536
G. Hanna and A. Sergi
and ρ B (Q, P ), the Wigner distribution function of the bath (i.e. the resonant
harmonic mode), is given by
ρ B (Q, P ) =
tanh(βω/2)
π
exp
−
2 tanh(βω/2)
ω
P 2
2
+
ω 2 Q 2
2
. (14.35)
To simulate the quantum-classical Liouville dynamics of the Hamiltonian in
Eq. (14.33), a generalization of the SSTP algorithm to time-dependent Hamiltonians
is needed. To this end, one can introduce a time-dependent basis in terms of the
eigenstates of ˆ
h(Q, t) = ˆ
H (Q, P , t) − P 2 /2M (defined by ˆ
h(Q, t)|α; Q, t =
E α (Q, t)|α; Q, t, where |α; Q, t and E α (Q, t) are the αth eigenstate and eigenvalue, respectively). In this basis, the evolution of an observable χ takes the form
χ αα (Q, P , t) = T
⎧
⎨
⎩
ββ
e
i
t
t 0
dτ L t (τ )
αα ,ββ
⎫
⎬
⎭
χ ββ (Q, P , t 0 ) , (14.36)
where t 0 is the initial time, T is the time-ordering operator and
iL
t
αα ,ββ (t) = i ˜
L αα ,ββ (t) + K
t
αα ,ββ (t) .
(14.37)
In the above equation, the operator K t is responsible for the nonadiabatic transitions
in the subsystem due to the interaction with the external field, and its matrix
elements are given by
K
t
αα ,ββ = = ˙
α|βδ α β + +β| ˙
α
δ αβ ,
(14.38)
where the dot denotes a time derivative. The matrix elements of the operator i ˜
L
have a similar form to those implied in Eq. (14.7) and are given by
i ˜
L αα ,ββ (t) = i ˜
L
0
αα (t)δ αβ δ α β + ˜
K αα ,ββ (t) ,
(14.39)
where i ˜
L 0
αα (t) = i ˜
ω αα (t) + i ˜
L αα (t), with the Bohr frequency ˜
ω αα (Q, t) =
[E α (Q, t)−E α (Q, t)]/ ¯
h and the classical-like Liouville operator i ˜
L αα = (P /M)·
(∂/∂Q) + (1/2)[ ˜
F α
W (t) + ˜
F α
W (t)]·∂/∂P ( ˜
F α
W (Q, t), is the time-dependent HellmanFeynman force derived from the adiabatic energy surface E α (Q, t)); the operator ˜
K
is responsible for nonadiabatic transitions in the subsystem due to the interaction
with the bath and its matrix elements are given by
˜
K αα ,ββ (t) = ˜
T α→β (t)δ α β + ˜
T
∗
α →β (t)δ αβ ,
(14.40)
where
˜
T α→β (t) =
P
M
· d αβ (Q, t)
1 +
1
2
E αβ (t)d αβ (Q, t)
P
M · d αβ (Q, t)
∂
∂P
,
(14.41)
G. Hanna and A. Sergi
and ρ B (Q, P ), the Wigner distribution function of the bath (i.e. the resonant
harmonic mode), is given by
ρ B (Q, P ) =
tanh(βω/2)
π
exp
−
2 tanh(βω/2)
ω
P 2
2
+
ω 2 Q 2
2
. (14.35)
To simulate the quantum-classical Liouville dynamics of the Hamiltonian in
Eq. (14.33), a generalization of the SSTP algorithm to time-dependent Hamiltonians
is needed. To this end, one can introduce a time-dependent basis in terms of the
eigenstates of ˆ
h(Q, t) = ˆ
H (Q, P , t) − P 2 /2M (defined by ˆ
h(Q, t)|α; Q, t =
E α (Q, t)|α; Q, t, where |α; Q, t and E α (Q, t) are the αth eigenstate and eigenvalue, respectively). In this basis, the evolution of an observable χ takes the form
χ αα (Q, P , t) = T
⎧
⎨
⎩
ββ
e
i
t
t 0
dτ L t (τ )
αα ,ββ
⎫
⎬
⎭
χ ββ (Q, P , t 0 ) , (14.36)
where t 0 is the initial time, T is the time-ordering operator and
iL
t
αα ,ββ (t) = i ˜
L αα ,ββ (t) + K
t
αα ,ββ (t) .
(14.37)
In the above equation, the operator K t is responsible for the nonadiabatic transitions
in the subsystem due to the interaction with the external field, and its matrix
elements are given by
K
t
αα ,ββ = = ˙
α|βδ α β + +β| ˙
α
δ αβ ,
(14.38)
where the dot denotes a time derivative. The matrix elements of the operator i ˜
L
have a similar form to those implied in Eq. (14.7) and are given by
i ˜
L αα ,ββ (t) = i ˜
L
0
αα (t)δ αβ δ α β + ˜
K αα ,ββ (t) ,
(14.39)
where i ˜
L 0
αα (t) = i ˜
ω αα (t) + i ˜
L αα (t), with the Bohr frequency ˜
ω αα (Q, t) =
[E α (Q, t)−E α (Q, t)]/ ¯
h and the classical-like Liouville operator i ˜
L αα = (P /M)·
(∂/∂Q) + (1/2)[ ˜
F α
W (t) + ˜
F α
W (t)]·∂/∂P ( ˜
F α
W (Q, t), is the time-dependent HellmanFeynman force derived from the adiabatic energy surface E α (Q, t)); the operator ˜
K
is responsible for nonadiabatic transitions in the subsystem due to the interaction
with the bath and its matrix elements are given by
˜
K αα ,ββ (t) = ˜
T α→β (t)δ α β + ˜
T
∗
α →β (t)δ αβ ,
(14.40)
where
˜
T α→β (t) =
P
M
· d αβ (Q, t)
1 +
1
2
E αβ (t)d αβ (Q, t)
P
M · d αβ (Q, t)
∂
∂P
,
(14.41)