Chemistry Reference
In-Depth Information
2. The use of hybrid functionals leads to systematically higher excitation
energies. On average, this is an improvement over the GGA results, which
are systematically too low. However, while diffuse excitations benefit from
mixing in some exact exchange due to a reduction of the self-interaction
error, valence excitation energies are not always improved, as is obvious for
the 1
1
B
3
u
and 2
1
B
3
u
valence states.
3. The 1
1
B
2
u
state is erroneously predicted below the 1
1
B
3
u
state by all density
functionals in Table 4, which is a potentially serious problem for
applications in photochemistry. This is not corrected by hybrid mixing.
4. The configuration-interaction singles (CIS) method, which uses a Hartree-
Fock reference that is computationally as expensive as hybrid TDDFT,
produces errors that are substantially larger than the hybrid TDDFT
results, especially for valence states. The coupled-cluster (CC) and CASPT2
methods are usually more accurate but are far more CPU expensive, and
they scale prohibitively as the system size grows. The cost of CASPT2 scales
exponentially with the number of correlated electrons, while the cost of
CC2 grows with the fifth power of the system size. This severely limits the
application of these methods to larger molecules.
The 1
1
B
2
u
excitation is polarized along the short axis of the naphthalene
molecule. In Platt's nomenclature of excited states of polycyclic aromatic
hydrocarbons (PAHs), 1
1
B
2
u
corresponds to the
1
L
a
state, which has more
ionic character than the 1
1
B
3
u
(or
1
L
b
) state. Parac and Grimme have pointed
out
216
that GGA functionals underestimate the excitation energy of the
1
L
a
state in PAHs considerably. Our example of naphthalene agrees with this
observation. The 1
1
B
2
u
excitation is computed to be 0.4-0.5 eV too low in
energy by LSDA and GGA functionals, leading to an incorrect ordering of
the first two singlet excited states.
Influence of the Ground-State Potential
From the very earliest calculations of transition frequencies,
189,191
it was
recognized that the inaccuracy of standard density functional approximations
(LDA, GGA, hybrids) for the ground-state XC potential leads to inaccurate KS
eigenvalues. Because the approximate KS potentials have incorrect asymptotic
behavior (they decay exponentially, rather than as
r
, as seen in Figure 3
and its discussion), the KS orbital eigenvalues are insufficiently negative, the
ionization threshold is far too low, and the Rydberg states are often unbound.
Many methods have since been developed to asymptotically correct
potentials
217,218
to correct this unfortunate behavior. Any corrections to the
ground-state potential are dissatisfying, however, as the resulting potential is
not
a functional derivative of an energy functional. Even mixing one approx-
imation for
v
XC
ð
r
Þ
1
=
and another for
f
XC
has become popular in an attempt to
rectify the problem. A more satisfying route to asymptotically correct
