Density Functional Theory pp 11-31 | Cite as
Coordinate Scaling Requirements for Approximating Exchange and Correlation
Abstract
In order to continue to better approximate the exact exchange-correlation density functional, Exc[n], for electronic structure calculations, sets of exact coordinate scaling and convexity requirements are reviewed for the purpose of testing approximations. The scaling requirements dictate how Exc[n] must transform when n(x, y, z) is replaced by αβγn(αx,βy,γz). High-density and low-density limits are explored. Simple formulas are shown which isolate the kinetic component of Exc[n] and which generate Exc[n] from just its electron-electron repulsion component, or exchange-correlation hole, at a coupling constant of unity. The scaling and convexity conditions are derived with help from the constrained-search formulation and through the development of a coupling-constant perturbation expansion. The local density approximation and various gradient approximations are compared against the exact conditions, with emphasis upon where the conditions are respected and where the conditions are violated. Numerical results are presented.
Keywords
Local Density Approximation Correlation Energy Electronic Structure Calculation Virial Theorem Uniform ScalingPreview
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References
- 1.R.M. Dreizler and E.K.U. Gross, Density Functional Theory, Springer-Verlag, Berlin-New York, 1990.MATHCrossRefGoogle Scholar
- 2.R.G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989.Google Scholar
- 3.R.O. Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989).ADSCrossRefGoogle Scholar
- 4.Entire issue of Advances in Quantum Chemistry, edited by S.B. Trickey, Academic Press, San Diego, 1990.Google Scholar
- 5.E.S. Kryachho and E.V. Ludena, Density Functinal Theory of Many-Electron Systems, Kluwer Press, Dordrecht, 1990.CrossRefGoogle Scholar
- 6.N.H. March, Electron Density Theory of Atoms and Molecules, Academic Press, London, 1992.Google Scholar
- 7.J.K. Labanowski and J.W. Andzelm, editors, Density Functional Methods in Chemistry, Springer-Verlag, New York, 1991.Google Scholar
- 8.R.M. Dreizler and J. da Providencia, editors, Density Functional Methods in Physics, Plenum Press, New York, 1985.Google Scholar
- 9.N.H. March and B.M. Deb, editors, The Single-Particle Density in Physics and Chemistry, Academic Press, New York, 1987.Google Scholar
- 10.R. Erdahl and V.H. Smith, Jr., editors, Density Matrices and Density Functionals, Reidel Press, Dordrecht, 1987.Google Scholar
- 11.M. Levy and J.P. Perdew, Phys. Rev. A. 32, 2010 (1985).ADSCrossRefGoogle Scholar
- 12.M. Levy, W. Yang, and R.G. Parr, J. Chem. Phys. 83, 2334 (1985).ADSCrossRefGoogle Scholar
- 13.W. Yang, page 499 in Ref. 10.Google Scholar
- 14.M. Levy, Int. J. Quantum Chem. S23, 617 (1989).Google Scholar
- 15.H. Ou-Yang and M. Levy, Phys. Rev. A 42, 155 (1990).ADSCrossRefGoogle Scholar
- 16.M. Levy and H. Ou-Yang, Phys. Rev. A 42, 651 (1990).ADSCrossRefGoogle Scholar
- 17.M. Levy, page 69 in Ref. 4.Google Scholar
- 18.L.C. Wilson and M. Levy, Phys. Rev. B 41, 12930 (1990).ADSCrossRefGoogle Scholar
