14 Simulating Quantum Dynamics in Classical Nanoscale Environments
525
14.2.4 Transition Filtering
To obtain reliable expectation values beyond short times using the SSTP algorithm,
experience has shown that one requires extremely large ensembles of trajectories.
This is mainly due to the fact that the statistical weights associated with the MC
sampling of the nonadiabatic transitions grow very rapidly in time, making it
extremely difficult to obtain converged expectation values at longer times. More
specifically, each trajectory acquires a statistical weight,
N
j =1 W j (where W j
is the MC weight for time step j ), which factors into the expectation value. In
practice, these weights grow exponentially with time [12], thereby requiring an
exponentially growing number of trajectories to converge expectation values at
longer times. To slow down the growth of these weights and thereby reduce the
number of trajectories required for convergence at longer times, observable cutting
[23] and transition filtering [12, 13, 15, 66, 79, 80] schemes have been used, both
separately and together, with varying degrees of success.
In observable cutting, one sets an upper bound on the magnitude of the weight at
each time step of the simulation, thereby ensuring that the magnitude of the weight
never becomes too large. Then, one computes the expectation value subject to this
upper bound. To determine the value of the upper bound that will yield a satisfactory
expectation value, one must repeat the calculation with increasingly larger values
of the upper bound. Using this scheme, however, it is difficult to know if one has
converged to the true result. In other words, if the upper bound is too small, the
result may appear stable but may be inaccurate. Conversely, if the upper bound is
too large, the result may be too noisy to be interpretable.
In contrast, transition filtering reduces the statistical noise by disallowing those
nonadiabatic transitions that would lead to large statistical weights. In this subsection, we will outline the transition filtering approach proposed and implemented
by Hanna et al. in Refs. [12, 13, 15]. In Refs. [12, 13], the scheme was laid out
and applied to two relatively simple systems (viz. 2 quantum states coupled to
20 classical DOF and 3 quantum states coupled to 1 classical DOF), yielding
substantial reductions in the numbers of trajectories required for convergence and
smoother and more accurate results than those obtained with observable cutting.
In Ref. [15], the scheme was applied to a model containing six quantum and six
classical DOF, in order to investigate the feasibility and efficacy of the scheme for
treating subsystems with larger numbers of states (as the number of trajectories
scales with the number of states in the quantum subsystem). In this case, it was
possible to obtain numerically stable results with more than one order of magnitude
fewer trajectories than the number of trajectories used to generate the results without
any filtering. This application will be discussed in more detail in Sect. 14.3.1.
We now outline the transition filtering scheme. To begin with, we define the
following transition probability for the MC sampling of nonadiabatic transitions
along an SSTP trajectory [12]
525
14.2.4 Transition Filtering
To obtain reliable expectation values beyond short times using the SSTP algorithm,
experience has shown that one requires extremely large ensembles of trajectories.
This is mainly due to the fact that the statistical weights associated with the MC
sampling of the nonadiabatic transitions grow very rapidly in time, making it
extremely difficult to obtain converged expectation values at longer times. More
specifically, each trajectory acquires a statistical weight,
N
j =1 W j (where W j
is the MC weight for time step j ), which factors into the expectation value. In
practice, these weights grow exponentially with time [12], thereby requiring an
exponentially growing number of trajectories to converge expectation values at
longer times. To slow down the growth of these weights and thereby reduce the
number of trajectories required for convergence at longer times, observable cutting
[23] and transition filtering [12, 13, 15, 66, 79, 80] schemes have been used, both
separately and together, with varying degrees of success.
In observable cutting, one sets an upper bound on the magnitude of the weight at
each time step of the simulation, thereby ensuring that the magnitude of the weight
never becomes too large. Then, one computes the expectation value subject to this
upper bound. To determine the value of the upper bound that will yield a satisfactory
expectation value, one must repeat the calculation with increasingly larger values
of the upper bound. Using this scheme, however, it is difficult to know if one has
converged to the true result. In other words, if the upper bound is too small, the
result may appear stable but may be inaccurate. Conversely, if the upper bound is
too large, the result may be too noisy to be interpretable.
In contrast, transition filtering reduces the statistical noise by disallowing those
nonadiabatic transitions that would lead to large statistical weights. In this subsection, we will outline the transition filtering approach proposed and implemented
by Hanna et al. in Refs. [12, 13, 15]. In Refs. [12, 13], the scheme was laid out
and applied to two relatively simple systems (viz. 2 quantum states coupled to
20 classical DOF and 3 quantum states coupled to 1 classical DOF), yielding
substantial reductions in the numbers of trajectories required for convergence and
smoother and more accurate results than those obtained with observable cutting.
In Ref. [15], the scheme was applied to a model containing six quantum and six
classical DOF, in order to investigate the feasibility and efficacy of the scheme for
treating subsystems with larger numbers of states (as the number of trajectories
scales with the number of states in the quantum subsystem). In this case, it was
possible to obtain numerically stable results with more than one order of magnitude
fewer trajectories than the number of trajectories used to generate the results without
any filtering. This application will be discussed in more detail in Sect. 14.3.1.
We now outline the transition filtering scheme. To begin with, we define the
following transition probability for the MC sampling of nonadiabatic transitions
along an SSTP trajectory [12]