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Fourth order gradient corrections to the free energy of noninteracting many-electron systems at finite temperature
Volume 108A, number 2
PHYSICS LETTERS
18 March 1985
FOURTH ORDER GRADIENT CORRECTIONS TO THE FREE ENERGY OF NONINTERACTING MANY-ELECTRON SYSTEMS AT FlNlTE TEMPERATURE D.J.W. GELDART and E. SOMMER Department of Physics, Dalhousie University,Halifax, Nova Scot& Canada B3H 355 Received 19 December 1984
The fourth order gradient expansion corrections to the free energy of a noninteracting many-electron system at finite temperature in an external potential are obtained. The required coefficients are conveniently expressed in terms of FermiDirac integrals.
The Thomas-Fermi (TF) or statistical model provides a widely used approximate description of matter in various forms. In the extended Thomas-Fermi (ETF) model, one attempts to improve upon the purely local character of TF theory by including low order gradient corrections to the kinetic energy and to exchange and correlation energies. There are various reasons for interest in ETF theory. Its structure is conceptually simple and also universal, being a form of density functional theory [ 11. It is a systematic expansion about the classical limit and, even though asymptotic, is improvable in principle by standard methods if sufficient knowledge of the expansion coefficients becomes available. It is much simpler in practice than methods based on self-consistency of single particle orbitals [2] but, of course, is less accurate. Accordingly, the class of problems where ETF is particularly useful typically involves reduced symmetry and often finite temperature or situations where exploratory work is advisable. For many-electron systems in their ground state, gradient corrections to the local density approximation are known up to sixth order for the kinetic energy [3-51. The exchange and correlation energy gradient corrections are known only to second order in powers of gradients [6-121. In applications, it is found that best results are obtained when the gradient series for the kinetic energy is truncated after the fourth order terms [ 13- 151. It is desirable to extend these results so as to be able to treat systems at finite 0.375-9601/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
temperature, particularly for use in descriptions of equations of state. Thus far, only the second order gradient expansion coefficient for free particles has been available at finite temperatures [ 16,171. There is no information at all so far concerning nonlocal contributions to interaction effects at finite temperature. To extend these results we have carried out the calculation of gradient corrections to the free energy of noninteracting electrons up to fourth order in powers of density gradients. The method of calculation is described here but, as the calculations are extremely lengthy, essentially only the main results are given. Consider a system of free particles in an external potential T/. The grand potential s2@, T,) is calculated as a series in powers of V as a function of chemical potential p and temperature T by standard methods [ 181. The electron density, n(r) = no + Z(r), where no is the average density, is similarly expanded as a series in powers of the potential. The series for n in powers of I’is then inverted to yield a functional Taylor expansion for Fourier components V(q) in powers of Fourier components E(q’). It is necessary to express the free energy F = -/3 ln Q where f3= l/k,T, in terms of the physical density so the chemical potential must also be expanded in powers of V and then converted to a series in powers of K. In this way, all explicit and implicit E dependence of the free energy can be exhibited, after considerable algebra, in
Volume
108A, number
PHYSICS
2
where Fk is proportional to n-k. The second order term involves only the linear response (Lindhard) function while higher order terms involve nonlinear response functions in a complicated way. We now truncate the series after k = 4 and generate a gradient expansion by assuming that Z(r) contains only low wavenumber Fourier components so that all combinations of response functions can be expanded in powers of wavenumbers. After lengthy algebra, we find that the series in Z can be partially resummed in such a way that quantities involving no also enter with appropriate derivatives and powers of Z so that all coefficients are evaluated at a local density no + i?(r) and no disappears from the problem, as expected. The result is F[n]
