Chemistry Reference
In-Depth Information
sometimes produce a single excitation in about the right region, in place
of two lines, where a double excitation has mixed strongly with a single
excitation.
When a double excitation lies close to a single excitation, elementary
quantum mechanics shows that
f
XC
must have a strong frequency depen-
dence.
277
Post-adiabatic TDDFT methodologies have been developed
277,282,283
recently for including a double excitation when it is close to an optically active
single excitation, and such post-adiabatic methodologies appear to work well
for small dienes.
277,284
By going beyond linear response, one might hope that
nontrivial double excitations are naturally included in, e.g., ALDA. It has
been proven that, in the higher order response in ALDA, the double excitations
occur simply as the sum of single excitations.
285
Thus, the scientific community
does not currently know how best to approximate these excitations. The pro-
blem with higher order excitations is particularly severe for quantum wells,
where the external potential is parabolic, leading to multiple near degeneracies
between levels of excitation.
282
For example, based on a HF reference, the 2
1
A
g
state of naphthalene has,
according to the RICC2 results, a considerable admixture of double excita-
tions. This admixture is consistent with the fact that the CIS method yields
an excitation energy that is too high by 1.5 eV compared to experiment.
The TDDFT results are much closer to experiment, yet still too high by several
tenths of an electron volt.
Polymers
An early triumph of the VK functional was its success in predicting the
static polarizabilities of long-chain conjugated polymers. These polarizabilites
are greatly underestimated by both LDA and GGA functionals, with the error
growing rapidly with the number of monomer units in the polymer.
289
In con-
trast, HF does rather well and does not overpolarize. The VK correction to
LDA yields excellent results in many (but not all) cases, showing that a cur-
rent-dependent functional can sometimes correct the overpolarization pro-
blem. Naturally, orbital-dependent functionals also account for this
effect,
290
but they do at a much higher computational cost. A comparison
of the methods is given in Figure 14.
Solids
When using TDDFT to calculate the optical response of insulators, local
approximations have again been shown to fail badly. Most noticeably, local
approximations do not describe excitonic effects,
291
or the exciton spectrum
within the band gap. Moreover, the computed gap is usually much smaller
than experiment because adiabatic approximations cannot change the gap
size from its KS value.
