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The surface exchange energy of a metal for the full range of wave vector fluctuations
Volume 89A, number
1
PHYSICS LETTERS
19 April 1982
THE SURFACE EXCHANGE ENERGY OF A METAL FOR THE FULL RANGE OF WAVE VECTOR FLUCTUATIONS Mark RASOLT Solid State Division, Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA Received 12 January 1982 Revised manuscript received 22 February 1982
The exchange energy for a finite barrier model of realistic density profile variation is studied as a function of the full range of wave vector fluctuations. The deviation between the exact and approximate treatments is examined with particular attei~tionto the small wave vector behavior.
The decomposition of the exchange and correlation energy (E~~) of a uniform electron gas in terms of its individual wave vectors [1,2] q presents a detail structure of this quantity over the full range of excitations. For a metal surface, where we assume translational symmetry only along the surface plane, new excitations indexed by the wave vector number q11 are created [3—6].Such excitations would include the modification of the particle—hole continuum and the bulk plasmons as well as the creation of surface plasmons [6]. We, therefore, expect that as in the bulk case [1,2] the wave vector decomposition of the surface Exc in terms of q11 would also detail the importance of the various excitations. Of particular interest here is the extent to which approximate schemes account for Exc in different ranges of In this note we present, as a first step toward that goal, the wave vector decomposition for the exact surface exchange energy (Ex) of a finite barrier model (FBM) and compare it with two common approximate treatments. A similar decomposition in terms of the spherical wave vector q for the infinite barrier model (IBM) was presented in refs. [7,8] and for Exc in terms of q11 also in the IBM [9]. However, the surface density variation in the IBM is much too rapid, compared to a realistic metallic density profile, to allow for a meaningful assessment of the approximations discussed below [10—121 The exact exchange and correlation energy ~ can be expressed in terms of the response function Xx(rl iw) [13,14]. For a system with translational symmetry perpendicular to the z-axis 2 Exc _~Af—_--_~ dq11 fdzi fdz 2ire 2—__ ~—q~~Iz1_z2i dX(f —~x~(q11,z1,z2,iw)+n~(q110,z1)~(z1_z2)). -
-
,
f
(1) Here X is the coupling constant and n~(r)the electron density. For exchange alone the coupling constant can be trivially integrated [13]. The final expression for the surface exchange energy Crhf ~E~/2A was first presented by Mahan [15] for the FBM. In fig. 1 we display 7h1(q) defined as dq11 (2) ~ Research sponsored by the Division of Materials Sciences, US Department of Energy under Contract No. W-7405-eng-26 with the Union Carbide Corporation.
0 031-9163/82/0000—0000/ $02.75 © 1982 North-Holland
27
Volume 89A, number 1
PHYSICS LETTERS
19 April 1982
D —
—
~!
— S..
YL~.\\
5/i,
-
\~ S~o
: I~’H F
—5
—
—
Fig. 1. 7(q
11), in atomic units, as a unction of q~.The full hf is
0
I
I
1
2
labeled yj-.~,the local density approximation is labeled by ‘rid’ and the local density plus gradient is labeled ‘fldg• The surface energy in au is the area under these curves i.e., o = f (dq11/k~) X
‘y(q~).
