Exchange and correlation potentials in the one-electron equation

0038-1098/83 $3.00 + .00 Pergamon Press Ltd.

Solid State Communications, Vol. 47, No. 8, pp. 641-644, 1983. Printed in Great Britain.

EXCHANGE AND CORRELATION POTENTIALS IN THE ONE-ELECTRON EQUATION H. Suehiro, K. Awa and H. Yasuhara* Department of Physics, Hokkaido University, Sapporo 060, Japan

(Received 21 March 1983 by J. Kanamori) The self-energy operator of an electron liquid is examined by including the particle-hole and particle-particle ladder types of vertex corrections to the simplest approximation discussed by Hedin and Lundqvist. It is found that the real part of the self-energy correction Z (p, ep) is much more independent of the wavenumber p than that in the simplest approximation because of the strong short-range correlations at metallic densities. The quasi-particle mass ratio m*/m in the present approximation gives values smaller than 1 in the whole metallic density region. THE ONE-ELECTRON THEORY has extensively been used in interpreting various electronic properties of solids. On the basis of the variational principle of the ground-state energy functional of non-uniform manyelectron systems [1], Kohn and Sham [2] have derived the one-electron Schr6dinger-like equation which is very convenient for the practical use. In a somewhat pragmatical manner this theory has been generalized to the spin-polarized case [3]. The spin-density functional formalism thus obtained has been employed with the local spin-density approximation which requires a knowledge of electron correlations only of a uniform electron liquid. This somewhat phenomenological theory has remarkably succeeded in the numerical estimation of the various electronic properties of not only simple metals and semiconductors but also even insulators. Strictly speaking, these formalisms originally are intended for the calculation of the ground-state energy and the charge distribution. The problem of the excitation spectrum in solids is one of the most difficult but important problems whose developments are now awaited. In an attempt to interpret the interesting structure in the dynamical structure factor observed for simple metals [4, 5] we have recently obtained the deeper understanding of the electron correlations of an electron liquid at metallic densities. Then it is timely to make an attempt to attack the fundamental problem above. The purpose of the present communication is to stress the importance of the short-range correlations in the estimation of the self-energy operator Z(p, e) in the Dyson equation which describes the motion of the quasiparticle in simple metals. Hedin and Lundqvist [6] discussed an approximate form of the self-energy Z(p, e)

which is of the first order in the RPA dynamically screened Coulomb interaction, but they completely neglected vertex corrections in it. It is now recognized that those types of vertex corrections which consist of an infinite series of particle-particle ladder interactions are indispensable for the adequate description of spinantiparallel correlation due to the short-range Coulomb repulsion in the metallic density regime [7-13]. For the qualitative and quantitative discussion of the selfenergy, it is therefore necessary to take into account the local field corrections to the RPA arising not only from spin-parallel correlation but also spin-antiparaUel correlation. We have here examined the following expression for the self-energy Z(p, e) of an electron liquid. This expression has been obtained through the inclusion of both types of local field corrections in the RPA expression for the self-energy, and its imaginary part has successfully been used in the interpretation of the interesting structure in the dynamical structure factor.

~,(p, e)

if dqdw [1--C(q)lv(q) J (2rr)4 1 + [1 -- G(q)] v(q)lr(°)(q, 6o) x G(°)(p --q, e -- w),

(1)

where Gt°)(p, e) is the bare Green function and v(q), the Coulomb interaction and ~r(°)(q, w) denotes the free polarization function. Two factors, G(q) and C(q) in equation (1) represent the spin-averaged and spinantiparaUel local field corrections, respectively. If one puts G(q) =C(q) = 0 in equation (1), then equation (1) is reduced to the expression discussed in detail by Hedin and Lundqvist [6]. The two local field factors G(q) and C(q) are determined as follows [4, 5] :

-- G(q)v(q) = ~(I(p, p';q) -- v(q))pp,

* College of Arts and Sciences, Tohoku University, Sendal 980, Japan.

