152
L. González et al.
where V BO
α (R(t)) are the Born-Oppenheimer PES, μ βα (R(t)) is the transition
dipole moment and E(t) the external field. Equations (7.10) and (7.17) apply exactly
as before. Given a reference state ψ BO
α at time t, the nuclear equation of motion will
be
M k
¨
R k = −
ψ
BO
α (r; R)
∇
R k
H
el
Ψ
BO
α (r; R)
,
(7.19)
where explicit time-dependence is omitted. In principle, one could incorporate other
couplings, for instance, the spin-orbit couplings, as additional non-diagonal terms in
H el . In the current FISH scheme [71], Eqs. (7.9), (7.10), (7.17) and (7.19) determine
the coupled electron-nuclear dynamics.
It has been shown that the choice of Born-Oppenheimer electronic states as the
reference state for the SH quantum force is sufficiently accurate to represent the dynamics of systems driven by different laser pulses [71]. However, in this approach
the gradients are evaluated directly in the laser-free potentials, and strong field effects like the Stark effect cannot be properly described. When the reference state
departs largely from the true electronic wave function, the ensemble of trajectories
needed to statistically reproduce the observed effects hugely raises. As previously
stated, when strong fields act on the molecule, the mean-field trajectory can actually
reproduce quite well the dynamics of the system, particularly in simpler systems.
In the SH approach, this Ehrenfest-like trajectory can be incorporated in a better
reference wave function. This is the basis of the SHARC method [74].
In SHARC, instead of expanding the wave function in the Born-Oppenheimer
basis [Eq. (7.8)], one uses a “fully adiabatic” basis,
Ψ
el
r, t; R(t)
=
α
a α (t)ψ
a
α
r, t; R(t)
,
(7.20)
that serves as a reference state. Notice that in general this reference state will be
explicitly time-dependent. To simplify the notation, we will thereafter neglect the
implicit time-dependence of R(t). This adiabatic basis is chosen to diagonalize the
Hamiltonian in the presence of the additional couplings not included in the BornOppenheimer approximation, such as the spin-orbit and electron-radiation coupling:
ψ
a
α (r, t; R) =
β
U αβ (t; R)ψ
BO
β (r; R).
(7.21)
U is the unitary rotation for each nuclear position and instant of time that makes
H
a
βα (t) =
ψ
a
β (r, t; R)
H
el
ψ
a
α (r, t; R)
= V
a
α (R, t)δ βα .
(7.22)
In the context of strong laser-molecule interaction, these truly adiabatic potentials, which are explicitly time-dependent, are called light-induced potentials (LIPs)
V a
α (R, t) = V LIP
α (R, t) and exhibit very interesting properties, including bond hardening and bond softening effects [19, 81–86]. In principle, distinct SHARC approaches could be followed depending on whether the adiabatic basis diagonalizes
the full Hamiltonian at each instant of time, including the laser and spin-orbit couplings, or the laser coupling only.
L. González et al.
where V BO
α (R(t)) are the Born-Oppenheimer PES, μ βα (R(t)) is the transition
dipole moment and E(t) the external field. Equations (7.10) and (7.17) apply exactly
as before. Given a reference state ψ BO
α at time t, the nuclear equation of motion will
be
M k
¨
R k = −
ψ
BO
α (r; R)
∇
R k
H
el
Ψ
BO
α (r; R)
,
(7.19)
where explicit time-dependence is omitted. In principle, one could incorporate other
couplings, for instance, the spin-orbit couplings, as additional non-diagonal terms in
H el . In the current FISH scheme [71], Eqs. (7.9), (7.10), (7.17) and (7.19) determine
the coupled electron-nuclear dynamics.
It has been shown that the choice of Born-Oppenheimer electronic states as the
reference state for the SH quantum force is sufficiently accurate to represent the dynamics of systems driven by different laser pulses [71]. However, in this approach
the gradients are evaluated directly in the laser-free potentials, and strong field effects like the Stark effect cannot be properly described. When the reference state
departs largely from the true electronic wave function, the ensemble of trajectories
needed to statistically reproduce the observed effects hugely raises. As previously
stated, when strong fields act on the molecule, the mean-field trajectory can actually
reproduce quite well the dynamics of the system, particularly in simpler systems.
In the SH approach, this Ehrenfest-like trajectory can be incorporated in a better
reference wave function. This is the basis of the SHARC method [74].
In SHARC, instead of expanding the wave function in the Born-Oppenheimer
basis [Eq. (7.8)], one uses a “fully adiabatic” basis,
Ψ
el
r, t; R(t)
=
α
a α (t)ψ
a
α
r, t; R(t)
,
(7.20)
that serves as a reference state. Notice that in general this reference state will be
explicitly time-dependent. To simplify the notation, we will thereafter neglect the
implicit time-dependence of R(t). This adiabatic basis is chosen to diagonalize the
Hamiltonian in the presence of the additional couplings not included in the BornOppenheimer approximation, such as the spin-orbit and electron-radiation coupling:
ψ
a
α (r, t; R) =
β
U αβ (t; R)ψ
BO
β (r; R).
(7.21)
U is the unitary rotation for each nuclear position and instant of time that makes
H
a
βα (t) =
ψ
a
β (r, t; R)
H
el
ψ
a
α (r, t; R)
= V
a
α (R, t)δ βα .
(7.22)
In the context of strong laser-molecule interaction, these truly adiabatic potentials, which are explicitly time-dependent, are called light-induced potentials (LIPs)
V a
α (R, t) = V LIP
α (R, t) and exhibit very interesting properties, including bond hardening and bond softening effects [19, 81–86]. In principle, distinct SHARC approaches could be followed depending on whether the adiabatic basis diagonalizes
the full Hamiltonian at each instant of time, including the laser and spin-orbit couplings, or the laser coupling only.