for the full range of q11. (We remark that the decomposition in terms of q11 has no one-to-one correspondence to the decomposition in q; the two could differ qualitatively at small q11 and q.) We return to these results shortly. We next turn to approximate forms of eq. (1). Since the treatment of Exc in eq. (1) for an arbitrary nonuniform electron gas is presently inaccessible such approximate forms are essential. The functional density formalism provides such framework [16,17] and one popular approximation is given below [16,17] 2 + ...}. (3) ~ ~fd3r {Axc[fl(r)1 +Bxc[n(r)] I Vn(r)1 For the FBM the accuracy of eq. (3) with exchange alone was first examined in ref. [11]. Here we present a detail study of eq. (3) for the full range of q 11 fluctuations. The corresponding forms for ‘yld(qII) within the local density approximation [neglecting the second term in eq. (3)] and 7ldg(~II)within the gradient approximation [includingthe second term in eq. (3)] are given by: fdz[’y~(q0, n(r)) with
—
y~(q11,n0)]
(4)
2/21r)2k~n(r)q 7~(q11,n(r)) = (e
11O<(q11 2kF(r)) {—[2k~(r)/q11] tans [(4k~(r) 2p(r) —q~)1’2+ 2k~(r)]/q 1’~} + [~ —q~/l6k~.(r)] ln{[(4k 11} [4k~(r) —q~1 where kF(r) = [31r2n(r)}1”2 and for 7~(q 11,n0) we replace n(r) by n0, rs = (3I4~o)u13and —
—
‘
—~
28
(5)
Volume 89A, number ‘Y1dg(~~i)= 7ld(qII)
1
PHYSICS LETTERS
19
April 1982 (6)
+ Yg(~~~)
with 2/2161r3)fdz O<(q
7g(qi~)= (q
11k~e
2kF(r)) ~[Vn(r)]2/[n(r)]2} 11
—
+2 k~(r)/(4k~(r) q~)312}. (7) 11}+ kF(r)/(4k~(r) q~)~’ The appropriate density n (r) n (z) (and its derivative) entering eqs. (3) —(7) is easily calculated for the FBM. We set the barrier height equal to the Fermi energy. This choice produced an excellent representation for the true density profile of a jellium metal with rs = 3 [11]. In fig. 1 we include the final results for 7ld(q) and to be compared with We first note the obvious significant improvement between the full ‘y~and ‘y1d 5(q11)over [rather ‘y~(q11)]over 2kF. The overall surface energies GM (i.e., the integral q than0iag were the whole range from 0 to 11) and and exposes shown previously to beq11in= excellent agreement [11]. Fig. 1 gives a much more detailed comparison the deviation between a~and crldg for the different regions of q 11. It shows (as expected) that the gradient expansion (which depends on the local density variation) tends to have greater difficulty in reproducing the long wave length fluctuations and in particular the finite negative intercept of limq i~O 7~(q11).It is of interest to examine this difference. Returning to eq. (1) we can always write x of a noninteracting electron gas as 0(q 0(q x 11,z1,z2,iw)’x~(q11,z1—z2,iw)+iJi 1,z1,z2,ico), (8)
X
{—
~ ln{ [(4k~.(r) q~)~’2 + 2k~(r)]/q —
~
—
—
where x~is the response function of a nomnteracting uniform electron gas and ~ contains all the modification (or geometrical detail in the noninteracting x°)due to the presence of the surface. We can now show, by inserting eq. (8) into eq. (1) (after some lengthy analysis) that the fmite interceptin 7~(q11)comes entirely from x~° and is given by 3. (9) lim 7~(q,1)= (9/l6ir)(1/4 1/ir)ç In other words the contribution from Vi0 to the small limit of ‘yhj{q~~) goes to zero. However, the gradient expansions [like eq. (3)] represent the effect of turning on an external potential on a uniform electron gas [18,19] and, therefore, have a one-to-one correspondence with Indeed from eqs. (3)—(7) and the asymptotic behavior of n(z), far away from the surface of the FBM both 7~(q 11)and ‘vg(~~~) must tend to zero (see also fig. 1) in agreement with the contribution of ~° to7g‘yM(q11)~ (We remark that the[7,8].) sharp From termination of n(z) at the does that anomalous to that model the above results we IBM conclude produce an infinite small q limit for as the LDA alone) based on electron gas forms are not likely to account propany corrections to the LDA (as well erly for the small q 11 limit of The extent of the deviations, in the small q11 region, between the exact 7~(q11)and ‘ykjg(q~~), of a realistic density profile (fig. 1), is of particular importance in view of many recent efforts to account for these excitations as corrections to 7ldg(~~~) [14,19]. The effect of correlations will surely modify significantly these results, nevertheless fig. 1 does provide an order of magnitude estimate for the importance of the region of small q11 excitations, and the extent to which these are accounted for by ‘~Mg(~11). In fact, when correlations are included the leading small q11 limit has a totally different origin [20]. The leading exchange term now gets screened resulting in the usual surface plasmon contribution, which is known to go to zero linearly in q11 [3,4,14]. The gradient contribution, however, now tends to a constant [8,201 [i.e., the contribution from ~ in eq. (8) probably dominates at small q11] leading to the expectation that the small q11 limit would now be better represented by approximations like eq. (3). —