+~(I(p,p';q)--v(q)--I(p,p';p--p' +q))pp,, 641

(2)

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EXCHANGE AND CORRELATION POTENTIALS

-- C(q)v(q) = (l(p, p'; q)

--

v(q))pp,.

Vol. 47, No. 8

(3) , ~ -1.g

Here the particle-particle ladder interaction I(p, p'; q) is the solution of the following integral equation, I ( p , p ' ; q) = v(q) + f ~ dk -~

~=4

uJ 2~SLATER

6. LN

PRESENT

v (q - - k )

x [1 - - f ( p + k)] [1 - - f ( p ' --k)] I(p, p'; k),

RPA

-2.0

-

(4)

e p - - e p + k -~- e p ' - - ep,_ k

where f(p) is the Fermi distribution function at zero temperature and ep = h 2p2/2m. The angular bracket (" • ")pp, in equations (2) and (3) denotes an appropriately averaged value over p and p' within Fermi spheres. The second term in the r.h.s, of equation (2) is the local field correction from the spin-parallel correlation. In estimating that term it is sufficient to take only the lowest order term in the Coulomb interaction, since there are a large amount of cancellation between the higher order direct and exchange processes; this amounts to adopting the usual Hubbard approximation. On the other hand the local field correction arising from spin-antiparallel correlation has been estimated using an approximate solution of equation (4) obtained by one of us (H.Y.) [ 7 - 9 ] . The particle-particle ladder interactions substantially have the effect of weakening those interaction parts which are associated with the large momentum transfers (q >1PF, PF: the Fermi wavenumber), but this series is rather inappropriate for the treatment of the small momentum transfer interactions. Therefore an appropriate extrapolation form of G(q) or C(q) which has been devised in the previous work is also used for q <~PF. In the high density region the local field correction from spin-antiparaUel correlation is almost negligible, but it becomes increasingly significant as the electron density decreases and in the metallic density region attains to the magnitude as large as that of the local field correction from spin-parallel correlation. The quasi-particle pole can self-consistently be determined by solving the Dyson equation, provided that the self-energy E(p, e) is constructed from selfconsistent internal Green functioris. However, in the present case where the self-energy Z(p, e) is evaluated using the bare internal Green function, it is more appropriate to use the first iterative solution obtained by substituting an unperturbed pole e = ep in the selfenergy [14]. On the basis of the Fermi liquid theory, Rice [ 15 ] obtained the self-energy expression from the ground-state energy calculated with the Hubbard dielectric function [ 16]. His expression can be recovered by simply omitting the local field correction from spin-

0

i

r

I

0.5

1.0

1.5

2.0

P/PF

Fig. l. The calculated values of Re Z(p, ep) in units of the Fermi energy eF in the RPA, the Rice approximation, and the present approximation are shown as a function of PiPe for rs = 4.0. The horizontal line corresponds to a 2/3 Slater approximation.

antiparallel correlation in equation (1). The expression of equation ( i ) m a y be expected to embody essential features of electron correlations in the metallic density region, though it has been obtained not by means of the straightforward diagrammatic expansion in the dynamically screened Coulomb interaction but with the aid of a somewhat intuitive physical consideration. In Fig. 1 we show the calculated values of the real part of Y.(p, ep) in units of eF (eF: the Fermi energy) as a function of the reduced wavenumber P/PF for the typical metallic density appropriate to sodium (r s = 4.0, rs: the usual electron density parameter). For comparison the calculated curves of Re ~ (p, ep) in the RPA and the Rice approximation are also drawn there. For the whole wavenumber region shown there the curve of Re F_,(p, ep) in the Rice approximation is uniformly shifted upwards by about 0.1 eF from the RPA one. The curve of Re Z(p, ep) in the present approximation is further shifted upwards from the Rice approximation by a considerable amount and is much more flattened, compared with the RPA and the Rice curves. It is precisely the local field corrections due to the short-range Coulomb repulsion that makes the quantity Re F_,(p, ep) surprisingly independent of P/PF over a wide range of the magnitude of PiPe. In other words, the use of the well-known Slater approximation and its modification by Kohn and Sham can be well justified by taking into account not only the longrange screening effect properly described by the RPA but also the strong short-range correlation effect. As is obvious from Fig. 1, there appears a dip in each calculated curve around PIPE ~-- 1.8 ~ 1.9. The dip appears precisely at a value of the wavenumber ratio where the

