14 Simulating Quantum Dynamics in Classical Nanoscale Environments
519
numerically, which first entails a numerical diagonalization of the Hamiltonian
matrix. However, the resulting eigenvectors are determined uniquely up to a
sign, which leads to an arbitrariness in the signs of the nonadiabatic coupling
vectors. These signs must be corrected by requiring that the eigenvectors change
continuously from one point to the next along a trajectory in configuration space.
However, the situation is further complicated when one is interested in computing
time-dependent expectation values based on an ensemble of trajectories (which are
typically initialized at different points in the configuration space). In this chapter,
we outline a recently proposed method [14] for ensuring eigenvector sign continuity
across an ensemble of trajectories in surface-hopping simulations.
To illustrate the utility of the aforementioned methods for simulating the
dynamics of mixed quantum-classical systems, we present two applications. The
first one demonstrates how transition filtering can be used to facilitate the calculation
of amide I mode populations in a six-site model of an alpha-helical polypeptide,
after a vibrational excitation at one end of the chain. The second one involves
calculations of populations and coherences in a reduced model of a plasmonic
metamolecule in an external field, using a generalized version of the SSTP algorithm
for simulations of systems with time-dependent Hamiltonians.
The chapter is organized as follows. In Sect. 14.2, we present the QCLE, discuss
how it can be solved via the SSTP algorithm, and explain how expectation values of
observables can be computed. Recently proposed schemes for transition filtering and
eigenvector sign correction are also outlined in this section. In Sect. 14.3, we present
two recent applications of the aforementioned techniques to simulating vibrational
energy transfer in an alpha-helical polypeptide and the field-driven dynamics of a
plasmonic metamolecule. Our summary and future outlook are given in Sect. 14.4.
14.2 Mixed Quantum-Classical Liouville Dynamics
14.2.1 The Quantum-Classical Liouville Equation
We begin by considering a quantum subsystem, whose Hamiltonian operator is ˆ
H S ,
embedded in a classical environment, whose Hamiltonian function is H B (Q, P ) =
P 2
2M + V (Q). The coordinates (Q, P ) denote the positions and momenta, respectively, of the atoms or molecular fragments (characterized by masses M and
interacting according to the potential energy function V (Q)) in the environment.
The coupling between the quantum subsystem and classical environment is governed by the operator ˆ
H C (Q), which is also a function of the configuration of the
environment. (In this chapter, we do not consider situations where the subsystem
interacts with a magnetic field, such that the coupling Hamiltonian depends only on
the configuration.) Hence, the Hamiltonian of the total system is given by
ˆ
H (Q, P ) = ˆ
H S + H B (Q, P ) + ˆ
H C (Q) ,
(14.1)
519
numerically, which first entails a numerical diagonalization of the Hamiltonian
matrix. However, the resulting eigenvectors are determined uniquely up to a
sign, which leads to an arbitrariness in the signs of the nonadiabatic coupling
vectors. These signs must be corrected by requiring that the eigenvectors change
continuously from one point to the next along a trajectory in configuration space.
However, the situation is further complicated when one is interested in computing
time-dependent expectation values based on an ensemble of trajectories (which are
typically initialized at different points in the configuration space). In this chapter,
we outline a recently proposed method [14] for ensuring eigenvector sign continuity
across an ensemble of trajectories in surface-hopping simulations.
To illustrate the utility of the aforementioned methods for simulating the
dynamics of mixed quantum-classical systems, we present two applications. The
first one demonstrates how transition filtering can be used to facilitate the calculation
of amide I mode populations in a six-site model of an alpha-helical polypeptide,
after a vibrational excitation at one end of the chain. The second one involves
calculations of populations and coherences in a reduced model of a plasmonic
metamolecule in an external field, using a generalized version of the SSTP algorithm
for simulations of systems with time-dependent Hamiltonians.
The chapter is organized as follows. In Sect. 14.2, we present the QCLE, discuss
how it can be solved via the SSTP algorithm, and explain how expectation values of
observables can be computed. Recently proposed schemes for transition filtering and
eigenvector sign correction are also outlined in this section. In Sect. 14.3, we present
two recent applications of the aforementioned techniques to simulating vibrational
energy transfer in an alpha-helical polypeptide and the field-driven dynamics of a
plasmonic metamolecule. Our summary and future outlook are given in Sect. 14.4.
14.2 Mixed Quantum-Classical Liouville Dynamics
14.2.1 The Quantum-Classical Liouville Equation
We begin by considering a quantum subsystem, whose Hamiltonian operator is ˆ
H S ,
embedded in a classical environment, whose Hamiltonian function is H B (Q, P ) =
P 2
2M + V (Q). The coordinates (Q, P ) denote the positions and momenta, respectively, of the atoms or molecular fragments (characterized by masses M and
interacting according to the potential energy function V (Q)) in the environment.
The coupling between the quantum subsystem and classical environment is governed by the operator ˆ
H C (Q), which is also a function of the configuration of the
environment. (In this chapter, we do not consider situations where the subsystem
interacts with a magnetic field, such that the coupling Hamiltonian depends only on
the configuration.) Hence, the Hamiltonian of the total system is given by
ˆ
H (Q, P ) = ˆ
H S + H B (Q, P ) + ˆ
H C (Q) ,
(14.1)