~
29
Volume 89A, number 1
PHYSICS LETTERS
References [1] J. Hubbard, Proc. R. Soc. (London) A240 (1957) 539. [2] P. Nozieres and D. Pines, Phys. Rev. 111(1958)445. [31 J. Schmit and A.A. Lucas, Solid State Commun. 11(1972)415. [4] R.A. Craig, Phys. Rev. B6 (1972) 1134. [5] RH. Ritchie, Surf. Sci. 34 (1973) 1. [6] A. Griffin, H. Kranz and J. Harris, J. Phys. F4 (1974) 1744. [71 M. Rasolt, G. Malmstrom and D.J.W. Geldart, Phys. Rev. B20 (1979) 3012. [8] M. Rasolt and D.J.W. Geldart, Phys. Rev. B21 (1980) 3158. [9] E. Wikborg and J.E. Inglesfield, Solid State Commun. 16 (1975) 335. [101 J.S.Y. Wang and M. Rasolt, Phys. Rev. B13 (1976) 5330. [11] M. Rasolt, J.S.Y. Wang and L.M. Kahn, Phys. Rev. B15 (1977) 580. [12] C.Q. Ma and V. Salmi, Phys. Rev. B16 (1977) 4249. [13] J. Harris and R.O. Jones, J. Phys. F4 (1974) 1170. [14] D.C. Langreth and J.P. Perdew, Phys. Rev. B15 (1977) 2884. [15] C.D. Mahan, Phys. Rev. B12 (1975) 5585. [16] P. Hohenberg and W. Kohn,Phys. Rev. B136 (1964) 864. [17]W. Kohn and L.J. Sham, Phys. Rev. A145 (1965) 1133. [18] S.K. Ma and K. Brueckner, Phys. Rev. 165 (1968) 18. [19] M. Rasolt and D.J.W. Geldart, Phys. Rev. Lett. 35 (1975) 1234. [20]M. Rasolt and D.J.W. Geldart, Phys. Rev. B, to be published.
30
19 April 1982
1
PHYSICS LETTERS
19 April 1982
THE SURFACE EXCHANGE ENERGY OF A METAL FOR THE FULL RANGE OF WAVE VECTOR FLUCTUATIONS Mark RASOLT Solid State Division, Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA Received 12 January 1982 Revised manuscript received 22 February 1982
The exchange energy for a finite barrier model of realistic density profile variation is studied as a function of the full range of wave vector fluctuations. The deviation between the exact and approximate treatments is examined with particular attei~tionto the small wave vector behavior.
The decomposition of the exchange and correlation energy (E~~) of a uniform electron gas in terms of its individual wave vectors [1,2] q presents a detail structure of this quantity over the full range of excitations. For a metal surface, where we assume translational symmetry only along the surface plane, new excitations indexed by the wave vector number q11 are created [3—6].Such excitations would include the modification of the particle—hole continuum and the bulk plasmons as well as the creation of surface plasmons [6]. We, therefore, expect that as in the bulk case [1,2] the wave vector decomposition of the surface Exc in terms of q11 would also detail the importance of the various excitations. Of particular interest here is the extent to which approximate schemes account for Exc in different ranges of In this note we present, as a first step toward that goal, the wave vector decomposition for the exact surface exchange energy (Ex) of a finite barrier model (FBM) and compare it with two common approximate treatments. A similar decomposition in terms of the spherical wave vector q for the infinite barrier model (IBM) was presented in refs. [7,8] and for Exc in terms of q11 also in the IBM [9]. However, the surface density variation in the IBM is much too rapid, compared to a realistic metallic density profile, to allow for a meaningful assessment of the approximations discussed below [10—121 The exact exchange and correlation energy ~ can be expressed in terms of the response function Xx(rl iw) [13,14]. For a system with translational symmetry perpendicular to the z-axis 2 Exc _~Af—_--_~ dq11 fdzi fdz 2ire 2—__ ~—q~~Iz1_z2i dX(f —~x~(q11,z1,z2,iw)+n~(q110,z1)~(z1_z2)). -
-
,
f
(1) Here X is the coupling constant and n~(r)the electron density. For exchange alone the coupling constant can be trivially integrated [13]. The final expression for the surface exchange energy Crhf ~E~/2A was first presented by Mahan [15] for the FBM. In fig. 1 we display 7h1(q) defined as dq11 (2) ~ Research sponsored by the Division of Materials Sciences, US Department of Energy under Contract No. W-7405-eng-26 with the Union Carbide Corporation.
