7 Ultrafast Laser-Induced Processes Described by Ab Initio Molecular
147
to obtain different branching ratios for competing pathways involving several PES.
In the presence of non-adiabatic phenomena, it is usual to combine MD with the
surface hopping (SH) method [48–50] so that the system is allowed to hop from one
PES to another, as required to describe non-adiabatic photochemistry. Of course,
SH requires the calculation of non-adiabatic coupling (NAC) terms within the given
method, a task which is straightforward within ab initio methods [51] but has been
derived only in the last decade for TD-DFT [52–55]. Several reviews highlight
many of the applications that can be treated with the trajectory SH method [56–58].
In contrast, the use of MD in the presence of spin-orbit couplings is much less extended [59–62]. Moreover, up to our knowledge, as of 2011, there are no on-the-fly
ab initio MD studies explicitly including both non-adiabatic and spin-orbit couplings in an ab initio framework. This shortcoming is mainly motivated by the fact
that intersystem crossing (ISC) between PES of different multiplicity is typically believed to occur in a much longer time scale than internal conversion (IC) via conical
intersections. Recent investigations, e.g. in benzene, revealed however that ISC can
compete with IC in an ultrafast time scale [63, 64]. Therefore, it seems mandatory
to have methods which can describe multidimensional dynamics in the presence of
both non-adiabatic and spin-orbit couplings [65]. Additionally, to achieve quantitative agreement with experiments, it is desirable to couple nuclear and electronic
degrees of freedom with an electromagnetic field, so that laser transitions can also
be directly modeled in theory, as they happen in experiment. Some approaches have
been derived to incorporate time-dependent external fields with MD [66–73], but
there is no general scheme able to incorporate all types of couplings (non-adiabatic,
spin-orbit, dipole couplings, etc.) into MD.
In this chapter we present a novel semiclassical scheme to integrate the equations
of motion of electrons and nuclei in molecules, including all degrees of freedom and
all types of couplings, that expands the electronic wave function in a relatively small
basis of eigenstates of the Hamiltonian, and treats the nuclear motion as a classical
trajectory unfolding on a single electronic state. The main novelty in the present
approach consists in the evaluation of the transition probability between electronic
states, which is performed by SH techniques in the adiabatic representation, essentially treating on the same footing both non-adiabatic beyond Born-Oppenheimer
transitions (ISC, IC) and laser-induced crossings. The numerical method is called
Surface-Hopping in the Adiabatic Representation including arbitrary Couplings, abbreviated SHARC [74, 75]. As it will be shown later in this chapter, the choice of the
given approximations and representation is particularly advantageous in evaluating
the dynamics when the laser field and NACs are strong.
The rest of this chapter is organized as follows. First, the basic ideas of semiclassical dynamics and in particular SH are explained in detail. SHARC is introduced
and put into the context of the related FISH (Field-Induced SH) [71] method, which
has been successfully used recently to interpret some control experiments [76–79].
In the following sections, the performance of SHARC will be illustrated in two
scenarios depending on the way that the light interaction is considered. In the first
scenario, the dynamics of the system is simulated starting in the electronic excited
state assuming an instantaneous excitation. In the second scenario, the interaction
147
to obtain different branching ratios for competing pathways involving several PES.
In the presence of non-adiabatic phenomena, it is usual to combine MD with the
surface hopping (SH) method [48–50] so that the system is allowed to hop from one
PES to another, as required to describe non-adiabatic photochemistry. Of course,
SH requires the calculation of non-adiabatic coupling (NAC) terms within the given
method, a task which is straightforward within ab initio methods [51] but has been
derived only in the last decade for TD-DFT [52–55]. Several reviews highlight
many of the applications that can be treated with the trajectory SH method [56–58].
In contrast, the use of MD in the presence of spin-orbit couplings is much less extended [59–62]. Moreover, up to our knowledge, as of 2011, there are no on-the-fly
ab initio MD studies explicitly including both non-adiabatic and spin-orbit couplings in an ab initio framework. This shortcoming is mainly motivated by the fact
that intersystem crossing (ISC) between PES of different multiplicity is typically believed to occur in a much longer time scale than internal conversion (IC) via conical
intersections. Recent investigations, e.g. in benzene, revealed however that ISC can
compete with IC in an ultrafast time scale [63, 64]. Therefore, it seems mandatory
to have methods which can describe multidimensional dynamics in the presence of
both non-adiabatic and spin-orbit couplings [65]. Additionally, to achieve quantitative agreement with experiments, it is desirable to couple nuclear and electronic
degrees of freedom with an electromagnetic field, so that laser transitions can also
be directly modeled in theory, as they happen in experiment. Some approaches have
been derived to incorporate time-dependent external fields with MD [66–73], but
there is no general scheme able to incorporate all types of couplings (non-adiabatic,
spin-orbit, dipole couplings, etc.) into MD.
In this chapter we present a novel semiclassical scheme to integrate the equations
of motion of electrons and nuclei in molecules, including all degrees of freedom and
all types of couplings, that expands the electronic wave function in a relatively small
basis of eigenstates of the Hamiltonian, and treats the nuclear motion as a classical
trajectory unfolding on a single electronic state. The main novelty in the present
approach consists in the evaluation of the transition probability between electronic
states, which is performed by SH techniques in the adiabatic representation, essentially treating on the same footing both non-adiabatic beyond Born-Oppenheimer
transitions (ISC, IC) and laser-induced crossings. The numerical method is called
Surface-Hopping in the Adiabatic Representation including arbitrary Couplings, abbreviated SHARC [74, 75]. As it will be shown later in this chapter, the choice of the
given approximations and representation is particularly advantageous in evaluating
the dynamics when the laser field and NACs are strong.
The rest of this chapter is organized as follows. First, the basic ideas of semiclassical dynamics and in particular SH are explained in detail. SHARC is introduced
and put into the context of the related FISH (Field-Induced SH) [71] method, which
has been successfully used recently to interpret some control experiments [76–79].
In the following sections, the performance of SHARC will be illustrated in two
scenarios depending on the way that the light interaction is considered. In the first
scenario, the dynamics of the system is simulated starting in the electronic excited
state assuming an instantaneous excitation. In the second scenario, the interaction