and so the ground state features a cusp at the perpendicular conformation. The S 1 ,
which is essentially represented by a single excitation, is virtually superimposed
with the CASSCF(2,2)/MCQDPT2 result. The dressed TD-DFT (Fig. 17b) includes
the double excitation, but the surfaces of S 0 and S 2 appear as diabatic states because
the ground- to excited-state coupling term is missing. This is largely fixed by
introducing the Brillouin corrections (Fig. 17c). The ground state is now in very
good agreement with the CASSCF(2,2)/MCQDPT2 S 0 state, although the degeneracy of S 1 and S 2 at 90
is still not fully captured. Thus the picture given by
Brillouin-corrected LR-TD-DFT is qualitatively correct with respect to the multireference results.
5 Effective Exchange-Correlation (xc) Kernel
We now have the tools to deduce an MBPT expression for the TD-DFT xc-kernel. It
should be emphasized that this is not a new exercise but that we seem to be the only
ones to do so within the PP formalism. We think this may have the advantage of
making a rather complicated subject more accessible to Quantum Chemists already
familiar with the PP formalism.
The problem of constructing xc-correlation objects such as the xc-potential v xc
and the xc-kernel f xc (ω) from MBPT for use in DFT has been termed “ab initio
DFT” by Bartlet [70, 71]. At the exchange-only level, the terms optimized effective
potential (OEP) [72, 73] or exact exchange [74, 75] are also used and OEP is also
used to include the correlated case [76, 77]. At first glance, nothing much is gained.
For example, the calculated excitation energies and oscillator strengths in ab initio
TD-DFT must be, by construction, exactly the same as those from MBPT. This
approach does not give explicit functionals of the density (though it may be thought
of as giving implicit functionals). However it does allow us to formulate expressions for and to calculate purely (TD-) DFT objects and hence it can provide insight
into, and computational checks of, the behavior of illusive objects such as v xc and
f xc (ω).
Here we concentrate on the latter, namely the xc-kernel. Previous work along
these lines has been carried out for the kernel by directly taking the derivative of
the OEP energy expression with the constraint that the orbitals come from a
local potential. This was first done by G€ orling in 1998 [60] for the full timedependent exchange-only problem. In 2002, Hirata et al. redid the derivation for
the static case [78]. Later, in 2006, a diagrammatic derivation of the static result was
given by Bokhan and Bartlett [71], and the functional derivative of the kernel g x
has been treated by Bokhan and Bartlett in the static exchange-only case [79].
In this section, we take a somewhat different and arguably more direct approach
than that used in the previously mentioned articles, in that we make direct use of the
fundamental relation
MBPT Insights About and Corrections to TD-DFT
39
which is essentially represented by a single excitation, is virtually superimposed
with the CASSCF(2,2)/MCQDPT2 result. The dressed TD-DFT (Fig. 17b) includes
the double excitation, but the surfaces of S 0 and S 2 appear as diabatic states because
the ground- to excited-state coupling term is missing. This is largely fixed by
introducing the Brillouin corrections (Fig. 17c). The ground state is now in very
good agreement with the CASSCF(2,2)/MCQDPT2 S 0 state, although the degeneracy of S 1 and S 2 at 90
is still not fully captured. Thus the picture given by
Brillouin-corrected LR-TD-DFT is qualitatively correct with respect to the multireference results.
5 Effective Exchange-Correlation (xc) Kernel
We now have the tools to deduce an MBPT expression for the TD-DFT xc-kernel. It
should be emphasized that this is not a new exercise but that we seem to be the only
ones to do so within the PP formalism. We think this may have the advantage of
making a rather complicated subject more accessible to Quantum Chemists already
familiar with the PP formalism.
The problem of constructing xc-correlation objects such as the xc-potential v xc
and the xc-kernel f xc (ω) from MBPT for use in DFT has been termed “ab initio
DFT” by Bartlet [70, 71]. At the exchange-only level, the terms optimized effective
potential (OEP) [72, 73] or exact exchange [74, 75] are also used and OEP is also
used to include the correlated case [76, 77]. At first glance, nothing much is gained.
For example, the calculated excitation energies and oscillator strengths in ab initio
TD-DFT must be, by construction, exactly the same as those from MBPT. This
approach does not give explicit functionals of the density (though it may be thought
of as giving implicit functionals). However it does allow us to formulate expressions for and to calculate purely (TD-) DFT objects and hence it can provide insight
into, and computational checks of, the behavior of illusive objects such as v xc and
f xc (ω).
Here we concentrate on the latter, namely the xc-kernel. Previous work along
these lines has been carried out for the kernel by directly taking the derivative of
the OEP energy expression with the constraint that the orbitals come from a
local potential. This was first done by G€ orling in 1998 [60] for the full timedependent exchange-only problem. In 2002, Hirata et al. redid the derivation for
the static case [78]. Later, in 2006, a diagrammatic derivation of the static result was
given by Bokhan and Bartlett [71], and the functional derivative of the kernel g x
has been treated by Bokhan and Bartlett in the static exchange-only case [79].
In this section, we take a somewhat different and arguably more direct approach
than that used in the previously mentioned articles, in that we make direct use of the
fundamental relation
MBPT Insights About and Corrections to TD-DFT
39