0 031-9163/82/0000—0000/ $02.75 © 1982 North-Holland
27
Volume 89A, number 1
PHYSICS LETTERS
19 April 1982
D —
—
~!
— S..
YL~.\\
5/i,
-
\~ S~o
: I~’H F
—5
—
—
Fig. 1. 7(q
11), in atomic units, as a unction of q~.The full hf is
0
I
I
1
2
labeled yj-.~,the local density approximation is labeled by ‘rid’ and the local density plus gradient is labeled ‘fldg• The surface energy in au is the area under these curves i.e., o = f (dq11/k~) X
‘y(q~).
for the full range of q11. (We remark that the decomposition in terms of q11 has no one-to-one correspondence to the decomposition in q; the two could differ qualitatively at small q11 and q.) We return to these results shortly. We next turn to approximate forms of eq. (1). Since the treatment of Exc in eq. (1) for an arbitrary nonuniform electron gas is presently inaccessible such approximate forms are essential. The functional density formalism provides such framework [16,17] and one popular approximation is given below [16,17] 2 + ...}. (3) ~ ~fd3r {Axc[fl(r)1 +Bxc[n(r)] I Vn(r)1 For the FBM the accuracy of eq. (3) with exchange alone was first examined in ref. [11]. Here we present a detail study of eq. (3) for the full range of q 11 fluctuations. The corresponding forms for ‘yld(qII) within the local density approximation [neglecting the second term in eq. (3)] and 7ldg(~II)within the gradient approximation [includingthe second term in eq. (3)] are given by: fdz[’y~(q0, n(r)) with
—
y~(q11,n0)]
(4)
2/21r)2k~n(r)q 7~(q11,n(r)) = (e
11O<(q11 2kF(r)) {—[2k~(r)/q11] tans [(4k~(r) 2p(r) —q~)1’2+ 2k~(r)]/q 1’~} + [~ —q~/l6k~.(r)] ln{[(4k 11} [4k~(r) —q~1 where kF(r) = [31r2n(r)}1”2 and for 7~(q 11,n0) we replace n(r) by n0, rs = (3I4~o)u13and —
—
‘
—~
28
(5)
Volume 89A, number ‘Y1dg(~~i)= 7ld(qII)
1
PHYSICS LETTERS
19
April 1982 (6)
+ Yg(~~~)
with 2/2161r3)fdz O<(q
7g(qi~)= (q
11k~e
2kF(r)) ~[Vn(r)]2/[n(r)]2} 11
—
+2 k~(r)/(4k~(r) q~)312}. (7) 11}+ kF(r)/(4k~(r) q~)~’ The appropriate density n (r) n (z) (and its derivative) entering eqs. (3) —(7) is easily calculated for the FBM. We set the barrier height equal to the Fermi energy. This choice produced an excellent representation for the true density profile of a jellium metal with rs = 3 [11]. In fig. 1 we include the final results for 7ld(q) and to be compared with We first note the obvious significant improvement between the full ‘y~and ‘y1d 5(q11)over [rather ‘y~(q11)]over 2kF. The overall surface energies GM (i.e., the integral q than0iag were the whole range from 0 to 11) and and exposes shown previously to beq11in= excellent agreement [11]. Fig. 1 gives a much more detailed comparison the deviation between a~and crldg for the different regions of q 11. It shows (as expected) that the gradient expansion (which depends on the local density variation) tends to have greater difficulty in reproducing the long wave length fluctuations and in particular the finite negative intercept of limq i~O 7~(q11).It is of interest to examine this difference. Returning to eq. (1) we can always write x of a noninteracting electron gas as 0(q 0(q x 11,z1,z2,iw)’x~(q11,z1—z2,iw)+iJi 1,z1,z2,ico), (8)
X
{—
~ ln{ [(4k~.(r) q~)~’2 + 2k~(r)]/q —
~
—
—
