An Introduction to the Theory of Crystalline Elemental Solids and ...

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[15] W. Kohn and L. Sham, Phys. Rev. 140, 1133 (1965).
[16] This may also be called “Pauli correlation” because it correlates the dynamics of electrons
of like spin. Obviously electrons also repel each other because of their charge.
[17] A distinction between “empirical” and “non-empirical” DFT functionals is also common.
Empirical in this context covers those functionals that are designed in order to reproduce
experimental results or the results of high-level quantum chemical calculations for quantities
such as atomization energies and bond lengths. Typically “training sets” comprised of a
large number of well-defined energies and structures of small gas phase molecules are used
when making this comparison. The other essentially opposing, non-empirical, strategy is
to design functionals that abide by a number of well-known physical constraints which the
exact E xc functional should obey and do not include parameters other than fundamental
constants.
[18] M. Gell-man and K. A. Brueckner, Phys. Rev. 106, 364 (1957).
[19] D. M. Ceperley and B. J. Alder, Phys. Rev. Lett 45, 566 (1980).
[20] J. P. Perdew and A. Zunger, Phys. Rev. B 23, 5048 (1981).
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[22] S. J. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58, 1200 (1980).
[23] A note on the nomenclature behind DFT functionals may be appropriate. Functionals are
most often named after their authors’ in such a way that the name “PBE” comes from a
paper by Perdew, Burke, and Ernzerhof. Occasionally the year the paper was published
is appended. And if the functional is one of the so-called hydrid functionals, discussed
in section 3.2(c) it is common to add the number of parameters used when mixing the
Hartree-Fock exchange with the DFT exchange and correlation. For example, PBE0, is a
hybrid functional with zero mixing parameters.
[24] O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976).
[25] O. Gunnarsson, M. Jonson, and B. I. Lundqvist, Solid State Commun. 24, 765 (1977).
[26] T. Ziegler, A. Rauk, and E. J. Baerends, Theor Chim. Acta 43, 261 (1977).
[27] K. Burke, J. P. Perdew, and M. Ernzerhof, J. Chem. Phys. 109, 3760 (1998).
[28] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).
[29] PBE and its off-spring differ only in how they treat electron exchange. The exchange
component of GGA-PBE is
∫
E P BE
x [n] =
nɛ xc−unif (n(r)) Fx
P BE (n, ∇n(r)) d 3 r
where F P BE
x
is an exchange enhancement factor given by
Fx
P BE
κ
[n, ∇n(r)] = 1 + κ −
1 + µs 2 /κ
with the dimensionless reduced gradient, s = |∇n|/2n(3π 2 n) 1/3 . In the original PBE κ =
0.804. In revPBE κ = 1.245. In RPBE F x [n, ∇n(r)] = 1 + κ − κexp(−µs 2 /κ), and in
PBE-WC µs 2 takes a more complex form.
[30] Y. Zhang and W. Yang, Phys. Rev. Lett. 80, 890 (1998).
[31] B. Hammer, L. B. Hansen, and J. K. Nørskov, Phys. Rev. B 59, 7413 (1999).
[32] Z. Wu and R. E. Cohen, Phys. Rev. B 73, 235116 (2006).
[33] J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Scuseria, Phys. Rev. Lett. 91, 146401
(2003).
[34] The philosophy behind the hybrid functionals is simple and rooted in the adiabatic connection
formula, which is a rigorous ab initio formula for the exchange-correlation energy