Abstract
Real-space analysis decomposes the exchange-correlation energy of a many-electron system into contributions from all possible interelectronic separations u. The density-gradient expansion of the exchange-correlation hole surrounding an electron has a characteristic structure. Its zeroth-order term, the local spin density (LSD) approximation, is a good approximation to both the hole and its cusp at u = 0. When the electron density varies slowly over space, addition of each successive term of higher order in ∇ improves the description of the hole at small u, but worsens it at large u. Starting with the second-order gradient expansion, we cut off the spurious large-u contributions in a way that restores the negativity and normalization constraints on the exchange hole, and the normalization constraint on the correlation hole. This procedure defines numerical generalized gradient approximations (GGA’s) for the exchange and correlation energies, using no empirical input. We report the results of this construction in detail. This numerical GGA satisfies the most important exact conditions respected by LSD, plus several more (but not all) exact conditions currently known. The PW91 functional is an analytic fit to this functional, designed to respect several further exact conditions including the Lieb-Oxford bound.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
W. Kohn and L.J. Sham, Phys. Rev. 140, A 1133 (1965).
R.O. Jones and O. Gunnarsson, Rev. Mod. Phys. 61, 689 (1989).
R.G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules (Oxford, New York, 1989).
R.M. Dreizler and E.K.U. Gross, Density Functional Theory (Springer-Verlag, Berlin, 1990).
D. M. Ceperley and B. J. Alder, Phys. Rev. Lett. 45, 566 (1980).
W. E. Pickett and J. Q. Broughton, Phys. Rev. B 48, 14859 (1993).
G. Ortiz and P. Ballone, Phys. Rev. B 50, 1391 (1994).
S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200 (1980).
J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).
J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992).
M. Rasolt and H.L. Davis, Phys. Lett. A 86, 45 (1981).
M. Rasolt and D.J.W. Geldart, Phys. Rev. B 34, 1325 (1986).
L. J. Sham, in Computational Methods in Band Theory, edited by P. M. Marcus, J. F. Janak, and A. R. Williams (Plenum, New York, 1971), p. 458.
S.-K. Ma and K.A. Brueckner, Phys. Rev. 165, 18 (1968).
L. Kleinman and S. Lee, Phys. Rev. B 37, 4634 (1988).
J.P. Perdew and Y. Wang, Phys. Rev. B 33, 8800 (1986).; 40, 3399 (1989) (E).
J.P. Perdew, Phys. Rev. B 33, 8822 (1986).; 34, 7406 (1986) (E).
D.C. Langreth and J.P. Perdew, Phys. Rev. B 21, 5469 (1980).
J.P. Perdew, Phys. Rev. Lett. 55, 1665 (1985); 55, 2370 (1985) (E).
D.C. Langreth and M.J. Mehl, Phys. Rev. B 28, 1809 (1983).
C.D. Hu and D.C. Langreth, Phys. Scr. 32, 391 (1985).
K. Burke, J. P. Perdew, and M. Levy, in Modern Density Functional Theory: A Tool for Chemistry, edited by J. M. Seminario and P. Politzer (Elsevier, Amsterdam, 1995).
J. P. Perdew, J. A. Chevary, S. H. Vosko, K. A. Jackson, M. R. Pederson, D.J. Singh, and C. Fiolhais, Phys. Rev. B 46, 6671 (1992).; 48, 4978 (1993) (E).
N. A. W. Holzwarth and Y. Zeng, Phys. Rev. B 49, 2351 (1994).
L. Stixrude, R. E. Cohen, and D. J. Singh, Phys. Rev. B 50, 6442 (1994).
P. Dufek, P. Blaha, and K. Schwarz, Phys. Rev. B 50, 7279 (1994).
P. Söderlind, O. Eriksson, J.M. Willis, and B. Johannson, Phys. Rev. B 50, 7291 (1994).
G. Kresse, J. Furthmüller, and J. Hafner, Phys. Rev. B 50, 13181 (1994).
M. Causá and A. Zupan, Chem. Phys. Letters 220, 145 (1994).
A. Khein, D.J. Singh, and C.J. Umrigar, Phys. Rev. B 51, 4105 (1995).
D. Porezag and M.R. Pederson, J. Chem. Phys. 102, 9345 (1995).
J.C. Grossman, L. Mitas, and K. Raghavachari, Phys. Rev. Lett. 75, 3870 (1995); 76, 1006 (1996) (E).
