9.9.7 Fewest-Switches Surface Hopping

As discussed in Section 9.8, 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 nonadiabatic simulations is Tully’s “fewest-switches” surface-hopping (FSSH) algorithm.¹¹⁴⁰ 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 nonadiabatic coupling [Eq. (9.7)]. 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 9.8.1). 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,^{1092, 1095, 620} 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.