Abstract
We have demonstrated that the nodal structure of the wave functions can have a great effect on the accuracy of the LSD approximation, and we have identified classes of problems where the LSD results must be treated by caution. For states with the minimum number of nodal planes consistent with the orthogonality of the orbitals, the LSD approximation usually leads to a moderate overestimate of the exchange-correlation energy. For states with additional nodal planes the exchange-correlation energy is often greatly overestimated. In atoms, the depopulation of s-orbitals can lead to large errors, and similar effects may be expected in bonding situations where sp or sd hybridization reduces the s occupancy. More work in this area is essential.
We have studied a very simple but analytically solvable model for which the exact functionals can be constructed explicitly using the Levy constrained search method. The results illustrate how the discontinuity in the exchange-correlation potential is built up as the chemical potential is moved through the gap. We have further studied a linear chain ("semiconductor") model, for which we have obtained an exact numerical solution. We have constructed the exact exchange-correlation potential for this model and find that this potential can have a finite discontinuity even when the chain length is extrapolated to oo. For “realistic” parameters the discontinuity is small. A large discontinuity is obtained for parameters which lead to a ground state of a charge density wave type. Earlier a large discontinuity was found for this model in a parameter range which leads to a Mott insulator.
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References
P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).
W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965).
For a survey, see U. von Barth and A.R. Williams, Theory of the Inhomomogeneous Electron Gas, Eds. S. Lundqvist and N.H. March (Plenum, New York, 1983).
O. Gunnarsson and R.O. Jones, J. Chem. Phys. 72, 5357 (1985).
J. Harris and R.O. Jones, J. Chem. Phys. 68, 3316 (1978).
G.S. Painter and F. Averill, Phys. Rev. B26, 1781 (1982).
O. Gunnarsson and R.O. Jones, Phys. Rev. B31, 7588 (1985); R.O. Jones and O. Gunnarsson, Phys. Rev. Lett. 55, 107 (1985).
J.P. Perdew and M. Levy, Phys. Rev. Lett. 51, 1884 (1983); see also J.P. Perdew, R.G. Parr, M. Levy and J.L. Balduz, Phys. Rev. Lett. 49, 1691 (1982).
L.J. Sham and M. Schlüter, Phys. Rev. Lett. 51, 1888 (1983).
O. Gunnarsson and K. Schönhammer, Phys. Rev. Lett. 56, 1968 (1986).
See, e.g., J.C. Slater, Quantum Theory of Molecules and Solids (McGraw-Hill, New York, 1974) Vol. IV.
U. von Barth, Phys. Rev. A20, 1963 (1979).
J.H. Wood, J. Phys. B13, 1 (1980).
See, e.g., J.C. Slater, Quantum Theory of Atomic Structure (McGraw-Hill, New York, 1960), Vol. II Appendix 21. By the Xα approximation we mean LSD with exchange only, and not the additional approximations involved in the scattering wave Xα method.
I. Lindgren and K. Schwarz, Phys. Rev. A5, 542 (1972); K. Schwarz, Phys. Rev. A 5, 2466 (1972).
Hartree-Fock energies were taken from E. Clementi and C. Roetti, At. Data Nucl. Data Tables 14, 177 (1974), and from G. Verhaegen and C. Moser, J. Phys. B 3, 478 (1980).
O. Gunnarsson and B.I. Lundqvist, Phys. Rev. B13, 4274 (1976).
S. Vosko, L. Wilk and M. Nusair, Can. J. Phys. 58, 1200 (1980).
R.O. Jones, J. Chem. Phys. 71, 1300 (1979).
J.F. Janak, Phys. Rev. B 18, 7165 (1978).
K. Schönhammer and O. Gunnarsson (to be published).
M. Levy, Proc. Nat. Acad. Sci. USA 76, 6062 (1979); Phys. Rev. A 26, 1200 (1982).
B.N. Parlett, The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, N.J.), p. 257, (1980).
U. von Barth, in The Electronic Structure of Complex Systems, Eds. P. Phariseau and W.M. Temmerman (Plenum, New York, 1984).
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© 1987 Springer-Verlag
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Gunnarsson, O., Jones, R.O., Schönhammer, K. (1987). Density-functional formalism: V xc, discontinuities, and the local density approximation. In: Yussouff, M. (eds) Electronic Band Structure and Its Applications. Lecture Notes in Physics, vol 283. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3540180982_3
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DOI: https://doi.org/10.1007/3540180982_3
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