=l
d3r-{ P’(r)n(r) +fln(r))
+ B(n(r))[Vn(r)J2
+ C(n(r))[V2n(r)]
+ ~(n(r)>V2n(r)[Vn(r)]
2
2 + E(n(r))[Vn(r)14
+ .. . )
(2) In (2),f(n) is the free energy of a uniform system of density n at temperature T. The lowest order gradient coefficient B(n) = (fi2/2m)[X2/12X;] was previously known
(3) [16,17]. The new results are
C(n) = (fi2/2m) [X$72X; D(n) = (h2/kz) - X$36X;]
- X,/120X,2]
[-X4/180X:
(4)
t 1 1X,X3/360Xit
,
(5)
E(n) = (W92m) [-X,/1440X; +X$72X:
,
+ X2Xa/lSOX:
- x,2x3/45G +X2,/288X:]
,
(6)
where X, = (~/~&lY~o(Po~
104
T)
(7)
LETTERS
18 March
1985
is a standard Ferm- Dirac integral. These quantities are easily obtainable as convenient expansions about either the degenerate (T--f 0) or the classical limit and numerical studies based on these results are feasible. Further discussion of these results with applications will be given elsewhere and we also hope to provide a similar extension to finite temperature of nonlocal (gradient) interaction contributions to the free energy. References 111 P. Hohenberg and W. Kahn, Phys. Rev. 136 (1964) B864. [2] W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) Al 133. [3] D.A. Kirzhnits, Zh. Eksp. Teor. Fiz. 32 (1957) 115 [Sov. Phys. JETP 5 (1957) 64. [4] C.H. Hodges, Can. J. Phys. 51 (1973) 1428. [5] D.R. Murphy, Phys. Rev. A24 (1981) 1682. [6] L.J. Sham, in: Computational methods in band theory, eds. P.M. Marcus, J.F. Janak and A.R. Williams (Plenum, New York, 1971) p. 458. [7] D.J.W. Geldart and M. Rasolt, report (unpublished). National Research Council, Ottawa, Canada. [8] D.J.W. Geldart, M. Rasolt and CO. Ambladh, Solid State Commun. 16 (1975) 243. [9] L. Kleinman, Phys. Rev. BlO (1974) 2221. [lo] S.K. Ma and K. Brueckner, Phys. Rev. 165 (1968) 18. [ll] M. Rasolt and D.J.W. Geldart, Phys. Rev. Lett. 35 (1975) 1234. [12] D.J.W. Geldart and M. Rasolt, Phys. Rev. B13 (1976) 1477. 1131 J. S.-Y. Wang and M. Rasolt, Phys. Rev. B13 (1976) 5330. [ 141 C.Q. Ma and V. Sahni, Phys. Rev. B16 (1977) 4249. [1.5] M.L. Plumer and D.J.W. Geldart, J. Phys. Cl6 (1983) 677. [16] F. Perrot, Phys. Rev. A20 (1979) 586. [17] M. Brack, Phys. Rev. Lett. 53 (1984) 119. [ 181 A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinski, Methods of quantum field theory in statistical physics (Prentice-Hall, Englewood Cliffs, 1963).
PHYSICS LETTERS
18 March 1985
FOURTH ORDER GRADIENT CORRECTIONS TO THE FREE ENERGY OF NONINTERACTING MANY-ELECTRON SYSTEMS AT FlNlTE TEMPERATURE D.J.W. GELDART and E. SOMMER Department of Physics, Dalhousie University,Halifax, Nova Scot& Canada B3H 355 Received 19 December 1984
The fourth order gradient expansion corrections to the free energy of a noninteracting many-electron system at finite temperature in an external potential are obtained. The required coefficients are conveniently expressed in terms of FermiDirac integrals.
The Thomas-Fermi (TF) or statistical model provides a widely used approximate description of matter in various forms. In the extended Thomas-Fermi (ETF) model, one attempts to improve upon the purely local character of TF theory by including low order gradient corrections to the kinetic energy and to exchange and correlation energies. There are various reasons for interest in ETF theory. Its structure is conceptually simple and also universal, being a form of density functional theory [ 11. It is a systematic expansion about the classical limit and, even though asymptotic, is improvable in principle by standard methods if sufficient knowledge of the expansion coefficients becomes available. It is much simpler in practice than methods based on self-consistency of single particle orbitals [2] but, of course, is less accurate. Accordingly, the class of problems where ETF is particularly useful typically involves reduced symmetry and often finite temperature or situations where exploratory work is advisable. For many-electron systems in their ground state, gradient corrections to the local density approximation are known up to sixth order for the kinetic energy [3-51. The exchange and correlation energy gradient corrections are known only to second order in powers of gradients [6-121. In applications, it is found that best results are obtained when the gradient series for the kinetic energy is truncated after the fourth order terms [ 13- 151. It is desirable to extend these results so as to be able to treat systems at finite 0.375-9601/85/$03.30 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
