Computational Chemistry

7.2.3.4d Meta-Generalized Gradient Approximation Functionals
(meta-GGA, MGGA)

We saw that functionals which use the first derivative of the electron density function,
GGA functionals (Section7.2.3.4c), are usually an improvement over ones relying
only onritself. One might therefore suspect that further improvement could be
obtained by invoking the second derivative ofr(∂^2 /∂x^2 +∂^2 /∂y^2 +∂^2 /∂z^2 )r¼
r^2 r. This is the Laplacian of the electron density function (so important in AIM
theory, Section 5.5.4). Functionals which use the second derivative ofrare called
meta-gradient corrected (meta-GGA, MGGA); meta¼beyond. This approach seems
to offer some improvement, but functionals that depend on the Laplacian ofrpresent
computational problems. One way to sidestep this is to make the MGGA functional
dependent not onritself but on thekinetic energy densityt, obtained by summing the
squares of the gradients of the Kohn–Sham MOs:

tðrÞ¼

1

2

occupiedX

i¼ 1

jrcKSi ðrÞj^2 (7.28)

This varies withressentially the same as does the Laplacian ofr[ 56 ]. Examples
of MGGA functionals are thetHCTH (Hamprecht, Cohen, Tozer, Handy) and
the B98 (Becke1998). MGGA functionals are, in the sense of GGA ones, local.
A detailed discussion of the theory and mathematics behind MGGA functionals is
given in reference [ 48 ], where they are said to “in general perform well for atomiza-
tion energies”, and PKZP and KCIS are designated the best MGGA performers.

7.2.3.4e Hybrid GGA (HGGA) Functionals: The Adiabatic Correction
Method (ACM)

These are functionals to which Hartree–Fock exchange has been added. The
justification for this lies in the adiabatic connection method (ACM) [ 17 ]. In
wavefunction theory, an adiabatic process is one in which the wavefunction
remains on the same PES, i.e. the variables that define it change smoothly as the
process evolves. The process seamlessly connects two states without crossing into
another electronic state. The ACM shows that the exchange-correlation energy
EXC(r) can be taken as a weighted sum of the DFT exchange-correlation energy
and HF exchange energy. This is the justification ofhybridDFT functionals (hybrid
DFT methods have been called ACM methods), which include an energy contribu-
tion from HF-type electron exchange, calculated from the KS wavefunction of the
noninteracting electrons. Those electrons have no coulomb interaction, but being,
after all, still electrons with a spin of one-half, like all good fermions they show
“Pauli repulsion” (Section 5.2.3.5), represented by the exchange K integral
(Eq.5.22). Hybrid functionals are functionals (of the GGA level or higher) that
contain HF exchange, the correction energy to the classical coulomb repulsion.

464 7 Density Functional Calculations