643

EXCHANGE AND CORRELATION POTENTIALS

Vol. 47, No. 8 I

I

I

\ 6_

//

-1.0

rs=4

rr

i

rs=6 -2.0

1.02 0

-3.0

0.5

1.0

1.5

2.0

P/PF

Fig. 2. The calculated values of Re Z(p, ev) in the present approximation are shown as a function of PiPe for r~ = 2.0, 4.0 and 6.0.

corresponding imaginary part of the self-energy of equation (1) abruptly starts to increase because of the opening of a damping channel due to plasmon emissions [4, 5]. The position of the dip is by an appreciable amount shifted to a smaller wavenumber by the inclusion of the local field correction to the RPA. The appearance of the dip can be traced back to the non-damping plasmon excitation near the Cut-off wavenumber k e. The magnitude of the dip is very sensitive to a damping width of the plasmon. This dip will be almost removed by an appropriate inclusion of the damping effects. Hedin and Lundqvist [6] showed the RPA curve of Re E(p, e,o) for wavenumber ratios up to 5.0. The value of Re E(p, ep) gradually approaches an asymptotic pv/p-behaviour as PIPe goes beyond about 2. The value of Re E(p, ep) estimated in the present approximation except for a sizable shift upwards will also show a similar behaviour in this wavenumber region. In Fig. 2 we show the calculated values of Re Z(p, %) in the present approximation for r~ = 2.0, 4.0 and 6.0. As can be seen in the review article by Hedin and Lundqvist [6], the calculated curves in the RPA are gradually inclined as r8 increases, over a considerable range of p/pF including PIPe = 1.0. The present calculations of Re Z(p, el,), on the other hand, exhibits almost no appreciable inclination of the curves in the whole metallic density region. This is because the local field correction from spin-antiparallel correlation increases remarkably as rs increases. Finally we show in Fig. 3 the values of the effective mass ratio m*/m as a function o f t s which is closely related to the inclination of Re ~ (p, ep) at p = PF. For comparison the calculated values of m*/m in the RPA and the Rice approximation are also drawn there. We have recalculated the mass ratio using the RPA and

SENT

i:iI Fig. 3. The calculated values of the effective mass ratio

m*/m in the RPA, the Rice approximation, and the present approximation are shown as a function of rs.

the Rice approximation; since Rice's numerical estimation [15] is not very accurate, we have in Fig. 3 shown recalculated values in the Rice approximation. As is well known, the high density expansion of m*/m is given as follows [17]:

m*/m =

Ion,( ) / 1---~- 2+ln

+""

,

(5)

a = (4/9rr) 1/3 = 0 . 5 2 1 0 6 . . . . In the high density region where the rs-expansion above is valid the mass ratio m*/m gives values smaller than 1. As r8 increases, the calculated values ofm*/m in the RPA exhibit a behaviour analogous to the high density expansion and exceed 1 for rs >t 2.23. The mass ratio m*/m in the Rice approximation is generally somewhat smaller than the RPA ratio but also exceeds 1 for r s ~> 2.7. The mass ratio m*/m in the present approximation is remarkably reduced and still remains smaller than 1 in the whole metallic density region; m*/m exceeds 1 at rs ~- 5.5. MacDonald, Dharma-wardana and Geldart [ 18] have calculated the mass ratio m*/m as a function of r, and have obtained a curve quite similar to ours; their ratio m*/m exceeds 1 at r s ~--5.6. They have, however, estimated the mass ratio by simply choosing the RPA diagram, the second-order exchange diagram, and the third-order direct and exchange particle-particle ladder diagrams with the

