As discussed in Section 9.7, optimization of minimum-energy
crossing points (MECPs) along conical seams, followed by optimization of
minimum-energy pathways that connect these MECPs to other points of interest on
ground- and excited-state potential energy surfaces, affords an appealing
one-dimensional picture of photochemical reactivity that is analogous to the
“reactant $\to $ transition state $\to $ product” picture of ground-state
chemistry. Just as the ground-state reaction is not obligated to proceed
*exactly* through the transition-state geometry, however, an excited-state
reaction need not proceed precisely through the MECP and the particulars of
nuclear kinetic energy can lead to deviations. This is arguably more of an
issue for excited-state reactions, where the existence of multiple conical
intersections can easily lead to multiple potential reaction mechanisms. AIMD
potentially offers a way to sample over the available mechanisms in order to
deduce which ones are important in an automated way, but must be extended in
the photochemical case to reactions that involve more than one Born-Oppenheimer
potential energy surface.

The most widely-used trajectory-based method for non-adiabatic simulations is
Tully’s “fewest-switches” surface-hopping (FSSH)
algorithm.^{Tully:1990, Hammes-Schiffer:1994} In this approach, classical
trajectories are propagated on a single potential energy surface, but can
undergo “hops” to a different potential surface in regions of near-degeneracy
between surfaces. The probability of these stochastic hops is governed by the
magnitude of the non-adiabatic coupling [Eq. (9.5)].
Considering the ensemble average of a swarm of trajectories then provides
information about, *e.g.*, branching ratios for photochemical reactions.

The FSSH algorithm, based on the AIMD code, is available in Q-Chem for any
electronic structure method where analytic derivative couplings are available,
which at present means CIS, TDDFT, and their spin-flip analogues (see
Section 9.7.1). The nuclear dynamics component of the
simulation is specified just as in an AIMD calculation. Artificial decoherence
can be added to the calculation at additional cost according to the augmented
FSSH (AFSSH) method,^{Subotnik:2011a, Subotnik:2011b, Landry:2012} which
enforces stochastic wave function collapse at a rate proportional to the
difference in forces between the trajectory on the active surface and position
moments propagated the other surfaces. At every time step, the component of
the wave function on each active surface is printed to the output file. These
amplitudes, as well as the position and momentum moments (if AFSSH is
requested), is also printed to a text file called SurfaceHopper located
in the $QC/AIMD sub-directory of the job’s scratch directory.

In order to request a FSSH calculation, only a few additional *$rem* variables
must be added to those necessary for an excited-state AIMD simulation. At
present, FSSH calculations can only be performed using Born-Oppenheimer
molecular dynamics (BOMD) method. Furthermore, the optimized velocity Verlet
(OVV) integration method is not supported for FSSH calculations.

FSSH_LOWESTSURFACE

Specifies the lowest-energy state considered in a surface hopping calculation.

TYPE:

INTEGER

DEFAULT:

None

OPTIONS:

$n$
Only states $n$ and above are considered in a FSSH calculation.

RECOMMENDATION:

None

FSSH_NSURFACES

Specifies the number of states considered in a surface hopping calculation.

TYPE:

INTEGER

DEFAULT:

None

OPTIONS:

$n$
$n$ states are considered in the surface hopping calculation.

RECOMMENDATION:

Any states which may come close in energy to the active surface should
be included in the surface hopping calculation.

FSSH_INITIALSURFACE

Specifies the initial state in a surface hopping calculation.

TYPE:

INTEGER

DEFAULT:

None

OPTIONS:

$n$
An integer between FSSH_LOWESTSURFACE and FSSH_LOWESTSURFACE
$+$ FSSH_NSURFACES $-1$.

RECOMMENDATION:

None

AFSSH

Adds decoherence approximation to surface hopping calculation.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

0
Traditional surface hopping, no decoherence.
1
Use augmented fewest-switches surface hopping (AFSSH).

RECOMMENDATION:

AFSSH will increase the cost of the calculation, but may improve accuracy
for some systems. See Refs. Subotnik:2011a, Subotnik:2011b, Landry:2012
for more detail.

AIMD_SHORT_TIME_STEP

Specifies a shorter electronic time step for FSSH calculations.

TYPE:

INTEGER

DEFAULT:

TIME_STEP

OPTIONS:

$n$
Specify an electronic time step duration of $n$/AIMD_TIME_STEP_CONVERSION
a.u. If $n$ is less than the nuclear time step variable TIME_STEP, the
electronic wave function will be integrated multiple times per nuclear time step,
using a linear interpolation of nuclear quantities such as the energy gradient and
derivative coupling. Note that $n$ must divide TIME_STEP evenly.

RECOMMENDATION:

Make AIMD_SHORT_TIME_STEP as large as possible while keeping the trace of
the density matrix close to unity during long simulations. Note that while specifying an
appropriate duration for the electronic time step is essential for maintaining accurate
wave function time evolution, the electronic-only time steps employ linear interpolation
to estimate important quantities. Consequently, a short electronic time step is not a
substitute for a reasonable nuclear time step.

FSSH_CONTINUE

Restart a FSSH calculation from a previous run, using the file 396.0. When this is enabled,
the initial conditions of the surface hopping calculation will be set, including the correct
wave function amplitudes, initial surface, and position/momentum moments (if AFSSH) from the
final step of some prior calculation.

TYPE:

INTEGER

DEFAULT:

0

OPTIONS:

$0$
Start fresh calculation.
$1$
Restart from previous run.

RECOMMENDATION:

None

$comment Only 50 AIMD steps are taken in this example, typically more would be required $end $molecule 0 1 C -1.620294 0.348677 -0.008838 C -0.399206 -0.437493 -0.012535 C -0.105193 -1.296810 -1.081340 H -0.789110 -1.374693 -1.905080 C 1.069016 -2.045054 -1.072304 H 1.292495 -2.701157 -1.889686 C 1.956240 -1.940324 0.002842 H 2.859680 -2.517019 0.008420 C 1.666259 -1.084065 1.071007 H 2.348104 -1.005765 1.894140 C 0.495542 -0.335701 1.065497 H 0.253287 0.325843 1.871866 O -1.931045 1.124872 0.911738 H -2.269528 0.227813 -0.865645 $end $rem JOBTYPE aimd EXCHANGE hf BASIS 3-21g CIS_N_ROOTS 3 SYMMETRY off SYM_IGNORE true CIS_SINGLETS false CIS_TRIPLETS true PROJ_TRANSROT true FSSH_LOWESTSURFACE 1 FSSH_NSURFACES 3 ! hop between T1 and T2 FSSH_INITIALSURFACE 1 ! start on T1 AFSSH 0 ! no decoherence CALC_NAC true AIMD_STEPS 50 TIME_STEP 14 AIMD_SHORT_TIME_STEP 2 AIMD_TIME_STEP_CONVERSION 1 ! Do not alter time_step duration AIMD_PRINT 1 AIMD_INIT_VELOC thermal AIMD_TEMP 300 # K AIMD_INTEGRATION vverlet FOCK_EXTRAP_ORDER 6 FOCK_EXTRAP_POINTS 12 $end