524
G. Hanna and A. Sergi
1. After specifying the initial density matrix, ρ k
0 k 0 (Q, P ), of the system, sample
the initial conditions from |ρ k
0 k 0 (Q, P )| using an appropriate scheme.
2. Propagate the bath DOF adiabatically on the adiabatic PES corresponding to the
pair of indices (k 0 k
0 ), i.e. (( k 0 + k
0
)/2, for a time interval t, and calculate the
phase factor W k
0 k 0 (0, ,t) for this interval.
3. According to Eq. (14.9) for K k 0 k
0 k 1 k
1
, determine stochastically whether or not a
nonadiabatic transition (k 0 k
0 ) → (k 1 k
1 ) takes place. One way of sampling the
nonadiabatic transitions involves first identifying which of the two terms in the
expression for K k 0 k
0 ,k 1 k
1
acts (viz. the first term changes the index k 0 → k 1 ,
while the second changes the index k
0 → k
1 ). This can be accomplished using
MC by assigning a 50% probability to selecting each term. Once one of these
terms has been selected, the actual transition is MC sampled with a probability
P kj =
t|d kj ·
P
M |
1 + t|d kj ·
P
M |
.
(14.17)
The probability of remaining in the same adiabatic state (i.e. no nonadiabatic
transition) is therefore
Q kj =
1
1 + t|d kj ·
P
M |
,
(14.18)
satisfying P kj + Q kj = 1. If the transition is accepted, then the momenta are
shifted according to Eqs. (14.12) and (14.13), and the value of the observable
becomes
χ k 1 k
1
((t) = 2W k 0 k
0
(0, ,t)d kj ·
P t
M
χ k 0 k
0
(Q t , P t +P )
1
P kj
. (14.19)
In the above equation, the factor of 2 is the MC weight associated with the
sampling of the k 0 → k 1 or k
0 → k
1 term in K k 0 k
0 ,k 1 k
1
(which each occur
with a probability of 50%), and 1/P kj is the MC weight associated with the
sampling of the transition. On the other hand, if the transition is rejected, there is
no momentum shift, and the value of the observable becomes
χ k 1 k
1
((t) = W k 0 k
0
(0, ,t)χ k 0 k
0
(Q t , P t )
1
Q kj
,
(14.20)
where 1/Q kj is the MC weight associated with no transition.
4. Repeat steps 2–3 until time t to calculate χ k N k
N
(t).
5. Multiply χ k N k
N
(t) by the sign of the density matrix, and average the resulting
value over the initial conditions to obtain ˆ
χ(t).
G. Hanna and A. Sergi
1. After specifying the initial density matrix, ρ k
0 k 0 (Q, P ), of the system, sample
the initial conditions from |ρ k
0 k 0 (Q, P )| using an appropriate scheme.
2. Propagate the bath DOF adiabatically on the adiabatic PES corresponding to the
pair of indices (k 0 k
0 ), i.e. (( k 0 + k
0
)/2, for a time interval t, and calculate the
phase factor W k
0 k 0 (0, ,t) for this interval.
3. According to Eq. (14.9) for K k 0 k
0 k 1 k
1
, determine stochastically whether or not a
nonadiabatic transition (k 0 k
0 ) → (k 1 k
1 ) takes place. One way of sampling the
nonadiabatic transitions involves first identifying which of the two terms in the
expression for K k 0 k
0 ,k 1 k
1
acts (viz. the first term changes the index k 0 → k 1 ,
while the second changes the index k
0 → k
1 ). This can be accomplished using
MC by assigning a 50% probability to selecting each term. Once one of these
terms has been selected, the actual transition is MC sampled with a probability
P kj =
t|d kj ·
P
M |
1 + t|d kj ·
P
M |
.
(14.17)
The probability of remaining in the same adiabatic state (i.e. no nonadiabatic
transition) is therefore
Q kj =
1
1 + t|d kj ·
P
M |
,
(14.18)
satisfying P kj + Q kj = 1. If the transition is accepted, then the momenta are
shifted according to Eqs. (14.12) and (14.13), and the value of the observable
becomes
χ k 1 k
1
((t) = 2W k 0 k
0
(0, ,t)d kj ·
P t
M
χ k 0 k
0
(Q t , P t +P )
1
P kj
. (14.19)
In the above equation, the factor of 2 is the MC weight associated with the
sampling of the k 0 → k 1 or k
0 → k
1 term in K k 0 k
0 ,k 1 k
1
(which each occur
with a probability of 50%), and 1/P kj is the MC weight associated with the
sampling of the transition. On the other hand, if the transition is rejected, there is
no momentum shift, and the value of the observable becomes
χ k 1 k
1
((t) = W k 0 k
0
(0, ,t)χ k 0 k
0
(Q t , P t )
1
Q kj
,
(14.20)
where 1/Q kj is the MC weight associated with no transition.
4. Repeat steps 2–3 until time t to calculate χ k N k
N
(t).
5. Multiply χ k N k
N
(t) by the sign of the density matrix, and average the resulting
value over the initial conditions to obtain ˆ
χ(t).