644

EXCHANGE AND CORRELATION POTENTIALS

statically screened Coulomb interaction. It is puzzling that a very similar behaviour of m*/m has been obtained from the quite different methods. In our opinion a meaningful remedy of the shortcomings of the RPA at metallic densities can be obtained only through the inclusion of infinite series of particle-hole and particleparticle ladder interactions which we have approximately cast into the two compact form of the local field corrections, G(q) and C(q) in the representation of the modified form for Z(p, e). MacDonald et al. [18] have not given numerical values of Re Z(p, ep) explicitly, but the overall aspects of their Re Z (p, ep) would be rather different from ours, considering that their secondand third-order correction have been estimated with the statically screened interaction and that the mass ratio m*/m has been evaluated according to the formula including an extra factor of renormalization constant z. It is, in fact, inferred that the inclination of their Re X(p, ep) at p = PF would be steeper than ours by a factor of z -x . A naive modification obtained by simply adding a few lowest order diagrams to the RPA one can not possibly lead to such a significant modification of the RPA expression as representing the reduction of the cut-off wavenumber k c. In this respect, the expression for the self-energy adopted by MacDonald et al. [18] seems to be questionable. For a further detailed study of the self-energy Z(p, e) it must be necessary to use the renormalized form of the internal Green function including the damping effects in constructing the self-energy expression. A careful re-examination of different sets of infinite series of diagrams and their suitable re-arrangement for the meaningful description of their physical implications will be required to obtain a further advanced approximation.

Acknowledgements - The authors would like to thank Professor T. Asahi for useful conversations and his

Vol. 47, No. 8

interest in this work. Thanks are also due to Professor H. Takayama for his encouraging discussions. One of the authors (H.Y.) is thankful to Professor H. Fukuyama and Dr. Y. Isawa for interesting discussions. Numerical calculations were made by Hitac M-200H at Hokkaido University Computing Center.

REFERENCES 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

P. Hohenberg & W. Kohn, Phys. Rev. 136, B864 (1964). W. Kohn&L.J. Sham, Phys. Rev. 140, Al133 (1965). U. von Barth & L. Hedin, J. Phys. C5, 1629 (1972). K. Awa, H. Yasuhara & T. Asahi, Solid State Commun. 38, 1285 (1981). K. Awa, H. ¥asuhara & T. Asahi, Phys. Rev. B25, 3670 (1982); 3687 (1982). L. Hedin & S. Imndqvist, Solid State Physics, (Edited by F. Seitz, D. Turnbull & Ehrenreich), Vol. 23, pp. 1-181. Academic Press (1969). H. Yasuhara, SolidState Commun. 11, 1481 (1972). H. Yasuhara, J. Phys. Soc. Japan 36,361 (1974). H. Yasuhara, Physica 78,420 (1974). B.B. Hede & J.P. Carbotte, Can. J. Phys. 50, 4512 (1972). D.N. Lowy &G.E. Brown, Phys. Rev. B12,2138 (1975). K. Awa & T. Asahi, J. Phys. Soc. Japan 48,757 (1980). R.F. Bishop & K.H. Ltihrmann, Phys. Rev. B26, 5523 (1982). D.F. DuBois, Ann. Phys. (New York) 7,174 (1959);8, 24 (1959). T.M. Rice, Ann. Phys. (New York) 31,100 (1965). J. Hubbard, Proc. Roy. Soc. (London) A240, 539 (1957); A243, 336 (1957). M. Gell-Mann, Phys. Rev. 106,364 (1957). A.H. MacDonald, M.W.C. Dharma-wardana & D.J.W. Geldart, J. Phys. F10, 1719 (1980).

NOTE ADDED IN PROOF. The absolute magnitudes of ReX(p, %) calculated by equation (1) seem to be somewhat too small; this is because we have overestimated the magnitude of the local field correction C(q) to appear in the numerator of equation (1). The main conclusions in the text need not be modified; they depend chiefly on the local field correction G(q) in the denominator of equation (1).