where x~is the response function of a nomnteracting uniform electron gas and ~ contains all the modification (or geometrical detail in the noninteracting x°)due to the presence of the surface. We can now show, by inserting eq. (8) into eq. (1) (after some lengthy analysis) that the fmite interceptin 7~(q11)comes entirely from x~° and is given by 3. (9) lim 7~(q,1)= (9/l6ir)(1/4 1/ir)ç In other words the contribution from Vi0 to the small limit of ‘yhj{q~~) goes to zero. However, the gradient expansions [like eq. (3)] represent the effect of turning on an external potential on a uniform electron gas [18,19] and, therefore, have a one-to-one correspondence with Indeed from eqs. (3)—(7) and the asymptotic behavior of n(z), far away from the surface of the FBM both 7~(q 11)and ‘vg(~~~) must tend to zero (see also fig. 1) in agreement with the contribution of ~° to7g‘yM(q11)~ (We remark that the[7,8].) sharp From termination of n(z) at the does that anomalous to that model the above results we IBM conclude produce an infinite small q limit for as the LDA alone) based on electron gas forms are not likely to account propany corrections to the LDA (as well erly for the small q 11 limit of The extent of the deviations, in the small q11 region, between the exact 7~(q11)and ‘ykjg(q~~), of a realistic density profile (fig. 1), is of particular importance in view of many recent efforts to account for these excitations as corrections to 7ldg(~~~) [14,19]. The effect of correlations will surely modify significantly these results, nevertheless fig. 1 does provide an order of magnitude estimate for the importance of the region of small q11 excitations, and the extent to which these are accounted for by ‘~Mg(~11). In fact, when correlations are included the leading small q11 limit has a totally different origin [20]. The leading exchange term now gets screened resulting in the usual surface plasmon contribution, which is known to go to zero linearly in q11 [3,4,14]. The gradient contribution, however, now tends to a constant [8,201 [i.e., the contribution from ~ in eq. (8) probably dominates at small q11] leading to the expectation that the small q11 limit would now be better represented by approximations like eq. (3). —
~
29
Volume 89A, number 1
PHYSICS LETTERS
References [1] J. Hubbard, Proc. R. Soc. (London) A240 (1957) 539. [2] P. Nozieres and D. Pines, Phys. Rev. 111(1958)445. [31 J. Schmit and A.A. Lucas, Solid State Commun. 11(1972)415. [4] R.A. Craig, Phys. Rev. B6 (1972) 1134. [5] RH. Ritchie, Surf. Sci. 34 (1973) 1. [6] A. Griffin, H. Kranz and J. Harris, J. Phys. F4 (1974) 1744. [71 M. Rasolt, G. Malmstrom and D.J.W. Geldart, Phys. Rev. B20 (1979) 3012. [8] M. Rasolt and D.J.W. Geldart, Phys. Rev. B21 (1980) 3158. [9] E. Wikborg and J.E. Inglesfield, Solid State Commun. 16 (1975) 335. [101 J.S.Y. Wang and M. Rasolt, Phys. Rev. B13 (1976) 5330. [11] M. Rasolt, J.S.Y. Wang and L.M. Kahn, Phys. Rev. B15 (1977) 580. [12] C.Q. Ma and V. Salmi, Phys. Rev. B16 (1977) 4249. [13] J. Harris and R.O. Jones, J. Phys. F4 (1974) 1170. [14] D.C. Langreth and J.P. Perdew, Phys. Rev. B15 (1977) 2884. [15] C.D. Mahan, Phys. Rev. B12 (1975) 5585. [16] P. Hohenberg and W. Kohn,Phys. Rev. B136 (1964) 864. [17]W. Kohn and L.J. Sham, Phys. Rev. A145 (1965) 1133. [18] S.K. Ma and K. Brueckner, Phys. Rev. 165 (1968) 18. [19] M. Rasolt and D.J.W. Geldart, Phys. Rev. Lett. 35 (1975) 1234. [20]M. Rasolt and D.J.W. Geldart, Phys. Rev. B, to be published.
30
19 April 1982