14 Simulating Quantum Dynamics in Classical Nanoscale Environments
523
leading to imaginary momentum changes. If this occurs, the nonadiabatic transition
is forbidden, and the trajectory is continued adiabatically. It should be noted that
the approximation made in the second line of Eq. (14.12) relies on | ¯
hω kj d kj c
−1
kj /2|
being small. Also, the total energy of the system is conserved along a trajectory
under the MJA.
The expectation value of ˆ
χ(Q, P , t) is given by
ˆ
χ(t)=
kk
dQdP χ k k (Q, P )ρ kk (Q, P , t)
=
(k 0 k
0 )...(k N k
N )
dQdP χ k
0 k 0 (Q, P )
N
j =1
(e
−i ˆ
Lt j ) k j −1 k
j −1 ,k j k
j
ρ k N k
N
(Q, P ),
(14.14)
or, in the Heisenberg representation, by
ˆ
χ(t) =
kk
dQdP χ kk (Q, P , t)ρ k k (Q, P )
=
(k 0 k
0 )...(k N k
N )
dQdPρ k
0 k 0 (Q, P )
N
j =1
(e
i ˆ
Lt j ) k j −1 k
j −1 ,k j k
j
χ k N k
N
(Q, P ).
(14.15)
The SSTP algorithm evaluates Eq. (14.15) via a hybrid molecular dynamics
(MD)/Monte Carlo (MC) approach, whereby the multidimensional sums over
the state indices are evaluated using MC sampling, the initial conditions of
the subsystem and bath are determined from ρ k
0 k 0 (Q, P ), and χ kk (Q, P , t) is
propagated in time using MD simulation.
We now detail the steps of the SSTP algorithm for evaluating Eq. (14.15) by
considering the evolution through a single time step (i.e. N = 1):
ˆ
χ((t) =
(k 0 k
0 )(k 1 k
1 )
dQdPρ k
0 k 0 (Q, P )(e
i ˆ
Lt j ) k 0 k
0 ,k 1 k
1
χ k 1 k
1
(Q, P ).
=
(k 0 k
0 )(k 1 k
1 )
dQdPρ k
0 k 0 (Q, P )W k
0 k 0 (t 0 , t 1 )e
iL k 0 k
0
t 1
×
δ k 0 k 1 δ k
0 k
1
+ tK k 0 k
0 ,k 1 k
1
χ k 1 k
1
(Q, P ).
(14.16)