The theoretical background for computation of these couplings has been
established by Chernyak and Mukamel [73] and, alternatively, by Baer [51]. Hu
and co-workers have further developed the Chernyak–Mukamel approach for
computation of first-order nonadiabatic coupling vectors between the ground state
and the first excited state [74–76]. Send and Furche have shown that the previous
result neglects molecular orbitals derivatives [77]. Their own derivation, including
such terms but still limited to couplings between the ground and the first excited
states, is implemented in Turbomole [78].
Because of the lack of analytical couplings between excited states in standard
quantum-chemistry programs, finite-difference couplings have been largely used in
surface hopping [32–34]. They are usually based on the approach proposed by
Hammes-Schiffer and Tully [31], who showed that the couplings can be written in
terms of wavefunction overlaps between sequential time steps during the dynamics –
see (7).
A particular problem with using TDDFT in this approach is the lack of an
explicit wavefunction for the electronic states. A common solution has been to
take the Casida’s ansatz for the state assignment [54] as an approximation to the
wavefunctions. In this case, the ground- and excited-state wavefunctions are given
(for a local functional) by
Ψ 0 ¼ KS
j i;
ð30Þ
Ψ L ¼
X
j, b
ffiffiffiffiffiffiffiffi ffi
Δε jb
ω L
r
F
jb
L jb
j i;
ð31Þ
where |KSi is the ground-state KS determinant and jjbi is the determinant with a
single excitation from j to b. With these wavefunctions, which are analogous to a
CIS expansion, the coupling terms σ LM can be evaluated according to (7) in terms of
atomic orbital overlap integrals [33]. Although the validity of Casida’s ansatz for
computation of couplings approach still needs to be extensively tested, it has been
shown that TDDFT dynamics computed with these couplings compares well with
dynamics based on CASSCF [79] and MRCI [80].
Equation (31) forms a non-orthogonal set, which can have consequences for the
evaluation of the couplings. Werner et al. [81] recommended the use the linearresponse coefficients without the square-root term in (31) as the CIS coefficients,
which forms an orthogonal set within TDA. An alternative solution is to orthogonalize the approximate wavefunctions given by (31) before computing the couplings. In practical terms, far from conical intersections with the ground state, either
of these approximations provides similar dynamical pictures. In some cases, negative excitation energies are obtained (see Sect. 3.3), and consequently the
wavefunctions are complex and imaginary couplings may be obtained. In such
situations, as we discuss later, it is not recommended to continue the SH/TDDFT
propagation.
The Casida wavefunctions given in (30) and (31) have been employed by
Tavernelli and co-workers to derive analytical nonadiabatic coupling vectors
426
M. Barbatti and R. Crespo-Otero
established by Chernyak and Mukamel [73] and, alternatively, by Baer [51]. Hu
and co-workers have further developed the Chernyak–Mukamel approach for
computation of first-order nonadiabatic coupling vectors between the ground state
and the first excited state [74–76]. Send and Furche have shown that the previous
result neglects molecular orbitals derivatives [77]. Their own derivation, including
such terms but still limited to couplings between the ground and the first excited
states, is implemented in Turbomole [78].
Because of the lack of analytical couplings between excited states in standard
quantum-chemistry programs, finite-difference couplings have been largely used in
surface hopping [32–34]. They are usually based on the approach proposed by
Hammes-Schiffer and Tully [31], who showed that the couplings can be written in
terms of wavefunction overlaps between sequential time steps during the dynamics –
see (7).
A particular problem with using TDDFT in this approach is the lack of an
explicit wavefunction for the electronic states. A common solution has been to
take the Casida’s ansatz for the state assignment [54] as an approximation to the
wavefunctions. In this case, the ground- and excited-state wavefunctions are given
(for a local functional) by
Ψ 0 ¼ KS
j i;
ð30Þ
Ψ L ¼
X
j, b
ffiffiffiffiffiffiffiffi ffi
Δε jb
ω L
r
F
jb
L jb
j i;
ð31Þ
where |KSi is the ground-state KS determinant and jjbi is the determinant with a
single excitation from j to b. With these wavefunctions, which are analogous to a
CIS expansion, the coupling terms σ LM can be evaluated according to (7) in terms of
atomic orbital overlap integrals [33]. Although the validity of Casida’s ansatz for
computation of couplings approach still needs to be extensively tested, it has been
shown that TDDFT dynamics computed with these couplings compares well with
dynamics based on CASSCF [79] and MRCI [80].
Equation (31) forms a non-orthogonal set, which can have consequences for the
evaluation of the couplings. Werner et al. [81] recommended the use the linearresponse coefficients without the square-root term in (31) as the CIS coefficients,
which forms an orthogonal set within TDA. An alternative solution is to orthogonalize the approximate wavefunctions given by (31) before computing the couplings. In practical terms, far from conical intersections with the ground state, either
of these approximations provides similar dynamical pictures. In some cases, negative excitation energies are obtained (see Sect. 3.3), and consequently the
wavefunctions are complex and imaginary couplings may be obtained. In such
situations, as we discuss later, it is not recommended to continue the SH/TDDFT
propagation.
The Casida wavefunctions given in (30) and (31) have been employed by
Tavernelli and co-workers to derive analytical nonadiabatic coupling vectors
426
M. Barbatti and R. Crespo-Otero