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G. Hanna and A. Sergi
14.4 Summary and Future Outlook
Quantum dynamical modelling of nanoscale systems containing large numbers of
DOF is essential for understanding their microscopic and macroscopic properties.
Adiabatic mixed quantum-classical dynamics simulations, which rely on the BornOppenheimer approximation, may be performed when there is a large separation
between the timescales of the quantum subsystem and classical environment.
However, for many chemical and biological processes of interest, this is not
the case, and, as a result, the dynamics is not confined to a single state of
the quantum subsystem. This has prompted the development of mixed quantumclassical methods for simulating nonadiabatic dynamics.
In this chapter, we discussed a hybrid MD/MC method known as the SSTP
algorithm, which is based on a solution of the QCLE. This method has been
shown to perform quite well on relatively simple model systems, but its extension
to systems containing multiple (i.e. >2) quantum DOF and strong subsystembath coupling has proven to be challenging. This is primarily due to the rapid
growth of MC weights (associated with the stochastic sampling of the nonadiabatic
transitions) in time, which render the expectation values difficult to converge. To
alleviate this problem, observable cutting and transition filtering techniques have
been put forward, which significantly reduce the number of trajectories required for
convergence of the expectation values. In this chapter, we discussed one transition
filtering scheme, which is designed to filter out the low-probability transitions
(which lead to the large MC weights) and thereby slow down the growth of the
weights. This scheme has been shown to yield substantial improvements in accuracy,
as compared to the other schemes, and to generate results with at least one order of
magnitude fewer trajectories than what would be required without the scheme. We
also discussed a simple procedure for ensuring sign continuity/consistency in the
evolution of eigenvectors within an ensemble of surface-hopping trajectories. This is
required because the on-the-fly numerical diagonalization of the Hamiltonian matrix
can yield eigenvectors with arbitrary signs. Without such a scheme, trajectory-based
calculations of expectation values in the adiabatic representation can be adversely
affected.
This chapter also covered two recent applications of the SSTP algorithm.
The first one demonstrated the efficacy of the aforementioned transition filtering
scheme for simulating vibrational energy transfer dynamics in systems with higher
dimensional Hilbert spaces and phase spaces. In particular, the time-dependent
populations of the amide I modes in a model of an alpha-helical polypeptide
(containing six quantum and six classical coordinates) were presented. The results
showed that it is possible to obtain smooth, reliable profiles with more than one
order of magnitude fewer trajectories compared to those obtained without any
filtering. In the second application, we presented a dynamical study of a model
for a plasmonic metamolecule interacting with an oscillatory driving field. In this
case, the Hamiltonian is explicitly time-dependent, and, thus, the SSTP algorithm
had to be generalized to account for this fact. Results showing the effects of the
G. Hanna and A. Sergi
14.4 Summary and Future Outlook
Quantum dynamical modelling of nanoscale systems containing large numbers of
DOF is essential for understanding their microscopic and macroscopic properties.
Adiabatic mixed quantum-classical dynamics simulations, which rely on the BornOppenheimer approximation, may be performed when there is a large separation
between the timescales of the quantum subsystem and classical environment.
However, for many chemical and biological processes of interest, this is not
the case, and, as a result, the dynamics is not confined to a single state of
the quantum subsystem. This has prompted the development of mixed quantumclassical methods for simulating nonadiabatic dynamics.
In this chapter, we discussed a hybrid MD/MC method known as the SSTP
algorithm, which is based on a solution of the QCLE. This method has been
shown to perform quite well on relatively simple model systems, but its extension
to systems containing multiple (i.e. >2) quantum DOF and strong subsystembath coupling has proven to be challenging. This is primarily due to the rapid
growth of MC weights (associated with the stochastic sampling of the nonadiabatic
transitions) in time, which render the expectation values difficult to converge. To
alleviate this problem, observable cutting and transition filtering techniques have
been put forward, which significantly reduce the number of trajectories required for
convergence of the expectation values. In this chapter, we discussed one transition
filtering scheme, which is designed to filter out the low-probability transitions
(which lead to the large MC weights) and thereby slow down the growth of the
weights. This scheme has been shown to yield substantial improvements in accuracy,
as compared to the other schemes, and to generate results with at least one order of
magnitude fewer trajectories than what would be required without the scheme. We
also discussed a simple procedure for ensuring sign continuity/consistency in the
evolution of eigenvectors within an ensemble of surface-hopping trajectories. This is
required because the on-the-fly numerical diagonalization of the Hamiltonian matrix
can yield eigenvectors with arbitrary signs. Without such a scheme, trajectory-based
calculations of expectation values in the adiabatic representation can be adversely
affected.
This chapter also covered two recent applications of the SSTP algorithm.
The first one demonstrated the efficacy of the aforementioned transition filtering
scheme for simulating vibrational energy transfer dynamics in systems with higher
dimensional Hilbert spaces and phase spaces. In particular, the time-dependent
populations of the amide I modes in a model of an alpha-helical polypeptide
(containing six quantum and six classical coordinates) were presented. The results
showed that it is possible to obtain smooth, reliable profiles with more than one
order of magnitude fewer trajectories compared to those obtained without any
filtering. In the second application, we presented a dynamical study of a model
for a plasmonic metamolecule interacting with an oscillatory driving field. In this
case, the Hamiltonian is explicitly time-dependent, and, thus, the SSTP algorithm
had to be generalized to account for this fact. Results showing the effects of the