the vacuum region tbe potential ~~,(r> is not a constant but rather the image potential. The orbitals in this region are then of the form [2] &(z) _ ~“~e-~k’. After determining J(q, z) for these orbitals, the (For an electron at the Fermi level v~, = (4/+/G)-‘.) contribution to G(k, k’; z) is readily seen to be either exponentially small or of 0(1/z), and o(l/z2) to V,‘(z). Therefore, in a fully-self-consistent calculation the asymptotic structure of the Slater potential would be the same as given by Eq. (10). (For completeness we note that the Slater potential has previously been studied for model-potential orbitals numerically by Harbola and Sahni [ 13,141, and analytically employing an approximate form of the Fermi hole by Juretschke [15,16].) Having derived the asymptotic structure of the Slater potential, we next determine the asymptotic structure of the exchange component U,(T) of the Kohn-Sham potential vX,(r> and of the exchange potential v,“P”(r) of the OPM. From the integral equation relating the exchange-correlation potential am,, and self-energy &(r, r’; E) derived by Sham [3], the asymptotic structure of v,,(r) is 1 r%(f) = 2*k(r)

1 /

dr’ &( r, r’; er) V,( 8) +

2Tk’ ( r) /

dr’!&*(Q&(r’,

r;

lr),

(‘1)

where the electron is at the Fermi level. The asymptotic structure of the exchange component vX(r> is obtained by substituting the self-energy &(r, f) = - y(r, r’)/2 1r - I-’ 1 into the above equation. The resulting expression is recognized to be the orbital-dependent potential [lo] v,.$T) due to the orbital-dependent Fermi hole (defined as P.&, r’) = 2& qkT (rYPJr’YPk(f )/F&r)) of the Hartree-Fock theory. Now the asymp(r) is also given by this potential for the highest occupied totic structure of the OPM exchange potential vXoPM orbital. Thus, the analytical determination of the asymptotic structure of this orbital-dependent potential will be that of both v,(t) and vyM (r). The orbital-dependent potential [s] V,.,(Z) corresponding to the Fermi level electron with momentum perpendicular to the surface can be rewritten as

vxI(Z) A 3k,/2T

2 = --j-i

34,(z)

0

dk C,(z)L*

dqJ(q,

z).

(‘2)

On the following the same steps as before in determining J(q, z) and 4 ,(z>, we obtain v~,,(x> = - ia,< j?>/x. Thus, the asymptotic structure of V,(T) and V,OPM(r) is ~V,“. The above derivation also shows for this nonuniform density system in which the energy spectrum is continuous, that to leading order, the orbital-dependent potentials V.&J for electrons within a shell of thickness (l/z) about the Fermi level are the same. Thus, their average taken over this shell, which is the exchange potential, is equivalent in leading order to the orbital-dependent potential for electrons at the Fermi level. In the case of discrete systems such as atoms, it is of course more readily apparent that the asymptotic structure is due to the highest occupied orbital electrons. In Fig. 1 we plot the Kohn-Sham exchange potential coefficient +cys( p> as a function of the barrier height parameter /3 for the metallic range of densities rs = 2-6. The relationship of the Wigner-Seitz radius rs, also shown in the figure, and the parameter /I is through fully-self-consistent calculations [ 14,171 within the local density approximation for exchange-correlation. For metallic densities ia,< p) ranges from 0.20 to 0.27. For j3 = r/z, the coefficient is exactly 4 and corresponds to a Wigner-Seitz radius of rs - 4.0 which is that of Na.

A. Solomarin, V. Sahni/Physics

Letters A 212 (1996) 263-269

WIGNER-SEITZ RADlus

v.

o.20 1.1

1.2

1.3

1.4

267

rs (au.)

1.5

1.6

BARRIERHEIGHTPARAMETER p=(W/E,)‘n Fig. 1. The Kohn-Sham exchange potential coefficient fas( p) as a function of the barrier height parameter /? = m, where W is the barrier height and Ed the Fermi energy. The relationship of the Wigner-Seitz radius rs to /3 is via self-consistent calculations in the local density approximation.

We note that the jellium model is stable for approximately this value of rs. Thus, the asymptotic structure of the exchange potentials v,(r) and vXoPM(r>f or simple metals is image-potential-like. From the figure it is also evident that the -x- ’ dependence of the correlation potential v,(r) is weak over the metallic range of densities. We note further that the results for the asymptotic structure of the exchange and correlation potentials derived are equally valid for the stabilized jellium or structureless pseudopotential model [ 181. As noted previously, the asymptotic structure of the exchange potential v,(t) has also been examined by Sham [3] and Eguiluz et al. [6] via the integral equation relating v,(t) to the self-energy &(r, f 1, and they have concluded that the structure _ - cu/x’. Although the conclusion of Sham is surprising in light of the proof that am_, of Fq. (12) decays as -x- ’ , the results of Eguiluz et al. on the other hand can be explained. In their numerical calculations these authors consider a slab configuration which they assume to be sufficiently thick to represent a semi-infinite crystal. Thus, consider a slab of thickness L sufficiently large to make the energy spectrum continuous. The integral J(q, z) of Eq. (4) then extends from -L, to 0, so that

J(q* z+”

z>-

ewqzq[

cos

S_ - eeqL cos( S_ +

k-L)] 7

q'+k!

