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G. Hanna and A. Sergi
approximation breaks down. However, mean-field and FSSH dynamics suffer from
several drawbacks. Both methods do not correctly describe the “quantum backreaction”, i.e. the effect due to the subsystem-environment coupling, whereby a
change in the environmental DOF modifies the subsystem Hamiltonian, which in
turn modifies the forces governing the motion of the environmental DOF. Meanfield dynamics is not capable of capturing decoherence, while FSSH can do so
after incorporating some ad hoc corrections [68, 71]. Both methods also struggle
to satisfy detailed balance. Due to the incorrect nature of the energy transfer
between the subsystem and environment, it is possible for the subsystem to approach
unrealistically high temperatures in mean-field dynamics [55]; in such cases, one
can employ quantum correction factors that modify the dynamics in such a way that
detailed balance is enforced [4]. In FSSH, it is possible to satisfy detailed balance
in certain limits if the so-called frustrated hops are allowed [64].
In this chapter, we focus on a mixed quantum-classical dynamics approach that
is based on the quantum-classical Liouville equation (QCLE) [2, 22, 33, 89], which
can be derived from the quantum Liouville equation by first taking its partial Wigner
transform [87] over the bath DOF (leading to a description of the subsystem and
bath DOF in terms of operators and phase space variables, respectively) and then
truncating the resulting equation after first order in ¯
h. The QCLE constitutes an ideal
starting point [31, 32] for deriving mixed quantum-classical dynamics algorithms
because (i) it can be rigorously derived from the more fundamental quantum
Liouville equation based on a well-defined and physically reasonable approximation, and (ii) it yields the exact quantum dynamics for quantum subsystems
that are bilinearly coupled to harmonic environments [45], which are frequently
used models in chemical and condensed matter physics research. A large number
of QCLE-based algorithms have been developed over the years [16–18, 23, 26–
28, 34, 36, 37, 40, 41, 44, 46, 47, 49, 53, 63, 82, 83]. This chapter will deal with a
particular algorithm known as sequential short-time propagation (SSTP) [23, 47],
due to its favourable compromise between accuracy and ease of implementation.
SSTP is a hybrid molecular dynamics/Monte Carlo algorithm, which relies on the
so-called momentum jump approximation [23, 65] and solves the QCLE in terms of
an ensemble of surface-hopping trajectories. However, a direct application of this
algorithm has proven to be very challenging due to severe numerical instabilities
that arise beyond short times when the subsystem-bath coupling strength is not
very weak. In this case, the statistical weights associated with the Monte Carlo
sampling of the nonadiabatic transitions (and that enter into the calculation of the
observable) grow very rapidly in time, thereby necessitating very large numbers
of trajectories to obtain converged expectation values. Transition filtering schemes
[12, 13, 15, 23, 66, 79, 80], which impede the growth of these weights by filtering
out the low-probability nonadiabatic transitions, have been shown to significantly
alleviate the numerical instabilities. We will discuss one such scheme [12, 13] in
this chapter.
In most mixed quantum-classical surface-hopping approaches (including the
SSTP algorithm), one must evaluate the adiabatic states and nonadiabatic couplings
between them at each time step of the simulation. In most cases, they are evaluated
G. Hanna and A. Sergi
approximation breaks down. However, mean-field and FSSH dynamics suffer from
several drawbacks. Both methods do not correctly describe the “quantum backreaction”, i.e. the effect due to the subsystem-environment coupling, whereby a
change in the environmental DOF modifies the subsystem Hamiltonian, which in
turn modifies the forces governing the motion of the environmental DOF. Meanfield dynamics is not capable of capturing decoherence, while FSSH can do so
after incorporating some ad hoc corrections [68, 71]. Both methods also struggle
to satisfy detailed balance. Due to the incorrect nature of the energy transfer
between the subsystem and environment, it is possible for the subsystem to approach
unrealistically high temperatures in mean-field dynamics [55]; in such cases, one
can employ quantum correction factors that modify the dynamics in such a way that
detailed balance is enforced [4]. In FSSH, it is possible to satisfy detailed balance
in certain limits if the so-called frustrated hops are allowed [64].
In this chapter, we focus on a mixed quantum-classical dynamics approach that
is based on the quantum-classical Liouville equation (QCLE) [2, 22, 33, 89], which
can be derived from the quantum Liouville equation by first taking its partial Wigner
transform [87] over the bath DOF (leading to a description of the subsystem and
bath DOF in terms of operators and phase space variables, respectively) and then
truncating the resulting equation after first order in ¯
h. The QCLE constitutes an ideal
starting point [31, 32] for deriving mixed quantum-classical dynamics algorithms
because (i) it can be rigorously derived from the more fundamental quantum
Liouville equation based on a well-defined and physically reasonable approximation, and (ii) it yields the exact quantum dynamics for quantum subsystems
that are bilinearly coupled to harmonic environments [45], which are frequently
used models in chemical and condensed matter physics research. A large number
of QCLE-based algorithms have been developed over the years [16–18, 23, 26–
28, 34, 36, 37, 40, 41, 44, 46, 47, 49, 53, 63, 82, 83]. This chapter will deal with a
particular algorithm known as sequential short-time propagation (SSTP) [23, 47],
due to its favourable compromise between accuracy and ease of implementation.
SSTP is a hybrid molecular dynamics/Monte Carlo algorithm, which relies on the
so-called momentum jump approximation [23, 65] and solves the QCLE in terms of
an ensemble of surface-hopping trajectories. However, a direct application of this
algorithm has proven to be very challenging due to severe numerical instabilities
that arise beyond short times when the subsystem-bath coupling strength is not
very weak. In this case, the statistical weights associated with the Monte Carlo
sampling of the nonadiabatic transitions (and that enter into the calculation of the
observable) grow very rapidly in time, thereby necessitating very large numbers
of trajectories to obtain converged expectation values. Transition filtering schemes
[12, 13, 15, 23, 66, 79, 80], which impede the growth of these weights by filtering
out the low-probability nonadiabatic transitions, have been shown to significantly
alleviate the numerical instabilities. We will discuss one such scheme [12, 13] in
this chapter.
In most mixed quantum-classical surface-hopping approaches (including the
SSTP algorithm), one must evaluate the adiabatic states and nonadiabatic couplings
between them at each time step of the simulation. In most cases, they are evaluated