530
G. Hanna and A. Sergi
Fig. 14.4), one sees continuous decays starting from O αα = ±1, followed by two
blips centred at O αα ≈ ±0.5, and a peak centred at O αα = 0 (which is presumably
due to character switches in the eigenstates after a nonadiabatic transition). The
discontinuities at O αα ≈ ±0.5 are suggestive of the onset of eigenvector rotations
near an avoided crossing or conical intersection and thereby point to an optimal
value that is slightly less than ≈ 0.5.
To validate the accuracy of the expectation values calculated using the streamlined version of the eigenvector sign correction procedure, one could repeat the
calculation with a reduced time step and compare the results to those obtained with
the larger time step. (For the system considered in Ref. [14], it was found that the
percentage of the total time that the m t (α) = 2 case arises for a particular choice
of decreases by about two orders of magnitude when the time step is decreased
by at least a factor of ten.) If there is no significant change in the results, then the
streamlined method is justified.
Finally, to speed up the generation of the ensemble of trajectories, one can split
up the work onto many computer processors. As a result, however, one can no
longer easily ensure eigenvector sign continuity across the entire ensemble, but
only within the sub-ensemble of trajectories generated on a given processor. In
Ref. [14], the authors confirmed that the final expectation values were not adversely
affected by the parallelization by checking what happens after randomizing the
initial eigenvector signs every n trajectories on each processor. This check revealed
that for the system considered, one must run at least ≈400 trajectories per processor
to recover the fully sign-corrected results.
14.3 Applications
14.3.1 Vibrational Energy Transfer in an Alpha-Helical
Polypeptide
We now present an application in which the utility of the previously discussed
transition filtering scheme is demonstrated for simulating vibrational energy transfer
in a six-state model of an alpha-helical polypeptide [15]. In this case, the vibrational
energy transfer is monitored by computing the time-dependent amide I mode
populations, following a vibrational excitation at one end of the polypeptide.
The aforementioned model, portrayed in Fig. 14.5, involves a one-dimensional
chain of hydrogen-bonded (H-bonded) peptide groups, constituting one of the three
peptide chains in an alpha-helical polypeptide. Adopting a mixed quantum-classical
description, the high-frequency amide I modes are treated quantum mechanically,
while the low-frequency H-bond displacement coordinates to which the amide I
modes are coupled are treated classically. The Hamiltonian of this system is given
by
H = H 0 |00| +
m,n
H mn |φ m φ n |,
(14.26)
G. Hanna and A. Sergi
Fig. 14.4), one sees continuous decays starting from O αα = ±1, followed by two
blips centred at O αα ≈ ±0.5, and a peak centred at O αα = 0 (which is presumably
due to character switches in the eigenstates after a nonadiabatic transition). The
discontinuities at O αα ≈ ±0.5 are suggestive of the onset of eigenvector rotations
near an avoided crossing or conical intersection and thereby point to an optimal
value that is slightly less than ≈ 0.5.
To validate the accuracy of the expectation values calculated using the streamlined version of the eigenvector sign correction procedure, one could repeat the
calculation with a reduced time step and compare the results to those obtained with
the larger time step. (For the system considered in Ref. [14], it was found that the
percentage of the total time that the m t (α) = 2 case arises for a particular choice
of decreases by about two orders of magnitude when the time step is decreased
by at least a factor of ten.) If there is no significant change in the results, then the
streamlined method is justified.
Finally, to speed up the generation of the ensemble of trajectories, one can split
up the work onto many computer processors. As a result, however, one can no
longer easily ensure eigenvector sign continuity across the entire ensemble, but
only within the sub-ensemble of trajectories generated on a given processor. In
Ref. [14], the authors confirmed that the final expectation values were not adversely
affected by the parallelization by checking what happens after randomizing the
initial eigenvector signs every n trajectories on each processor. This check revealed
that for the system considered, one must run at least ≈400 trajectories per processor
to recover the fully sign-corrected results.
14.3 Applications
14.3.1 Vibrational Energy Transfer in an Alpha-Helical
Polypeptide
We now present an application in which the utility of the previously discussed
transition filtering scheme is demonstrated for simulating vibrational energy transfer
in a six-state model of an alpha-helical polypeptide [15]. In this case, the vibrational
energy transfer is monitored by computing the time-dependent amide I mode
populations, following a vibrational excitation at one end of the polypeptide.
The aforementioned model, portrayed in Fig. 14.5, involves a one-dimensional
chain of hydrogen-bonded (H-bonded) peptide groups, constituting one of the three
peptide chains in an alpha-helical polypeptide. Adopting a mixed quantum-classical
description, the high-frequency amide I modes are treated quantum mechanically,
while the low-frequency H-bond displacement coordinates to which the amide I
modes are coupled are treated classically. The Hamiltonian of this system is given
by
H = H 0 |00| +
m,n
H mn |φ m φ n |,
(14.26)