L. Stixrude and R.E. Cohen, Science, 267, 1972 (1995).
B. Hammer and J.K. Nørskov, Nature 376, 238 (1995).
A. Gross, B. Hammer, M. Scheffler, and W. Brenig, Phys. Rev. Lett. 73, 3121 (1994).
N. Moll, M. Bockstedte, M. Fuchs, E. Pehlke, and M. Scheffler, Phys. Rev. B 52, 2550 (1995).
D.R. Hamann, Phys. Rev. Lett. 76, 660 (1996).
J.P. Perdew, R.G. Parr, M. Levy, and J.L. Balduz, Jr., Phys. Rev. Lett. 49, 1691 (1982).
J. P. Perdew, in Density Functional Methods in Physics, edited by R.M. Dreizler and J. da Providencia (Plenum, NY, 1985), p. 265.
K. Burke, M. Ernzerhof, and J.P. Perdew, Why semi-local functionals work: Accuracy of the on-top hole density, work in progress.
C. J. Umrigar and X. Gonze, in High Performance Computing and its Application to the Physical Sciences, Proceedings of the Mardi Gras 1993 Conference, edited by D. A. Browne et al (World Scientific, Singapore, 1993).
C. Filippi, C. J. Umrigar, and M. Taut, J. Chem. Phys. 100, 1290 (1994).
C. J. Umrigar and X. Gonze, Phys. Rev. A 50, 3827 (1994).
M. Slamet and V. Sahni, Phys. Rev. B 44, 10921 (1991).
A.D. Becke, Phys. Rev. A 38, 3098 (1988).
J.P. Perdew, in Electronic Structure of Solids’ 91, edited by P. Ziesche and H. Eschrig (Akademie Verlag, Berlin, 1991), page 11.
E. H. Lieb and S. Oxford, Int. J. Quantum Chem. 19, 427 (1981).
M. Levy, Int. J. Quantum Chem. S23, 617 (1989).
M. Levy, Phys. Rev. A 43, 4637 (1991).
A. Görling and M. Levy, Phys. Rev. A 45, 1509 (1992).
J.P. Perdew and K. Burke, Int. J. Quantum Chem. 57, 309 (1996).
C. Lee, W. Yang, and R.G. Parr, Phys. Rev. B 37, 785 (1988).
J. P. Perdew and Y. Wang, Phys. Rev. B 46, 12947 (1992).: 56, 7018 (1997) (E).
K. Burke and J. P. Perdew, Int. J. Quantum Chem. 56, 199 (1995).
E. Engel and S.H. Vosko, Phys. Rev. B 47, 13164 (1993). See Fig. 2.
J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996); 78, 1396 (1997) (E).
J.P. Perdew, K. Burke, and Y. Wang, Phys. Rev. B 54, 16533 (1996).
K. Burke, J.P. Perdew, and M. Ernzerhof, System-averaged exchange-correlation holes, work in progress.
D.C. Langreth and J.P. Perdew, Solid State Commun. 17, 1425 (1975).
D.C. Langreth and J.P. Perdew, Phys. Rev. B 15, 2884 (1977).
O. Gunnarsson and B.I. Lundqvist, Phys. Rev. B 13, 4274 (1976).
D.M. Ceperley and B.J. Alder, Phys. Rev. B 36, 2092 (1987).
K. Burke, J. P. Perdew, and D. C. Langreth, Phys. Rev. Lett. 73, 1283 (1994).
J.P. Perdew, A. Savin, and K. Burke, Phys. Rev. A 51, 4531 (1995).
