208
6 Charge and Energy Transfer Processes
charged species are screened by polar or polarizable media, especially in the longrange, as is the case with Förster coupling. It is often found that the stabilization of
localized excitations or redox states due to the environment, being markedly specific of their peculiar charge distribution, bonding and structure, makes the diabatic
representation even more suitable to treat ET and CT [1]. As already mentioned,
continuum models of the environment in many cases offer a satisfactory description
of these static effects and can also, more schematically, take into account the dynamic
polarizability of dielectric media. Specific interactions require to represent all parts
of the microenvironment at atomic scale, possibly treating (part of) the environment
by force fields (see Curutchet and Mennucci for a thorough discussion of different
options [4]).
The diabatic representation may also facilitate the integration of the TDSE to
simulate fast nonadiabatic processes. When two or more states are close in energy
and are weakly coupled, as is the case with similar molecules loosely interacting,
the system finds itself very often close to a crossing seam. Every time a wavepacket
or a trajectory goes through the crossing line between two diabatic PESs, since the
electronic coupling is weak, the nonadiabatic couplings g 12 and t 12 present narrow
and tall peaks, as in the NaCl avoided crossing of Sect. 5.1. For a trajectory, this means
the nonadiabatic coupling changes in time very suddenly, making the numerical
integration of the TDSE quite inaccurate unless very small time steps are used.
Even if the dynamics is treated in the adiabatic representation, an auxiliary “local
diabatization” can avoid this drawback [25]. In principle, the nonlocal nature of
quantum wavepackets should prevent the occurrence of such sudden events. However,
the evaluation of coupling matrix elements between static or traveling basis functions
in MCTDH methods suffers of similar drawbacks. Modeling the dependence of the
electronic structure on the internal coordinates by effective Hamiltonian in diabatic
representations is a good solution also in this case [26–28].
Several ways to define (quasi)-diabatic representations have been proposed. Most
of them consist in an orthogonal transformation (“rotation”) of the subset of adiabatic
states {. . . ψ k . . .} that are energetically accessible during the process under study, to
produce the diabatic basis:
{. . . η i . . .} = {. . . ψ k . . .}T .
(6.77)
The adiabatic to diabatic rotation T can be determined by various ad hoc localization criteria. A general diabatization method consists in constructing (normally
nonorthogonal) diabatic templates and in rotating the adiabatic basis to achieve maximum overlap with the templates [29, 30]. If localization is the goal, the templates
can be defined as products of X and Y wavefunctions, representing the ground and
excited states of the two subsystems. Once the rotation T has been computed, it can
be applied to the diagonal matrix of the adiabatic energies E to get the Hamiltonian
in the diabatic basis:
H
(dia)
= T
† E T .
(6.78)
The diagonal and off-diagonal elements of H are the diabatic state energies and the
interactions between localized excitations, respectively.
6 Charge and Energy Transfer Processes
charged species are screened by polar or polarizable media, especially in the longrange, as is the case with Förster coupling. It is often found that the stabilization of
localized excitations or redox states due to the environment, being markedly specific of their peculiar charge distribution, bonding and structure, makes the diabatic
representation even more suitable to treat ET and CT [1]. As already mentioned,
continuum models of the environment in many cases offer a satisfactory description
of these static effects and can also, more schematically, take into account the dynamic
polarizability of dielectric media. Specific interactions require to represent all parts
of the microenvironment at atomic scale, possibly treating (part of) the environment
by force fields (see Curutchet and Mennucci for a thorough discussion of different
options [4]).
The diabatic representation may also facilitate the integration of the TDSE to
simulate fast nonadiabatic processes. When two or more states are close in energy
and are weakly coupled, as is the case with similar molecules loosely interacting,
the system finds itself very often close to a crossing seam. Every time a wavepacket
or a trajectory goes through the crossing line between two diabatic PESs, since the
electronic coupling is weak, the nonadiabatic couplings g 12 and t 12 present narrow
and tall peaks, as in the NaCl avoided crossing of Sect. 5.1. For a trajectory, this means
the nonadiabatic coupling changes in time very suddenly, making the numerical
integration of the TDSE quite inaccurate unless very small time steps are used.
Even if the dynamics is treated in the adiabatic representation, an auxiliary “local
diabatization” can avoid this drawback [25]. In principle, the nonlocal nature of
quantum wavepackets should prevent the occurrence of such sudden events. However,
the evaluation of coupling matrix elements between static or traveling basis functions
in MCTDH methods suffers of similar drawbacks. Modeling the dependence of the
electronic structure on the internal coordinates by effective Hamiltonian in diabatic
representations is a good solution also in this case [26–28].
Several ways to define (quasi)-diabatic representations have been proposed. Most
of them consist in an orthogonal transformation (“rotation”) of the subset of adiabatic
states {. . . ψ k . . .} that are energetically accessible during the process under study, to
produce the diabatic basis:
{. . . η i . . .} = {. . . ψ k . . .}T .
(6.77)
The adiabatic to diabatic rotation T can be determined by various ad hoc localization criteria. A general diabatization method consists in constructing (normally
nonorthogonal) diabatic templates and in rotating the adiabatic basis to achieve maximum overlap with the templates [29, 30]. If localization is the goal, the templates
can be defined as products of X and Y wavefunctions, representing the ground and
excited states of the two subsystems. Once the rotation T has been computed, it can
be applied to the diagonal matrix of the adiabatic energies E to get the Hamiltonian
in the diabatic basis:
H
(dia)
= T
† E T .
(6.78)
The diagonal and off-diagonal elements of H are the diabatic state energies and the
interactions between localized excitations, respectively.