(BP) of the CI [6]. At a CI, the non-adiabatic coupling between the nuclear and
electronic degrees of freedom becomes divergent [6, 9]; hence, the manifold of CI
points, the CI seam, plays a decisive role for the dynamics of the electronic states by
providing funnels for the efficient population transfer [9–12, 20]. As the BP defines
all possible directions of exiting the strong non-adiabatic coupling region [6], it
becomes a very important descriptor of the CI which, very similar to the transition
vector in transition state theory [21, 22], determines the branching of possible
products of the reaction occurring through the CI [23].
The correct theoretical description of CIs requires the use of computational
methods capable of describing various electron correlation effects in the ground
and the excited states with high and unbiased accuracy [10, 12, 24]. Naturally, since
the first realization of the importance of CIs for the dynamics of excited states [9,
10], the computational investigation of CIs has been the realm of multi-reference
(MR) methods of wavefunction theory (WFT), such as the complete active space
self-consistent field (CASSCF) [25–28], the CAS-based second-order perturbation
theory (CASPT2) [29], and the MR configuration interaction (MRCI) [30] methods.
Although capable of delivering highly accurate results, these methods are computationally demanding and can be used routinely to calculate small molecular
systems limited to just a handful of atoms. To go beyond this limitation and to
match the challenges presented by the rapidly expanding use of ultrafast spectroscopy in photochemistry and photobiology [31, 32] one needs to employ computational methods capable of realistically describing large molecules. As the methods
of density functional theory (DFT) [33, 34] offer a reasonably accurate description
of electron correlation effects (which are vital for molecular bonding and structure)
at a typical mean-field computational cost, these methods seem to represent a
viable alternative to MR-WFT approaches.
However, there is a caveat. Originally [33], DFT was formulated for ground
states only, and it was deemed inappropriate to apply it to variational calculation of
individual excited states [35–37]. A way around this problem is offered by the
response formalism implemented in the currently widely used linear-response
(LR) time-dependent DFT (TD-DFT) approach [38, 39]. LR-TD-DFT (or TD-DFT
for brevity) enables one to obtain the excitation energies from the poles of the
ground-state density-density response function [38, 39], thus enabling the use of
ground-state theory for obtaining excited states. Although capable of describing
crossings between excited states correctly, TD-DFT may experience difficulties
with the S 0 /S 1 Cls as the ground electronic state (S 0 ) is the variationally optimized
reference state for the response calculation and is thus described on a different
footing than the excited state (S 1 ), which is a response state. In particular, there is no
coupling between the ground and excited states in most approximate LR-TD-DFT
methods, which results in a wrong dimensionality of the S 0 /S 1 crossing seam, MÀ1
instead of MÀ2 [40–43].
An interesting modification of the original LR-TD-DFT formalism is
implemented in the spin-flip (SF) TD-DFT method which employs a variationally
optimized high-spin open-shell reference state (e.g., triplet state) to access lower
spin states (e.g., singlets) by allowing one-electron transitions with simultaneous
Description of Conical Intersections with Density Functional Methods
447
electronic degrees of freedom becomes divergent [6, 9]; hence, the manifold of CI
points, the CI seam, plays a decisive role for the dynamics of the electronic states by
providing funnels for the efficient population transfer [9–12, 20]. As the BP defines
all possible directions of exiting the strong non-adiabatic coupling region [6], it
becomes a very important descriptor of the CI which, very similar to the transition
vector in transition state theory [21, 22], determines the branching of possible
products of the reaction occurring through the CI [23].
The correct theoretical description of CIs requires the use of computational
methods capable of describing various electron correlation effects in the ground
and the excited states with high and unbiased accuracy [10, 12, 24]. Naturally, since
the first realization of the importance of CIs for the dynamics of excited states [9,
10], the computational investigation of CIs has been the realm of multi-reference
(MR) methods of wavefunction theory (WFT), such as the complete active space
self-consistent field (CASSCF) [25–28], the CAS-based second-order perturbation
theory (CASPT2) [29], and the MR configuration interaction (MRCI) [30] methods.
Although capable of delivering highly accurate results, these methods are computationally demanding and can be used routinely to calculate small molecular
systems limited to just a handful of atoms. To go beyond this limitation and to
match the challenges presented by the rapidly expanding use of ultrafast spectroscopy in photochemistry and photobiology [31, 32] one needs to employ computational methods capable of realistically describing large molecules. As the methods
of density functional theory (DFT) [33, 34] offer a reasonably accurate description
of electron correlation effects (which are vital for molecular bonding and structure)
at a typical mean-field computational cost, these methods seem to represent a
viable alternative to MR-WFT approaches.
However, there is a caveat. Originally [33], DFT was formulated for ground
states only, and it was deemed inappropriate to apply it to variational calculation of
individual excited states [35–37]. A way around this problem is offered by the
response formalism implemented in the currently widely used linear-response
(LR) time-dependent DFT (TD-DFT) approach [38, 39]. LR-TD-DFT (or TD-DFT
for brevity) enables one to obtain the excitation energies from the poles of the
ground-state density-density response function [38, 39], thus enabling the use of
ground-state theory for obtaining excited states. Although capable of describing
crossings between excited states correctly, TD-DFT may experience difficulties
with the S 0 /S 1 Cls as the ground electronic state (S 0 ) is the variationally optimized
reference state for the response calculation and is thus described on a different
footing than the excited state (S 1 ), which is a response state. In particular, there is no
coupling between the ground and excited states in most approximate LR-TD-DFT
methods, which results in a wrong dimensionality of the S 0 /S 1 crossing seam, MÀ1
instead of MÀ2 [40–43].
An interesting modification of the original LR-TD-DFT formalism is
implemented in the spin-flip (SF) TD-DFT method which employs a variationally
optimized high-spin open-shell reference state (e.g., triplet state) to access lower
spin states (e.g., singlets) by allowing one-electron transitions with simultaneous
Description of Conical Intersections with Density Functional Methods
447