Chemistry Reference
In-Depth Information
BEYOND STANDARD FUNCTIONALS
We surveyed and illustrated in the previous sections some of the many
present successful applications of TDDFT. In those applications, standard
approximations (local, gradient-corrected, and hybrid) were used for both the
ground-state calculation and the excitations, via the adiabatic approximation.
In this section, we survey several important areas in which the standard functio-
nals have been found to
fail
, and we explain what might be done about it.
The errors in standard TDDFT are due to locality in both space and time,
both of which are intimately related. In fact, all memory effects, i.e., depen-
dence on the history of the density,
185
implying a frequency dependence in
the XC kernel, can be subsumed into an initial-state dependence,
181
but proba-
bly not vice versa. Several groups are attempting to incorporate such effects
into new approximate functionals,
262-273
but none have yet shown universal
applicability.
The failure of the adiabatic approximation is most noticeable when high-
er order excitations are considered; these excitations are found to be missing in
the usual linear response treatment.
191
The failure of the local approximation
in space is evident, for example, when TDDFT is applied to extended systems,
e.g., polymers or solids. Local approximations yield short-ranged XC kernels
that become irrelevant compared to Hartree contributions in the long-wave-
length limit. The Coulomb repulsion between electrons generally requires
long-ranged
ð
1
=
r
Þ
exchange effects when long-wavelength response is being
calculated.
Several approaches to correcting these problems have been developed
and applied in places where the standard formulation has failed. These
approaches fall into two distinct categories. First, where approximations
that are local in the
density
fail, one can try approximations that are local
(or semilocal) in the
current density.
In fact, for TDDFT, the gradient expan-
sion, producing the leading corrections to ALDA,
only
works if the current is
the basic variable.
274
Using the gradient expansion itself is called the Vignale-
Kohn (VK) approximation,
275,276
and it has been tried on a variety of pro-
blems. The second category involves constructing orbital-dependent approxi-
mations with explicit frequency dependence.
277,278
This can work well for
specific cases, but it is difficult to construct general density functional approx-
imations from these examples. More importantly, solution of the OEP
equations is typically far more CPU expensive than for the simple KS
equations, making OEP impractical for large molecules.
Double Excitations
Casida
191
first pointed out that double excitations appear to be miss-
ing
279
from TDDFT linear response within any adiabatic approximation.
Experience
280,281
shows that, as in naphthalene, adiabatic TDDFT will
