14 Simulating Quantum Dynamics in Classical Nanoscale Environments
527
14.2.5 Eigenvector Sign Correction
In nonadiabatic dynamics methods such as the SSTP algorithm, the adiabatic states
and nonadiabatic couplings between them must be evaluated on-the-fly. Typically,
they are evaluated numerically, which first involves a numerical diagonalization of
the Hamiltonian matrix (represented in a convenient basis). The diagonalization
results in eigenvectors that are uniquely determined up to a sign, which leads to
arbitrary signs of the nonadiabatic coupling vectors. The signs may be corrected
by requiring that the eigenvectors change continuously from one time step to
the next. However, when a system is in the vicinity of an avoided crossing or
conical intersection between two PESs, it becomes more complicated to correctly
distinguish between physical and unphysical sign changes in the eigenvectors.
Previously, Errea et al. proposed a procedure for correcting the sign of an electronic
state, which involves tracking the sign of the overlap between the state evaluated at
neighbouring points in a grid of nuclear positions [19]. This procedure also accounts
for situations in which the system is in the vicinity of an avoided crossing or conical
intersections.
The procedure put forward in Ref. [19] focuses on ensuring the continuous
evolution of eigenvectors and nonadiabatic coupling vectors along a single trajectory
in configuration space. However, when computing time-dependent expectation
values based on an ensemble of trajectories (which are typically initialized at
different points in configuration space), one is faced with the issue of how to
ensure eigenvector sign continuity within the ensemble. In Ref. [14], building upon
elements of Errea et al.’s method, Hanna et al. proposed a simple procedure that
accomplishes this within the context of mixed quantum-classical surface-hopping
dynamics simulations. The utility of this procedure was demonstrated by calculating
both adiabatic state populations in a reduced model of a condensed phase protoncoupled electron transfer reaction and subsystem state populations in an excitonic
model of an alpha-helical chain undergoing vibrational energy transport, using the
SSTP algorithm. More specifically, the results showed that ensuring eigenvector
sign continuity across the ensemble leads to minor (but significant) improvements
in the accuracy of expectation values in some cases and to drastic improvements in
other cases.
We now outline the procedure proposed in Ref. [14]. To properly evaluate the
expression for a time-dependent expectation value in Eq. (14.15), the sign continuity
of the eigenvectors must be preserved across all of the points in the configurational
space (of the classical DOF) visited by an ensemble of trajectories, as opposed to
the points within a single trajectory. This may be accomplished by first sampling
the initial conditions of the classical DOF for the first microcanonical (i.e. NVE)
trajectory from a canonical (i.e. NVT) trajectory, in which the eigenvector signs are
tracked and corrected at each time step according to the following steps:
1. Compute O αα (t) = =α; Q t |α; Q t−t (i.e. the overlap of eigenvector α with
eigenvector α of the previous time step) for each α. When the classical
configuration Q t−t is not close to an avoided crossing or conical intersection
527
14.2.5 Eigenvector Sign Correction
In nonadiabatic dynamics methods such as the SSTP algorithm, the adiabatic states
and nonadiabatic couplings between them must be evaluated on-the-fly. Typically,
they are evaluated numerically, which first involves a numerical diagonalization of
the Hamiltonian matrix (represented in a convenient basis). The diagonalization
results in eigenvectors that are uniquely determined up to a sign, which leads to
arbitrary signs of the nonadiabatic coupling vectors. The signs may be corrected
by requiring that the eigenvectors change continuously from one time step to
the next. However, when a system is in the vicinity of an avoided crossing or
conical intersection between two PESs, it becomes more complicated to correctly
distinguish between physical and unphysical sign changes in the eigenvectors.
Previously, Errea et al. proposed a procedure for correcting the sign of an electronic
state, which involves tracking the sign of the overlap between the state evaluated at
neighbouring points in a grid of nuclear positions [19]. This procedure also accounts
for situations in which the system is in the vicinity of an avoided crossing or conical
intersections.
The procedure put forward in Ref. [19] focuses on ensuring the continuous
evolution of eigenvectors and nonadiabatic coupling vectors along a single trajectory
in configuration space. However, when computing time-dependent expectation
values based on an ensemble of trajectories (which are typically initialized at
different points in configuration space), one is faced with the issue of how to
ensure eigenvector sign continuity within the ensemble. In Ref. [14], building upon
elements of Errea et al.’s method, Hanna et al. proposed a simple procedure that
accomplishes this within the context of mixed quantum-classical surface-hopping
dynamics simulations. The utility of this procedure was demonstrated by calculating
both adiabatic state populations in a reduced model of a condensed phase protoncoupled electron transfer reaction and subsystem state populations in an excitonic
model of an alpha-helical chain undergoing vibrational energy transport, using the
SSTP algorithm. More specifically, the results showed that ensuring eigenvector
sign continuity across the ensemble leads to minor (but significant) improvements
in the accuracy of expectation values in some cases and to drastic improvements in
other cases.
We now outline the procedure proposed in Ref. [14]. To properly evaluate the
expression for a time-dependent expectation value in Eq. (14.15), the sign continuity
of the eigenvectors must be preserved across all of the points in the configurational
space (of the classical DOF) visited by an ensemble of trajectories, as opposed to
the points within a single trajectory. This may be accomplished by first sampling
the initial conditions of the classical DOF for the first microcanonical (i.e. NVE)
trajectory from a canonical (i.e. NVT) trajectory, in which the eigenvector signs are
tracked and corrected at each time step according to the following steps:
1. Compute O αα (t) = =α; Q t |α; Q t−t (i.e. the overlap of eigenvector α with
eigenvector α of the previous time step) for each α. When the classical
configuration Q t−t is not close to an avoided crossing or conical intersection