K. Burke and J.P. Perdew, Mod. Phys. Lett. B 9, 829 (1995).
E.K.U. Gross and R.M. Dreizler, Z. Phys. A 302, 103 (1981).
Y. Wang, J. P. Perdew, J. A. Chevary, L. D. MacDonald, and S. H. Vosko, Phys. Rev. A 41, 78 (1990).
J.P. Perdew, Phys. Lett. A 165, 79 (1992).
G.L. Oliver and J.P. Perdew, Phys. Rev. A 20, 397 (1979).
J. P. Perdew, in Condensed Matter Theories, Vol. 2, edited by P. Vashishta, R. K. Kalia, and R. F. Bishop (Plenum, NY, 1987), p. 89.
M. Ernzerhof, J.P. Perdew, and K. Burke, in Density Functional Theory, edited by R. Nalewajski (Spinger-Verlag, Berlin, 1996).
D.C. Langreth and J.P. Perdew, Phys. Lett. A 92, 451 (1982).
M. Ernzerhof, K. Burke, and J.P. Perdew, J. Chem. Phys. 105, 2798 (1996).
Y. Wang and J. P. Perdew, Phys. Rev. B 43, 8911 (1991).
J.P. Perdew, Physica B 172, 1 (1991).
Y. Wang and J. P. Perdew, Phys. Rev. B 44, 13298 (1991).
M. Levy and J. P. Perdew, Phys. Rev. B 48, 11638 (1993).
J. C. Kimball, Phys. Rev. A 7, 1648 (1973).
E. R. Davidson, Reduced Density Matrices in Quantum Chemistry (Academic Press, New York, 1976).
J.P. Perdew, M. Ernzerhof, K. Burke, and A. Savin, Int. J. Quantum Chem. 61, 197 (1997).
M. Levy and J.P. Perdew, Phys. Rev. A 32, 2010 (1985).
A. Zupan, J.P. Perdew, K. Burke, and M. Causá, Int. J. Quantum Chem. 61, 835 (1997).
E. Engel, J.A. Chevary, L.D. MacDonald, and S.H. Vosko, Z. Phys. D 23, 7 (1992).
D.J. Lacks and R.G. Gordon, Phys. Rev. A 47, 4681 (1993).
L. A. Eriksson, O. L. Malkina, V. G. Malkin, and D. R. Salahub, J. Chem. Phys. 100, 5066 (1994).
E. Clementi and C. Roetti, At. Data Nucl. Data Tables 14, 177 (1974).
D. C. Langreth and S. H. Vosko, Adv. in Quantum Chem. 21, 175 (1990).
D.J.W. Geldart, E. Dunlap, M.L. Glasser, and M.R.A. Shegelski, Solid State Commun. 88, 81 (1993).
S.J. Chakravorty, S.R. Gwaltney, E.R. Davidson, F.A. Parpia, and C. Froese Fischer, Phys. Rev. A 47, 3649 (1993).
A. Görling, M. Levy, and J. P. Perdew, Phys. Rev. B 47, 1167 (1993).
J.P. Perdew, Int. J. Quantum Chem. S 27, 93 (1993).
Y. Li and J.B. Krieger, Phys. Rev. A 41, 1701 (1990).
K. Burke, J.P. Perdew, and M. Ernzerhof, Int. J. Quantum Chem. 61, 287 (1997).
P. Söderlind, doctoral dissertation, Uppsala University (1994).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media New York
About this chapter
Cite this chapter
Burke, K., Perdew, J.P., Wang, Y. (1998). Derivation of a Generalized Gradient Approximation: The PW91 Density Functional. In: Dobson, J.F., Vignale, G., Das, M.P. (eds) Electronic Density Functional Theory. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-0316-7_7
Download citation
DOI: https://doi.org/10.1007/978-1-4899-0316-7_7
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4899-0318-1
Online ISBN: 978-1-4899-0316-7
eBook Packages: Springer Book Archive