- 19.H. Ou-Yang and M. Levy, Phys. Rev. Lett. 65, 1036 (1990).ADSCrossRefGoogle Scholar
- 20.H. Ou-Yang and M. Levy, Phys. Rev. A 44, 54 (1991).ADSCrossRefGoogle Scholar
- 21.M. Levy, Phys. Rev. A 43, 4637 (1991).ADSCrossRefGoogle Scholar
- 22.A. Görling and M. Levy, Phys. Rev. A 45, 1509 (1992).ADSCrossRefGoogle Scholar
- 23.M. Levy and J.P. Perdew, Phys. Rev. B 48, 11638 (1993).ADSCrossRefGoogle Scholar
- 24.A. Görling and M. Levy, Phys. Rev. B 47, 13105 (1993).ADSCrossRefGoogle Scholar
- 25.A. Görling, M. Levy, and J.P. Perdew, Phys. Rev. B 47, 1167 (1993).ADSCrossRefGoogle Scholar
- 26.Q. Zhao, M. Levy and R.G. Parr, Phys. Rev. A 47, 918 (1993).ADSCrossRefGoogle Scholar
- 27.M. Levy and A. Görling, Phil. Mag., in press.Google Scholar
- 28.M. Levy and J.P. Perdew, Int. J. Quantum Chem. 49, 539 (1994).CrossRefGoogle Scholar
- 29.J.P. Perdew, in Electronic Structure of Solids ‘91, edited by P. Ziesche and H. Eschrig, Academie Verlag, Berlin, 1991.Google Scholar
- 30.E.H. Lieb and S. Oxford, Int. J. Quantum Chem. 19, 427 (1981).CrossRefGoogle Scholar
- 31.D.C. Langreth and J.P. Perdew, Solid State Commun. 17, 1425 (1975).ADSCrossRefGoogle Scholar
- 32.O. Gunnarsson and B.I. Lundqvist, Phys. Rev. B 13, 4274 (1976).ADSCrossRefGoogle Scholar
- 33.D.C. Langreth and J.P. Perdew, Phys. Rev. B 15, 2884 (1976).ADSCrossRefGoogle Scholar
- 34.See Appendix A in Ref. [25].Google Scholar
- 35.Q. Zhao and R.G. Parr, Phys. Rev. A 46, 5320 (1992).ADSCrossRefGoogle Scholar
- 36.P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964).MathSciNetADSCrossRefGoogle Scholar
- 37.M. Levy, Proc. Natl. Acad. Sci (U.S.A.) 76, 6062 (1979).ADSCrossRefGoogle Scholar
- 38.M. Levy, Phys. Rev. A 26, 1200 (1982).ADSCrossRefGoogle Scholar
- 39.J.E. Harriman, Phys. Rev. A 24, 680 (1981).ADSCrossRefGoogle Scholar
- 40.E.H. Lieb, Int. J. Quantum Chem. 24, 224 (1983).CrossRefGoogle Scholar
- 41.J.K. Percus, Int. J. Quantum Chem. 13, 89 (1978).CrossRefGoogle Scholar
- 42.M. Gell-Mann and K.A. Brueckner, Phys. Rev. 106, 364 (1957).MathSciNetADSMATHCrossRefGoogle Scholar
- 43.W. Kohn and L.J. Sham, Phys. Rev. 140, A1133 (1965).MathSciNetADSCrossRefGoogle Scholar
- 44.E. Engel and S.H. Vosko, Phys. Rev. A 47, 2800 (1993).ADSCrossRefGoogle Scholar
- 45.F.W. Averill and G.S. Painter, Phys. Rev. B 24, 6795 (1981).ADSCrossRefGoogle Scholar
- 46.S.K. Ghosh and R.G. Parr, J. Chem. Phys. 82, 3307 (1985).ADSCrossRefGoogle Scholar
- 47.D.J. Lacks and R.G. Gordon, Phys. Rev. A 47, 4681 (1993).ADSCrossRefGoogle Scholar
- 48.A.D. Becke, Phys. Rev. A 33, 3098 (1988).ADSCrossRefGoogle Scholar
- 49.J.P. Perdew and Y. Wang, Phys. Rev. B 33, 8800 (1986).ADSCrossRefGoogle Scholar
- 50.D.C. Langreth and M.J. Mehl, Phys. Rev. B 28, 1809 (1993).ADSCrossRefGoogle Scholar
- 51.C. Lee, W. Yang, and R.G. Parr, Phys. Rev. B 37, 785 (1988).ADSCrossRefGoogle Scholar
- 52.C.J. Umrigar and X. Gonze, published in “High Performance Computing and its Application to the Physical Sciences”, proceedings of the Mardi Gras’ 93 Conference, edited by D.A. Browne et al., (World Scientific, 1993).Google Scholar
- 53.C.J. Umrigar and X. Gonze, to appear in Phys. Rev. A.Google Scholar