conical intersection, but the topography along the θ ¼ 90
line shows diverse
spurious minima.
Spurious minima were also observed in TDDFT and TDA with B3LYP [121,
122]. The minimum is at a qualitatively wrong position at TDDFT-B3LYP with
restricted open shell (TDDFT-RO). The same happens for TDDFT-BLYP. TDA
can qualitatively predict the conical intersection (θ ¼ 90
, φ > 70
). Somewhat
surprisingly, the simple HOMO–LUMO gap is the only other method predicting a
twisted-pyramidalized conical intersection (at θ ¼ 90
, φ ¼ 70
). All other methods
wrongly predict an extended S 1 /S 0 seam along the θ ¼ 90
line, starting from
φ ¼ 0
.
A disturbing feature which can be observed in TDDFT-B3LYP, TDDFT-UBSB3LYP, TDA-B3LYP, and TDDFT-LC-BLYP is that the excitation may become
negative near the crossing seam. This is not an exclusive problem with DFT-based
methods; it can be observed in other single-reference methods as well, such as
coupled cluster with either equation-of-motion or linear-response approaches.
As discussed in [123], near the degeneracy, self-interaction errors may cause the
HOMO to be less stable than the LUMO, leading to imaginary excitation energies
in TDDFT – see (27). This does not happen at TDA because the excitation energy is
given by terms such as those in (28). This feature has been said to represent an
improved stability of the TDA-based dynamics in comparison to that based on
TDDFT [123]. Nevertheless, as we can see in the negative gap regions in Fig. 3,
both methods are still unstable near the degeneracy. These negative excitations are
clearly caused by the mixing with the other singly-excited determinants, which
cannot be avoided either in TDDFT or in TDA.
In SDKS methods, the excitation energies are given by the bare KS gaps [61]. In
the case of the ethylene, we can see in Fig. 3 that the bare KS energy (given by the
HOMO–LUMO gap and neglecting double excitations near the twisted structure)
gives an adequate representation of the S 1 state and of the S 1 /S 0 gap. This good
behavior, however, should not be taken for granted. Maitra [60], in a critical
discussion of these approximation in the context of surface hopping, showed that
the bare KS energies may be far from adiabatic and closer to diabatic energies.
Ethylene is admittedly too harsh a case for DFT-based methods, as its dynamics
is deeply controlled by coordinates involved in the nonadiabatic events. SH/DFT
methods have been developed to deal with large molecular systems and in these
cases the dynamics may evolve in the configurational space spanning regions
relatively far from any muticonfigurational ground state. Under such situation, the
excited-state dynamics involving a large manifold of excited states can be well
simulated with DFT-based methods.
Take, for instance, the schematic dynamics illustrated in Fig. 4 (top). The
dynamics starts at a high excited state (here, the second state to simplify the
picture). Using a multireference method such as MRCI, we would observe a
relaxation to the first excited state (a), then oscillation around the minimum of
this state (b). From this minimum, the molecule can eventually fluoresce to the
ground state (c) or cross a barrier (d) to reach a conical intersection, from where it
relaxes to the ground state minimum (e).
Surface Hopping Dynamics with DFT Excited States
431
line shows diverse
spurious minima.
Spurious minima were also observed in TDDFT and TDA with B3LYP [121,
122]. The minimum is at a qualitatively wrong position at TDDFT-B3LYP with
restricted open shell (TDDFT-RO). The same happens for TDDFT-BLYP. TDA
can qualitatively predict the conical intersection (θ ¼ 90
, φ > 70
). Somewhat
surprisingly, the simple HOMO–LUMO gap is the only other method predicting a
twisted-pyramidalized conical intersection (at θ ¼ 90
, φ ¼ 70
). All other methods
wrongly predict an extended S 1 /S 0 seam along the θ ¼ 90
line, starting from
φ ¼ 0
.
A disturbing feature which can be observed in TDDFT-B3LYP, TDDFT-UBSB3LYP, TDA-B3LYP, and TDDFT-LC-BLYP is that the excitation may become
negative near the crossing seam. This is not an exclusive problem with DFT-based
methods; it can be observed in other single-reference methods as well, such as
coupled cluster with either equation-of-motion or linear-response approaches.
As discussed in [123], near the degeneracy, self-interaction errors may cause the
HOMO to be less stable than the LUMO, leading to imaginary excitation energies
in TDDFT – see (27). This does not happen at TDA because the excitation energy is
given by terms such as those in (28). This feature has been said to represent an
improved stability of the TDA-based dynamics in comparison to that based on
TDDFT [123]. Nevertheless, as we can see in the negative gap regions in Fig. 3,
both methods are still unstable near the degeneracy. These negative excitations are
clearly caused by the mixing with the other singly-excited determinants, which
cannot be avoided either in TDDFT or in TDA.
In SDKS methods, the excitation energies are given by the bare KS gaps [61]. In
the case of the ethylene, we can see in Fig. 3 that the bare KS energy (given by the
HOMO–LUMO gap and neglecting double excitations near the twisted structure)
gives an adequate representation of the S 1 state and of the S 1 /S 0 gap. This good
behavior, however, should not be taken for granted. Maitra [60], in a critical
discussion of these approximation in the context of surface hopping, showed that
the bare KS energies may be far from adiabatic and closer to diabatic energies.
Ethylene is admittedly too harsh a case for DFT-based methods, as its dynamics
is deeply controlled by coordinates involved in the nonadiabatic events. SH/DFT
methods have been developed to deal with large molecular systems and in these
cases the dynamics may evolve in the configurational space spanning regions
relatively far from any muticonfigurational ground state. Under such situation, the
excited-state dynamics involving a large manifold of excited states can be well
simulated with DFT-based methods.
Take, for instance, the schematic dynamics illustrated in Fig. 4 (top). The
dynamics starts at a high excited state (here, the second state to simplify the
picture). Using a multireference method such as MRCI, we would observe a
relaxation to the first excited state (a), then oscillation around the minimum of
this state (b). From this minimum, the molecule can eventually fluoresce to the
ground state (c) or cross a barrier (d) to reach a conical intersection, from where it
relaxes to the ground state minimum (e).
Surface Hopping Dynamics with DFT Excited States
431