temperature, particularly for use in descriptions of equations of state. Thus far, only the second order gradient expansion coefficient for free particles has been available at finite temperatures [ 16,171. There is no information at all so far concerning nonlocal contributions to interaction effects at finite temperature. To extend these results we have carried out the calculation of gradient corrections to the free energy of noninteracting electrons up to fourth order in powers of density gradients. The method of calculation is described here but, as the calculations are extremely lengthy, essentially only the main results are given. Consider a system of free particles in an external potential T/. The grand potential s2@, T,) is calculated as a series in powers of V as a function of chemical potential p and temperature T by standard methods [ 181. The electron density, n(r) = no + Z(r), where no is the average density, is similarly expanded as a series in powers of the potential. The series for n in powers of I’is then inverted to yield a functional Taylor expansion for Fourier components V(q) in powers of Fourier components E(q’). It is necessary to express the free energy F = -/3 ln Q where f3= l/k,T, in terms of the physical density so the chemical potential must also be expanded in powers of V and then converted to a series in powers of K. In this way, all explicit and implicit E dependence of the free energy can be exhibited, after considerable algebra, in
Volume
108A, number
PHYSICS
2
where Fk is proportional to n-k. The second order term involves only the linear response (Lindhard) function while higher order terms involve nonlinear response functions in a complicated way. We now truncate the series after k = 4 and generate a gradient expansion by assuming that Z(r) contains only low wavenumber Fourier components so that all combinations of response functions can be expanded in powers of wavenumbers. After lengthy algebra, we find that the series in Z can be partially resummed in such a way that quantities involving no also enter with appropriate derivatives and powers of Z so that all coefficients are evaluated at a local density no + i?(r) and no disappears from the problem, as expected. The result is F[n]
=l
d3r-{ P’(r)n(r) +fln(r))
+ B(n(r))[Vn(r)J2
+ C(n(r))[V2n(r)]
+ ~(n(r)>V2n(r)[Vn(r)]
2
2 + E(n(r))[Vn(r)14
+ .. . )
(2) In (2),f(n) is the free energy of a uniform system of density n at temperature T. The lowest order gradient coefficient B(n) = (fi2/2m)[X2/12X;] was previously known
(3) [16,17]. The new results are
C(n) = (fi2/2m) [X$72X; D(n) = (h2/kz) - X$36X;]
- X,/120X,2]
[-X4/180X:
(4)
t 1 1X,X3/360Xit
,
(5)
E(n) = (W92m) [-X,/1440X; +X$72X:
,
+ X2Xa/lSOX:
- x,2x3/45G +X2,/288X:]
,
(6)
where X, = (~/~&lY~o(Po~
104
T)
(7)
LETTERS
18 March
1985
is a standard Ferm- Dirac integral. These quantities are easily obtainable as convenient expansions about either the degenerate (T--f 0) or the classical limit and numerical studies based on these results are feasible. Further discussion of these results with applications will be given elsewhere and we also hope to provide a similar extension to finite temperature of nonlocal (gradient) interaction contributions to the free energy. References 111 P. Hohenberg and W. Kahn, Phys. Rev. 136 (1964) B864. [2] W. Kohn and L.J. Sham, Phys. Rev. 140 (1965) Al 133. [3] D.A. Kirzhnits, Zh. Eksp. Teor. Fiz. 32 (1957) 115 [Sov. Phys. JETP 5 (1957) 64. [4] C.H. Hodges, Can. J. Phys. 51 (1973) 1428. [5] D.R. Murphy, Phys. Rev. A24 (1981) 1682. [6] L.J. Sham, in: Computational methods in band theory, eds. P.M. Marcus, J.F. Janak and A.R. Williams (Plenum, New York, 1971) p. 458. [7] D.J.W. Geldart and M. Rasolt, report (unpublished). National Research Council, Ottawa, Canada. [8] D.J.W. Geldart, M. Rasolt and CO. Ambladh, Solid State Commun. 16 (1975) 243. [9] L. Kleinman, Phys. Rev. BlO (1974) 2221. [lo] S.K. Ma and K. Brueckner, Phys. Rev. 165 (1968) 18. [ll] M. Rasolt and D.J.W. Geldart, Phys. Rev. Lett. 35 (1975) 1234. [12] D.J.W. Geldart and M. Rasolt, Phys. Rev. B13 (1976) 1477. 1131 J. S.-Y. Wang and M. Rasolt, Phys. Rev. B13 (1976) 5330. [ 141 C.Q. Ma and V. Sahni, Phys. Rev. B16 (1977) 4249. [1.5] M.L. Plumer and D.J.W. Geldart, J. Phys. Cl6 (1983) 677. [16] F. Perrot, Phys. Rev. A20 (1979) 586. [17] M. Brack, Phys. Rev. Lett. 53 (1984) 119. [ 181 A.A. Abrikosov, L.P. Gorkov and I.E. Dzyaloshinski, Methods of quantum field theory in statistical physics (Prentice-Hall, Englewood Cliffs, 1963).