not only deviate strongly from the exact energies, but also provide a poor approximation for adiabatic surfaces [60].
Still aiming at maximum computational optimization, Prezhdo and co-workers
have also introduced the so-called “classical path approximation” (CPA) in
PYXAID [61]. The CPA implies that the nuclear dynamics is supposed to evolve
independently of the electrons, driven by kinetic effects. In practical terms, it means
that the nuclear dynamics is propagated in the ground state and then is used to
compute the nonadiabatic electronic events in the excited states using either the
fewest switches surface hopping or one of the other surface hopping algorithms
developed by the Prezhdo group [132]. The CPA may be especially inadequate for
systems undergoing significant chemical changes such as photofragmentation and
chemical reactions.
Another efficient implementation of SH/DFT is based on ROKS and was
developed by Doltsinis and Marx [137]. The spin-adapted wavefunctions and the
KS orbitals optimized for the excited-state density should in principle be an
improvement over the SDKS. SH/ROKS, however, is limited to a single excited
state, which is a very strong limitation for most realistic problems. Such an
approach is implemented in the CPMD program.
SH/LR-TDDFT has been pioneered by Tavernelli and Rothlisberger [34] and by
Mitric ´ and Bonac ˇic ´-Koutecky ´ [81]. Linear response should provide a better description of the excited-state surfaces than single determinant and ROKS approaches, at
higher computational costs naturally (see Sect. 3.1). TDDFT is also not limited to a
single surface as ROKS, but it cannot describe multiple excitations. In the framework of linear response, surface hopping dynamics has been investigated with
TDDFT, TDA, and TD-DFTB (see Sect. 5). In the CPMD program, this approach
is implemented based on plane waves, while in Newton-X, it is implemented on a
localized basis. It is also implemented in Turbomole, but limited to couplings
between the ground state and the first excited state. Ehrenfest dynamics [138],
another related semiclassical nonadiabatic dynamics method, can be performed
with TDDFT using the Octopus program [139].
In the case of Newton-X, the coefficients c L to compute the hopping probabilities
(10) can be obtained either by integrating (4) or by using the local diabatization
method (8). The program also allows the computation of surface hopping through
interfaces with different programs (Turbomole, Gaussian [140], Columbus [141,
142], Gamess [143]) and using different methods (TDA, TDDFT, ADC(2), CC2,
CASSCF, MRCI), making it particularly convenient for comparative analysis.
5 Case Studies
There are a large number of molecular systems which have been investigated with
SH/DFT. This section does not aim at providing a comprehensive review of them,
but instead at pointing out the main classes of problems in different fields, from
where the reader can search for more information.
434
M. Barbatti and R. Crespo-Otero
Still aiming at maximum computational optimization, Prezhdo and co-workers
have also introduced the so-called “classical path approximation” (CPA) in
PYXAID [61]. The CPA implies that the nuclear dynamics is supposed to evolve
independently of the electrons, driven by kinetic effects. In practical terms, it means
that the nuclear dynamics is propagated in the ground state and then is used to
compute the nonadiabatic electronic events in the excited states using either the
fewest switches surface hopping or one of the other surface hopping algorithms
developed by the Prezhdo group [132]. The CPA may be especially inadequate for
systems undergoing significant chemical changes such as photofragmentation and
chemical reactions.
Another efficient implementation of SH/DFT is based on ROKS and was
developed by Doltsinis and Marx [137]. The spin-adapted wavefunctions and the
KS orbitals optimized for the excited-state density should in principle be an
improvement over the SDKS. SH/ROKS, however, is limited to a single excited
state, which is a very strong limitation for most realistic problems. Such an
approach is implemented in the CPMD program.
SH/LR-TDDFT has been pioneered by Tavernelli and Rothlisberger [34] and by
Mitric ´ and Bonac ˇic ´-Koutecky ´ [81]. Linear response should provide a better description of the excited-state surfaces than single determinant and ROKS approaches, at
higher computational costs naturally (see Sect. 3.1). TDDFT is also not limited to a
single surface as ROKS, but it cannot describe multiple excitations. In the framework of linear response, surface hopping dynamics has been investigated with
TDDFT, TDA, and TD-DFTB (see Sect. 5). In the CPMD program, this approach
is implemented based on plane waves, while in Newton-X, it is implemented on a
localized basis. It is also implemented in Turbomole, but limited to couplings
between the ground state and the first excited state. Ehrenfest dynamics [138],
another related semiclassical nonadiabatic dynamics method, can be performed
with TDDFT using the Octopus program [139].
In the case of Newton-X, the coefficients c L to compute the hopping probabilities
(10) can be obtained either by integrating (4) or by using the local diabatization
method (8). The program also allows the computation of surface hopping through
interfaces with different programs (Turbomole, Gaussian [140], Columbus [141,
142], Gamess [143]) and using different methods (TDA, TDDFT, ADC(2), CC2,
CASSCF, MRCI), making it particularly convenient for comparative analysis.
5 Case Studies
There are a large number of molecular systems which have been investigated with
SH/DFT. This section does not aim at providing a comprehensive review of them,
but instead at pointing out the main classes of problems in different fields, from
where the reader can search for more information.
434
M. Barbatti and R. Crespo-Otero