initio” methods ofChapter 5(Section 5.5.2.2b). LSDA functionals are useful in
solid-state physics, but for molecular calculations have been largely replaced by
higher rungs of the ladder. The local spin density method has however been stoutly
defended by knowledgeable practitioners [ 47 ], who point out that it gives “remark-
ably accurate bond lengths”, that its atomization energy errors “can be dramatically
reduced” with one empirical parameter, and that “For chemistry without free atoms,
LSD is not such a bad starting point”. A recently developed, potentially very useful
local function is M06-L (below) [ 45 ]. Nevertheless, LSDA calculations have been
largely replaced by an approach that uses not just the electron density, but also its
gradient.
7.2.3.4c Gradient-Corrected Functionals: The Generalized Gradient
Approximation (GGA)
Most DFT calculations nowadays use exchange-correlation energy functionals
EXCthat utilize both the electron densityandits gradient, the first derivative ofr
with respect to position, (∂/∂xþ∂/∂yþ∂/∂z)r¼rr.Thesefunctionalsare
calledgradient-corrected,orsaidtousethegeneralized-gradient approximation
(GGA). They have also been callednonlocalfunctionals,incontrasttoLDAand
LSDA functionals, but it has been suggested [ 52 ]thatthetermnonlocalbe
avoided in referring to gradient-corrected functionals; recall the discussion of
“local” in Section7.2.3.4a.Theexchange-correlationenergyfunctionalcanbe
written as the sum of an exchange-energy functional and a correlation-energy
functional, both negative, i.e.EXC¼ExþEc;|Ex|ismuchbiggerthan|Ec|. For the
argon atomExis"30.19 hartrees, whileEcis only"0.72 hartrees, calculated by
the HF method [ 53 ]. Thus it is not surprising that gradient corrections have proved
more effective when applied to the exchange-energy functional, and a major
advance in practical DFT calculationswas the introduction of the B88 (Becke
1988) functional [ 54 ], a “new and greatly improved functional for the exchange
energy” [ 55 ]. Examples of gradient-corrected correlation-energy functionals are
the LYP (Lee-Yang-Parr) and the P86 (Perdew 1986) functionals. All these
functionals are commonly used with Gaussian-type (i.e. functions with exp
("r^2 )) basis functions for representing the KS orbitals (Eq.7.26). A calculation
with B88 for the exchange functionalEx,andLYPforthecorrelationfunctional
Ec,andthe6-31Gbasisset(Section5.3.3) would be designated as a B88-LYP/6-
31G or B88LYP/6-31G* calculation. Sometimes rather than the analytical func-
tions that constitute the standard Gaussian basis sets, numerical basis sets are
used. A numerical basis function is essentially a table of the values that an atomic
orbital wavefunction has at many pointsaround the nucleus, derived from best-fit
functions devised to pass through these points. These numerical functions can be
used instead of the analytical Gaussian-type functions ubiquitous in ab initio
calculations.
7.2 The Basic Principles of Density Functional Theory 463