Essentials of Computational Chemistry

8.4 EXCHANGE-CORRELATION FUNCTIONALS 259
in ρ has already been noted). In practice, the only functionals conforming to this definition
that have seen much application are those that derive from analysis of the uniform electron
gas (where the density has the same value at every position), and as a result LDA is often
used more narrowly to imply that it is these exchange and correlation functionals that are
being employed.
The distinction is probably best indicated by example. Following from Eq. (8.7) and
the discussion in Section 8.1.2, the exchange energy for the uniform electron gas can be
computed exactly, and is given by Eq. (8.23) with the constant α equal to 2
. However, the
3
Slater approach takes a value for α of 1, and the Xα model most typically uses 3
4 .Allof
these models have the same ‘local’ dependence on the density, but only the first is typically
referred to as LDA, while the other two are referred to by name as Slater (S) and Xα .
The LDA, Slater, and Xα methods can all be extended to the spin-polarized regime using
εx[ρ(r), ζ ] = ε 0 x [ρ(r)] + ε 1 x [ρ(r)] − ε0 x [ρ(r)] (1 + ζ) 4/3 + (1 − ζ) 4/3 − 2
2(21/3
− 1)
(8.26)
where the superscript-zero exchange energy density is given by Eq. (8.23) with the appropriate
value of α (referring here to the empirical constant, not the electron spin), and the
superscript-one energy is the analogous expression derived from consideration of a uniform
electron gas composed only of electrons of like spin. Noting that ζ = 0 everywhere for an
unpolarized system, it is immediately apparent that the second term in Eq. (8.26) is zero
for that special case. Systems including spin polarization (e.g., open-shell systems) must use
the spin-polarized formalism, and its greater generality is sometimes distinguished by the
sobriquet ‘local spin density approximation’ (LSDA).
As for the correlation energy density, even for the ‘simple’ uniform electron gas no
analytical derivation of this functional has proven possible (although some analytical details
about the zero- and infinite-density limits can be established). However, using quantum
Monte Carlo techniques, Ceperley and Alder (1980) computed the total energy for uniform
electron gases of several different densities to very high numerical accuracy. By subtracting
the analytical exchange energy for each case, they were able to determine the correlation
energy in these systems. Vosko, Wilk, and Nusair (1980) later designed local functionals
of the density fitting to these results (and the analytical low and high density limits). In
particular, they proposed one spin-polarized functional completely analogous to Eq. (8.26)
in terms of its dependence on ζ , but with the unpolarized and fully polarized energy densities
expressed (now in terms of rS instead of ρ, see Eq. (8.24)) as
ε i c (rS) = A
ln
2
rS
rS + b √ rS + c +
bx0
−
x2 0 + bx0
ln
+ c
2b
√ 4c − b 2 tan−1
( √ rS − x0) 2
rS + b √
+
rS + c
2(b + 2x0)
√
4c − b2 2 √ rS + b
√ 4c − b 2 tan−1
√
4c − b2 2 √ rS + b
(8.27)