body states can be computed. The sum-over-states expansions also facilitate interpretation of the resulting signal as the meaning of the various resonances is
transparent in this form. Both the correlation function and sum-over-states forms
may be displayed either in the time-domain (as a function of t 3 , t 2 , t 1 ) or the frequency
domain (as a function of the conjugate variables Ω 3 , Ω 2 , Ω 1 ). It is often useful to
employ a mixed representation, e.g., S(Ω 1 , t 2 , Ω 3 ), which is 2D in frequency and 1D
in time so that correlations between resonances observed at the two frequencies can
be observed and monitored as the time argument is allowed to vary. These techniques can therefore provide a high degree of selectivity and carry a rich abundance
of information on the electronic and nuclear structure and dynamics.
Our correlation function expressions (10) and (15) are given by the expectation
values with respect to |ψ 0 i. Alternatively, we may describe the system using the
density matrix
^
ρ ¼
X
i
P i ψ i
j i ψ i
h j;
ð30Þ
whose dynamics is determined by the Liouville equation
_
^
ρ ¼ Ài ^
H; ^
ρ
Â
à À i ^
H int ; ^
ρ
Â
Ã
:
ð31Þ
Here P i is the probability that the system is found in state |ψ i i. When all degrees of
freedom are treated at the Hamiltonian level, it is more convenient to remain in
Hilbert space rather than recasting in Liouville space (as is done in [1]). This
facilitates computations because Hilbert space has far fewer dimensions than the
associated Liouville space. In these cases, the above equations may still be utilized
formally with appropriate choice of the P i . In terms of the density matrix, the
expectation value of the dipole is given by
^
μ t
ð Þ
h
i Tr ^
μ t
ð Þ^ ρ t
ð Þ
½
;
ð32Þ
and we may expand ρ(t) perturbatively to arbitrary order in the interaction
Hamiltonian
^
ρ
n
ð Þ t
ð Þ ¼ i
ð Þ
n
ð
dr n . . . dr 1
ð t
t 0
dτ n . . .
ð τ 2
t 0
dτ 1 E r n ; τ n
ð
Þ. . . E r 1 ; τ 1
ð
Þ
 ^
μ τ n
ð Þ, . . . ; ^
μ τ 1
ð Þ, ^
ρ
½
½
. . .
½
:
ð33Þ
One can then include the effects of coupling to a bath by introducing further terms
to the equation of motion – see (30) – which represent the dissipation of system
excitations into the bath. One numerically inexpensive strategy to implement this
idea is the stochastic Liouville equation (SLE)
290
Y. Zhang et al.
transparent in this form. Both the correlation function and sum-over-states forms
may be displayed either in the time-domain (as a function of t 3 , t 2 , t 1 ) or the frequency
domain (as a function of the conjugate variables Ω 3 , Ω 2 , Ω 1 ). It is often useful to
employ a mixed representation, e.g., S(Ω 1 , t 2 , Ω 3 ), which is 2D in frequency and 1D
in time so that correlations between resonances observed at the two frequencies can
be observed and monitored as the time argument is allowed to vary. These techniques can therefore provide a high degree of selectivity and carry a rich abundance
of information on the electronic and nuclear structure and dynamics.
Our correlation function expressions (10) and (15) are given by the expectation
values with respect to |ψ 0 i. Alternatively, we may describe the system using the
density matrix
^
ρ ¼
X
i
P i ψ i
j i ψ i
h j;
ð30Þ
whose dynamics is determined by the Liouville equation
_
^
ρ ¼ Ài ^
H; ^
ρ
Â
à À i ^
H int ; ^
ρ
Â
Ã
:
ð31Þ
Here P i is the probability that the system is found in state |ψ i i. When all degrees of
freedom are treated at the Hamiltonian level, it is more convenient to remain in
Hilbert space rather than recasting in Liouville space (as is done in [1]). This
facilitates computations because Hilbert space has far fewer dimensions than the
associated Liouville space. In these cases, the above equations may still be utilized
formally with appropriate choice of the P i . In terms of the density matrix, the
expectation value of the dipole is given by
^
μ t
ð Þ
h
i Tr ^
μ t
ð Þ^ ρ t
ð Þ
½
;
ð32Þ
and we may expand ρ(t) perturbatively to arbitrary order in the interaction
Hamiltonian
^
ρ
n
ð Þ t
ð Þ ¼ i
ð Þ
n
ð
dr n . . . dr 1
ð t
t 0
dτ n . . .
ð τ 2
t 0
dτ 1 E r n ; τ n
ð
Þ. . . E r 1 ; τ 1
ð
Þ
 ^
μ τ n
ð Þ, . . . ; ^
μ τ 1
ð Þ, ^
ρ
½
½
. . .
½
:
ð33Þ
One can then include the effects of coupling to a bath by introducing further terms
to the equation of motion – see (30) – which represent the dissipation of system
excitations into the bath. One numerically inexpensive strategy to implement this
idea is the stochastic Liouville equation (SLE)
290
Y. Zhang et al.