2 Surface Hopping Overview
In surface hopping, the time propagation of the quantum wavepacket is approximated by a swarm of semiclassical trajectories evolving on Born–Oppenheimer
surfaces of multiple electronic states. Nonadiabatic events (wavepacket density
transfer between states; see [8] for an excellent review on this topic) are simulated
by a stochastic algorithm which allows each trajectory to jump to other states during
the propagation. Thus, the statistics over the ensemble of trajectories in terms of
fraction of trajectories in each electronic state in each time step is expected to be an
approximated representation of the wavepacket density distribution among the
excited states as a function of time. The method was conceptually proposed by
Nikitin [18] and was used first by Tully and Preston [19]. It has been recently
reviewed in [11, 20–22].
In the most common surface hopping approach, all nuclear coordinates are
driven by Newton’s equations of motion on a single adiabatic electronic state K.
For the coordinates R m with the associated nuclear mass M m , they are given by
d
2 R m
dt 2 ¼ À
1
M m
∂E K
∂R m
;
ð1Þ
where E K is the adiabatic potential energy of the current state K. Given a set of
initial positions and velocities, (1) is numerically integrated.
Along with the Newton’s equations, the probability for the system to hop to
another state L is evaluated. Diverse schemes for the evaluation of such probabilities have been developed [19, 23–30]. The most successful and popular approach
has been the fewest switches proposed by Tully in the early 1990s [28].
In the fewest switches, the number of hopping events within one time step Δt is
minimized. Under this condition, the hopping probability between states K and L is
P K!L ¼
Population increment in L due to flux from K during Δt
Population of K
:
ð2Þ
The population of each electronic state L is given in terms of the coefficients
c L (t) of the time-dependent wavefunction written as a linear combination of electronic time-independent electronic wavefunctions Ψ L :
φ r; R; t
ð
Þ¼
X
J
c J t
ð ÞΨ J r; R t
ð Þ
ð
Þ:
ð3Þ
The coefficients c J are obtained by solving a local approximation for the timedependent electronic Schro ¨dinger equation, given in the adiabatic representation by
[28]
Surface Hopping Dynamics with DFT Excited States
419
In surface hopping, the time propagation of the quantum wavepacket is approximated by a swarm of semiclassical trajectories evolving on Born–Oppenheimer
surfaces of multiple electronic states. Nonadiabatic events (wavepacket density
transfer between states; see [8] for an excellent review on this topic) are simulated
by a stochastic algorithm which allows each trajectory to jump to other states during
the propagation. Thus, the statistics over the ensemble of trajectories in terms of
fraction of trajectories in each electronic state in each time step is expected to be an
approximated representation of the wavepacket density distribution among the
excited states as a function of time. The method was conceptually proposed by
Nikitin [18] and was used first by Tully and Preston [19]. It has been recently
reviewed in [11, 20–22].
In the most common surface hopping approach, all nuclear coordinates are
driven by Newton’s equations of motion on a single adiabatic electronic state K.
For the coordinates R m with the associated nuclear mass M m , they are given by
d
2 R m
dt 2 ¼ À
1
M m
∂E K
∂R m
;
ð1Þ
where E K is the adiabatic potential energy of the current state K. Given a set of
initial positions and velocities, (1) is numerically integrated.
Along with the Newton’s equations, the probability for the system to hop to
another state L is evaluated. Diverse schemes for the evaluation of such probabilities have been developed [19, 23–30]. The most successful and popular approach
has been the fewest switches proposed by Tully in the early 1990s [28].
In the fewest switches, the number of hopping events within one time step Δt is
minimized. Under this condition, the hopping probability between states K and L is
P K!L ¼
Population increment in L due to flux from K during Δt
Population of K
:
ð2Þ
The population of each electronic state L is given in terms of the coefficients
c L (t) of the time-dependent wavefunction written as a linear combination of electronic time-independent electronic wavefunctions Ψ L :
φ r; R; t
ð
Þ¼
X
J
c J t
ð ÞΨ J r; R t
ð Þ
ð
Þ:
ð3Þ
The coefficients c J are obtained by solving a local approximation for the timedependent electronic Schro ¨dinger equation, given in the adiabatic representation by
[28]
Surface Hopping Dynamics with DFT Excited States
419