(13)

where the second term is the contribution from the surface at -L. Since in the asymptotic region ( z + m), both z- ’ , then qL and k_L -sz 1. Then expanding about qL and k_L one obtains to leading order

q and k_ -

J( q, z) - q2emqzL cos S_/(q2

+ k!),

(‘4)

which then leads to a l/z2 dependence for the exchange potential vX’,(r)as obtained by Eguiluz et al. Thus, to obtain the correct asymptotic structure of the exchange potential for the semi-infinite crystal via numerical solution of the Sham integral equation, the slab thickness assumed must be much greater than the distance of the asymptotic electron from the surface. However, due to screening, whose manifestation is the partial cancellation of the exchange and correlation holes in the metal bulk, the structure of the exchange-correlation potential vX,(r> in finite thickness slab calculations should be equivalent to that obtained for the semi-infinite case. In conclusion we note that in contrast to previous work where the image potential structure of the exchange-correlation potential vX.(r) was attributed entirely to either Pauli or Coulomb correlations, the present derivation shows the image tail to result from the combined effect of these correlations. The fact that the exchange v,(r) and correlation am’, potentials each depend upon the barrier height and Fermi energy, whereas the asymptotic structure of vX,(r) is the classical image potential can be explained as follows. The total charge of the Fermi-Coulomb and Fermi holes is --e where e is the electronic charge, whereas that of the Coulomb

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268

hole is zero. Furthermore, for asymptotic positions of the electron, the Fermi hole is delocalized [ 14,161 and spread throughout the crystal. As such the correlation potential V,(T) can be thought of as being comprised of two components. The first due to the delocalized part of the Coulomb hole (of charge +e> in the metal bulk, and the second due to that part of the Coulomb hole (of charge -e> localized about the surface region. The component due to the delocalized part of the Coulomb hole then screens out the exchange contribution to the image tail. It is the component due to the surface localized parf of the Coulomb hole that gives rise to image potential structure of vX,(r) making it independent of the parameters of the metal. Finally, knowledge of the exact asymptotic structure of the exchange and correlation potentials should prove valuable in the construction of approximate energy functionals and potentials. At present there are two approximate exchange potentials in the literature which possess the correct asymptotic structure at a metal surface. The first vi’)(r) is derived [I91 from the exact exchange energy functional E,[ p], but determined by obtaining the functional derivative for a restricted class of density variations. The resulting general expression is v,‘O’(r) = [2 dp( r)/dk,]

-I d[ p( ‘)V,“( ‘)1/d&

(15)

whose leading term can readily be seen to be fV,“. The second expression, due to Sham [3], is

(16) Following the same analytical procedure as described previously, the asymptotic structure of r~~‘~‘“(r)can also be shown to be - ic~s
References

[II P.

Hohenberg and W. Kohn, Phys. Rev. 136 (1964) B864, W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) Al 133. 121A.R. Williams and U. von Barth, in: Theory of the inhomogeneous electron gas, eds. by S. Lundqvist and N.H. March (Plenum, New York, 1983) p. 189; C.-O. Almbladh and U. von Barth, Phys. Rev. 31 (1985) 3231. [31 L.J. Sham, Phys. Rev. B 32 (1985) 3876. [41 M.K. Harbola and V. Sahni, Phys. Rev. Lett. 62 (1989) 489; V. Sahni and M.K. Harbola, Int. J. Quantum Chem. Symp. 24 (1990) 569. 151 J. Bardeen, Phys. Rev. 49 (1936) 653. l61A.G. Eguiluz et al., Phys. Rev. Lett. 68 (1992) 1359; Int. J. Quantum Chem. Symp. 26 (1992) 837. [71 M.K. Harbola and V. Sahni. Phys. Rev. B 39 (1989) 10437. Is1 M.K. Harbola and V. Sahni, Int. J. Quantum Chem. Symp. 27 (1993) 101. 191 J. Dobson. in: NATO ASI, Vol. 337. Density functional theory, eds. E.K.U. Gross and R.M. Dreizler (Plenum, New York, 1995) p. 393. 1101 J.C. Slater, Phys. Rev. 81 (1951) 385. [Ill R.T. Sharp and G.K. Horton, Phys. Rev. 90 (1953) 3876; J.D. Talman and W.F. Shadwick, Phys. Rev. A 14 (1976) 36. [I21 V. Sahni, C.Q. Ma and J.S. Flamholz, Phys. Rev. B 18 (1978) 393 I. 1131 M.K. Harbola and V. Sabni, Phys. Rev. B 36 (1987) 5024.

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Letters A 212 (1996) 263-269

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