5.5 Computational Note: Methods for Nonadiabatic Dynamics
169
5.5 Computational Note: Methods for Nonadiabatic
Dynamics
We consider here some of the most widely used methods for the description
of nonadiabatic processes, such as excited state decay, photoisomerizations, photodissociations, energy transfer, and charge transfer, taking place on an ultrafast
(sub-picosecond) or fast timescale (say, up to a few tens of ps). Slow nonadiabatic
transitions are better tackled by perturbation methods; see Chap. 3.
The numerical integration of the TDSE (2.78), in the form deriving from the adiabatic Born–Huang expansion (2.77), requires the knowledge of the PESs U k (Q)
and of the nonadiabatic couplings g kl (Q) and t kl (Q), in the whole region of nuclear
coordinates that is accessible to the wavepackets Θ k . Which electronic states must
be included in the expansion and which regions of the PESs will be explored by the
wavepackets depend on the initial conditions and particularly on the total energy, but
also on the duration of the dynamics of interest. As an alternative, we can express
the TDSE in the diabatic representation, Eq. (5.20), and then we need the diabatic
quantities H kl (Q)) for all the electronic states considered. In principle, this means
that the adiabatic or diabatic PESs and couplings must be evaluated a priori for a sufficiently large number of points of the nuclear configuration space and then expressed
as analytic functions of the internal coordinates Q. For polyatomic molecules, the
fitting or interpolation of the computed electronic quantities is a very cumbersome
task, even taking into account the simplifications introduced by the diabatic representation. In part for this reason, approximated approaches in which the nuclear
motion is described classically are very popular. In fact, due to the local character of
classical mechanics, a “direct” (or “on-the-fly”) strategy can be easily adopted, such
that the electronic calculations are performed when needed, during the integration
of the dynamical equations [17, 18]. In particular, for each time step in a nuclear
classical trajectory, one just needs to evaluate the electronic energies and interstate
couplings, plus the forces acting on the nuclei, at a single molecular geometry. Of
course, the classical approximation is justified on the basis of the relatively large
nuclear masses and is valid as far as quantum effects such as zero-point vibrations,
tunneling, and interference can be ignored (see Chap. 4).
In the following, we will distinguish between quantum wavepacket dynamics
(QWD) and classical trajectory approaches. In some QWD methods an effort is made
to keep the nuclear motion as local as possible, for example by using nonspreading
and traveling basis functions, such that the direct strategy can be applied. In classical
trajectory approaches, the quantum effects that are ignored in the first place can be
reintroduced by ad hoc provisions of the method.
5.5.1 Quantum Wavepacket Dynamics
Assuming that the electronic quantities (either the adiabatic U k , g kl (Q) and t kl (Q) or
the diabatic H kl (Q)) are known, the numerical integration of the TDSE for the nuclear
169
5.5 Computational Note: Methods for Nonadiabatic
Dynamics
We consider here some of the most widely used methods for the description
of nonadiabatic processes, such as excited state decay, photoisomerizations, photodissociations, energy transfer, and charge transfer, taking place on an ultrafast
(sub-picosecond) or fast timescale (say, up to a few tens of ps). Slow nonadiabatic
transitions are better tackled by perturbation methods; see Chap. 3.
The numerical integration of the TDSE (2.78), in the form deriving from the adiabatic Born–Huang expansion (2.77), requires the knowledge of the PESs U k (Q)
and of the nonadiabatic couplings g kl (Q) and t kl (Q), in the whole region of nuclear
coordinates that is accessible to the wavepackets Θ k . Which electronic states must
be included in the expansion and which regions of the PESs will be explored by the
wavepackets depend on the initial conditions and particularly on the total energy, but
also on the duration of the dynamics of interest. As an alternative, we can express
the TDSE in the diabatic representation, Eq. (5.20), and then we need the diabatic
quantities H kl (Q)) for all the electronic states considered. In principle, this means
that the adiabatic or diabatic PESs and couplings must be evaluated a priori for a sufficiently large number of points of the nuclear configuration space and then expressed
as analytic functions of the internal coordinates Q. For polyatomic molecules, the
fitting or interpolation of the computed electronic quantities is a very cumbersome
task, even taking into account the simplifications introduced by the diabatic representation. In part for this reason, approximated approaches in which the nuclear
motion is described classically are very popular. In fact, due to the local character of
classical mechanics, a “direct” (or “on-the-fly”) strategy can be easily adopted, such
that the electronic calculations are performed when needed, during the integration
of the dynamical equations [17, 18]. In particular, for each time step in a nuclear
classical trajectory, one just needs to evaluate the electronic energies and interstate
couplings, plus the forces acting on the nuclei, at a single molecular geometry. Of
course, the classical approximation is justified on the basis of the relatively large
nuclear masses and is valid as far as quantum effects such as zero-point vibrations,
tunneling, and interference can be ignored (see Chap. 4).
In the following, we will distinguish between quantum wavepacket dynamics
(QWD) and classical trajectory approaches. In some QWD methods an effort is made
to keep the nuclear motion as local as possible, for example by using nonspreading
and traveling basis functions, such that the direct strategy can be applied. In classical
trajectory approaches, the quantum effects that are ignored in the first place can be
reintroduced by ad hoc provisions of the method.
5.5.1 Quantum Wavepacket Dynamics
Assuming that the electronic quantities (either the adiabatic U k , g kl (Q) and t kl (Q) or
the diabatic H kl (Q)) are known, the numerical integration of the TDSE for the nuclear