and, consequently, of the excited-states based upon. In particular, the methods for
computing excitation energies discussed in Sect. 3.1 have in common the assumption that the ground state can be described by a single KS determinant. (For a
conceptual discussion of nondynamic electron correlation in DFT, see [95]. For a
discussion about conical intersection in DFT, see [96]).
The development of a density functional theory including nondynamic
(or “strong”) electron correlation – which Becke designated as the “last frontier”
in DFT [97] – has been pursued by many groups following different approaches.
Among these approaches, we may cite the use of restricted open-shell ground-state
representations [98], configuration ensembles with fractional occupations [91, 99–
101], configuration interaction [102, 103], multiconfigurational DFT [104], hybrid
multiconfiguration/(TD)DFT [105, 106], and spin-unrestricted broken-symmetry
(UBS) [107] approaches. Unfortunately, analytical energy gradients are not available for most of these methods, which rules out their use in surface hopping
dynamics.
All the troubles caused by nondynamic electron correlation are rather restricted
to the crossing between the ground and the first excited state. In the case of crossing
between excited states, however, if each of these states is well represented by single
excitations from a well-behaved ground state, the description of the crossing does
not present further problems.
SH/DFT has other potential problems (Fig. 2) besides nondynamical electron
correlation. First, the results are deeply dependent on the functional. Second,
double and higher excitations are not properly described by LR-TDDFT. Third,
diffuse and charge-transfer states may be poorly described by conventional functionals. All these problems, though, are not exclusive of dynamics simulations and
are part of routine investigations of excited states based on DFT. As such, they are
addressed in the same way, by systematic test of functionals, methodological
comparisons, and use of range-separated functionals.
To illustrate the current situation of excited state description for SH/DFT, we
show in Fig. 3 the potential-energy surfaces for the S 1 state and for the S 1 /S 0 gap of
ethylene along two important reaction coordinates, H 2 CCH 2 -torsion (θ) and CCH 2 -
pyramidalization (φ). All other coordinates are kept at their ground-state values
optimized at the same level as used for energy calculations. These surfaces were
computed with several different DFT-based methods and with ab initio MR-CISD,
which is taken as the reference method. With the exception of the DFT-MRCI
[102], computed with the SV(P) basis set [108], all other DFT-based surfaces were
computed with the 6-31G* basis set [109]. LC-BLYP [110–112] was computed
with μ ¼ 0.2 a 0
À1 [113]. TD-DFTB was based on analytic expressions for the matrix
elements [114]. MR-CISD was based on a small (two electron, two orbital)-space
[115] with the aug-cc-pVTZ basis set [116] assigned to C and cc-pVDZ assigned
to H.
Ethylene S 1 surface is a specially challenging problem, starting from (i) the
adequate computation of the excitation into the V state [6], then (ii) the description
of the right topography of the state, which includes a crossing with the Z state near
the twisted structure and a conical intersection at twisted-pyramidalized geometries
Surface Hopping Dynamics with DFT Excited States
429
computing excitation energies discussed in Sect. 3.1 have in common the assumption that the ground state can be described by a single KS determinant. (For a
conceptual discussion of nondynamic electron correlation in DFT, see [95]. For a
discussion about conical intersection in DFT, see [96]).
The development of a density functional theory including nondynamic
(or “strong”) electron correlation – which Becke designated as the “last frontier”
in DFT [97] – has been pursued by many groups following different approaches.
Among these approaches, we may cite the use of restricted open-shell ground-state
representations [98], configuration ensembles with fractional occupations [91, 99–
101], configuration interaction [102, 103], multiconfigurational DFT [104], hybrid
multiconfiguration/(TD)DFT [105, 106], and spin-unrestricted broken-symmetry
(UBS) [107] approaches. Unfortunately, analytical energy gradients are not available for most of these methods, which rules out their use in surface hopping
dynamics.
All the troubles caused by nondynamic electron correlation are rather restricted
to the crossing between the ground and the first excited state. In the case of crossing
between excited states, however, if each of these states is well represented by single
excitations from a well-behaved ground state, the description of the crossing does
not present further problems.
SH/DFT has other potential problems (Fig. 2) besides nondynamical electron
correlation. First, the results are deeply dependent on the functional. Second,
double and higher excitations are not properly described by LR-TDDFT. Third,
diffuse and charge-transfer states may be poorly described by conventional functionals. All these problems, though, are not exclusive of dynamics simulations and
are part of routine investigations of excited states based on DFT. As such, they are
addressed in the same way, by systematic test of functionals, methodological
comparisons, and use of range-separated functionals.
To illustrate the current situation of excited state description for SH/DFT, we
show in Fig. 3 the potential-energy surfaces for the S 1 state and for the S 1 /S 0 gap of
ethylene along two important reaction coordinates, H 2 CCH 2 -torsion (θ) and CCH 2 -
pyramidalization (φ). All other coordinates are kept at their ground-state values
optimized at the same level as used for energy calculations. These surfaces were
computed with several different DFT-based methods and with ab initio MR-CISD,
which is taken as the reference method. With the exception of the DFT-MRCI
[102], computed with the SV(P) basis set [108], all other DFT-based surfaces were
computed with the 6-31G* basis set [109]. LC-BLYP [110–112] was computed
with μ ¼ 0.2 a 0
À1 [113]. TD-DFTB was based on analytic expressions for the matrix
elements [114]. MR-CISD was based on a small (two electron, two orbital)-space
[115] with the aug-cc-pVTZ basis set [116] assigned to C and cc-pVDZ assigned
to H.
Ethylene S 1 surface is a specially challenging problem, starting from (i) the
adequate computation of the excitation into the V state [6], then (ii) the description
of the right topography of the state, which includes a crossing with the Z state near
the twisted structure and a conical intersection at twisted-pyramidalized geometries
Surface Hopping Dynamics with DFT Excited States
429