inversion of the spin (spin-flip transitions) [44–47]. In SF-TD-DFT, the ground
singlet state and the excited singlet states are obtained on the same footing as
response states. This enables some interaction between these states and reinstates
the correct dimensionality of the S 0 /S 1 conical intersection [40, 43, 48]. However,
the downside of SF-TD-DFT is a substantial spin-contamination of the so-obtained
singlet state, which may lead to the occurrence of mixed spin-symmetry states, thus
complicating the identification of physically meaningful excited states [48].
An alternative to the response formalism, for obtaining excited states in the
context of DFT, is offered by the ensemble DFT approaches [39, 49–52]. Ensemble
DFT, formulated in the seminal works of Lieb [53] (ground-state ensembles) and of
Gross, Oliveira, and Kohn [49] (ensembles of ground and excited states) enables
one to obtain excitation energies from the variational calculation, as contrasted with
the response states of LR-TD-DFT and SF-TD-DFT. A practical implementation of
ensemble DFT in the form of a generally applicable computational scheme was
achieved in the spin-restricted ensemble-referenced Kohn–Sham (REKS) method
[54–56] and its state-averaged (SA) extensions, SA-REKS [57] and SI-SA-REKS
[42, 58] (SI: state-interaction). In particular, the SI-SA-REKS method has proved
its ability to describe properly the S 0 /S 1 CIs by yielding the correct dimensionality
of the CI, MÀ2 [43, 58, 59]. Being a spin-restricted method which employs the
same spatial orbitals for electrons with opposite spins, REKS is free from the spincontamination which infests the SF-TD-DFT description, and yields a spin and
space (if present) symmetry adapted description of ground and excited states.
The aforementioned computational approaches (LR-TD-DFT, SF-TD-DFT, and
REKS) cover all the practically accessible implementations of DFT for describing
the ground and excited states of molecules. In this chapter, the ability of these
computational methods to describe CIs, and particularly S 0 /S 1 CIs, of molecules is
assessed in relation to probably the best description that the MR-WFT world can
offer, namely MRCISD (MRCI with single and double excitations). When using
approximate density functionals, LR-TD-DFT often yields good results for the
energies of one-electron valence transitions and it seems tempting to believe that
TD-DFT should be capable of yielding PES crossings as well. Although its inability
to do so was clearly spelled out in the past [40, 41], the use of TD-DFT for modeling
photodynamics and the underlying mechanistic features, such as CIs, seems to be
gaining ground [60–73]. The consequences of using TD-DFT for photodynamics
are reviewed in another chapter of this book;
1 here, we focus on the ability (or, more
precisely, the inability) of this formalism to provide a proper description of CI
topography while yielding seemingly reasonable geometries and relative energies
of the minimum-energy crossing points.
As an alternative to TD-DFT, we investigate the performance of SF-TD-DFT
and REKS methodologies which are known to yield the proper dimensionality of
the CI seam [43]. Besides the molecular geometry at the minimum-energy CI
1 See the chapter “Surface Hopping Dynamics with DFT Excited States” by M. Barbatti and
R. Crespo-Otero.
448
M. Huix-Rotllant et al.
singlet state and the excited singlet states are obtained on the same footing as
response states. This enables some interaction between these states and reinstates
the correct dimensionality of the S 0 /S 1 conical intersection [40, 43, 48]. However,
the downside of SF-TD-DFT is a substantial spin-contamination of the so-obtained
singlet state, which may lead to the occurrence of mixed spin-symmetry states, thus
complicating the identification of physically meaningful excited states [48].
An alternative to the response formalism, for obtaining excited states in the
context of DFT, is offered by the ensemble DFT approaches [39, 49–52]. Ensemble
DFT, formulated in the seminal works of Lieb [53] (ground-state ensembles) and of
Gross, Oliveira, and Kohn [49] (ensembles of ground and excited states) enables
one to obtain excitation energies from the variational calculation, as contrasted with
the response states of LR-TD-DFT and SF-TD-DFT. A practical implementation of
ensemble DFT in the form of a generally applicable computational scheme was
achieved in the spin-restricted ensemble-referenced Kohn–Sham (REKS) method
[54–56] and its state-averaged (SA) extensions, SA-REKS [57] and SI-SA-REKS
[42, 58] (SI: state-interaction). In particular, the SI-SA-REKS method has proved
its ability to describe properly the S 0 /S 1 CIs by yielding the correct dimensionality
of the CI, MÀ2 [43, 58, 59]. Being a spin-restricted method which employs the
same spatial orbitals for electrons with opposite spins, REKS is free from the spincontamination which infests the SF-TD-DFT description, and yields a spin and
space (if present) symmetry adapted description of ground and excited states.
The aforementioned computational approaches (LR-TD-DFT, SF-TD-DFT, and
REKS) cover all the practically accessible implementations of DFT for describing
the ground and excited states of molecules. In this chapter, the ability of these
computational methods to describe CIs, and particularly S 0 /S 1 CIs, of molecules is
assessed in relation to probably the best description that the MR-WFT world can
offer, namely MRCISD (MRCI with single and double excitations). When using
approximate density functionals, LR-TD-DFT often yields good results for the
energies of one-electron valence transitions and it seems tempting to believe that
TD-DFT should be capable of yielding PES crossings as well. Although its inability
to do so was clearly spelled out in the past [40, 41], the use of TD-DFT for modeling
photodynamics and the underlying mechanistic features, such as CIs, seems to be
gaining ground [60–73]. The consequences of using TD-DFT for photodynamics
are reviewed in another chapter of this book;
1 here, we focus on the ability (or, more
precisely, the inability) of this formalism to provide a proper description of CI
topography while yielding seemingly reasonable geometries and relative energies
of the minimum-energy crossing points.
As an alternative to TD-DFT, we investigate the performance of SF-TD-DFT
and REKS methodologies which are known to yield the proper dimensionality of
the CI seam [43]. Besides the molecular geometry at the minimum-energy CI
1 See the chapter “Surface Hopping Dynamics with DFT Excited States” by M. Barbatti and
R. Crespo-Otero.
448
M. Huix-Rotllant et al.