528
G. Hanna and A. Sergi
and the time step is sufficiently small, the eigenvectors are not expected to rotate
by much. In this case, the O αα ’s will be close to 1 (assuming that the eigenvectors
are normalized). However, if an eigenvector undergoes an arbitrary sign flip, then
the O αα ’s will be close to -1. In contrast, when Q t−t is close to an avoided
crossing or conical intersection, the eigenvectors may vary rapidly, leading to
O αα ’s that are substantially lower than 1 and possibly even negative. In this
case, if an eigenvector undergoes an arbitrary sign flip, then O αα could lie in
the interval −δ < O αα < δ, where |δ| is substantially lower than 1 (e.g. 0.7 or
lower).
2. Determine the value of m t (α) for all α’s, according to
m t (α) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0 if |O αα (t) − 1| < <
1 if |O αα (t) + 1| < <
2 otherwise
(14.24)
where is a threshold parameter (with 0 < < < 1) that depends on the size
of the time step, t, and the details of the PESs. The m t (α) = 0 and 1 cases
would typically arise when Q t−t is not in the vicinity of an avoided crossing
or conical intersection and the eigenvector undergoes a physical rotation and
unphysical sign flip, respectively. The m t (α) = 2 case would typically arise
when Q t−t is in the vicinity of an avoided crossing or conical intersection. In
this case, one must return the system to the end of the previous time step, divide
the current time step in half, and evolve the system with the reduced time step.
This process is repeated until m t (α) becomes either 0 or 1. A sensible choice of
(in combination with a sufficiently small t) should yield m t (α) = 0, 1 most
of the time (in order to minimize the number of time steps needed).
3. Correct the signs of all eigenvectors according to
|α; Q t
c
= (−1)
m t (α)
|α; Q t ,
(14.25)
i.e. if m t (α) = 0, then the sign is not changed, and if m t (α) = 1, then the sign is
flipped to ensure the continuous evolution of the eigenvector.
At the end of this NVT dynamics phase, the eigenvectors are stored in memory
(i) to continue the sign tracking and correction during the first NVE trajectory
in the ensemble and (ii) to continue the sign tracking and correction during the
second NVT dynamics phase preceding the second NVE trajectory. This procedure
is repeated for the remaining trajectories in the ensemble (see Fig. 14.3 for an
illustration of this procedure). In this way, the basis coefficients in |α; Q =
i c α
i (Q)|φ i (where {|φ i } is a basis set of choice), which may enter explicitly
into the sampling of ρ α α (Q, P ) and into the dynamics of χ αα (Q, P , t), will evolve
continuously throughout the configurational space.
For systems that do not spend a significant amount of time in the vicinity of
an avoided crossing or conical intersection, one can reduce the computational time
G. Hanna and A. Sergi
and the time step is sufficiently small, the eigenvectors are not expected to rotate
by much. In this case, the O αα ’s will be close to 1 (assuming that the eigenvectors
are normalized). However, if an eigenvector undergoes an arbitrary sign flip, then
the O αα ’s will be close to -1. In contrast, when Q t−t is close to an avoided
crossing or conical intersection, the eigenvectors may vary rapidly, leading to
O αα ’s that are substantially lower than 1 and possibly even negative. In this
case, if an eigenvector undergoes an arbitrary sign flip, then O αα could lie in
the interval −δ < O αα < δ, where |δ| is substantially lower than 1 (e.g. 0.7 or
lower).
2. Determine the value of m t (α) for all α’s, according to
m t (α) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
0 if |O αα (t) − 1| < <
1 if |O αα (t) + 1| < <
2 otherwise
(14.24)
where is a threshold parameter (with 0 < < < 1) that depends on the size
of the time step, t, and the details of the PESs. The m t (α) = 0 and 1 cases
would typically arise when Q t−t is not in the vicinity of an avoided crossing
or conical intersection and the eigenvector undergoes a physical rotation and
unphysical sign flip, respectively. The m t (α) = 2 case would typically arise
when Q t−t is in the vicinity of an avoided crossing or conical intersection. In
this case, one must return the system to the end of the previous time step, divide
the current time step in half, and evolve the system with the reduced time step.
This process is repeated until m t (α) becomes either 0 or 1. A sensible choice of
(in combination with a sufficiently small t) should yield m t (α) = 0, 1 most
of the time (in order to minimize the number of time steps needed).
3. Correct the signs of all eigenvectors according to
|α; Q t
c
= (−1)
m t (α)
|α; Q t ,
(14.25)
i.e. if m t (α) = 0, then the sign is not changed, and if m t (α) = 1, then the sign is
flipped to ensure the continuous evolution of the eigenvector.
At the end of this NVT dynamics phase, the eigenvectors are stored in memory
(i) to continue the sign tracking and correction during the first NVE trajectory
in the ensemble and (ii) to continue the sign tracking and correction during the
second NVT dynamics phase preceding the second NVE trajectory. This procedure
is repeated for the remaining trajectories in the ensemble (see Fig. 14.3 for an
illustration of this procedure). In this way, the basis coefficients in |α; Q =
i c α
i (Q)|φ i (where {|φ i } is a basis set of choice), which may enter explicitly
into the sampling of ρ α α (Q, P ) and into the dynamics of χ αα (Q, P , t), will evolve
continuously throughout the configurational space.
For systems that do not spend a significant amount of time in the vicinity of
an avoided crossing or conical intersection, one can reduce the computational time