We present a current, up-to-date review of the surface hopping methodology for solving nonadiabatic problems, 25 years after Tully published the fewest switches surface hopping algorithm. After reviewing the original motivation for and failures of the algorithm, we give a detailed examination of modern advances, focusing on both theoretical and practical issues. We highlight how one can partially derive surface hopping from the Schrödinger equation in the adiabatic basis, how one can change basis within the surface hopping algorithm, and how one should understand and apply the notions of decoherence and wavepacket bifurcation. The question of time reversibility and detailed balance is also examined at length. Recent applications to photoexcited conjugated polymers are discussed briefly.

In this article, we demonstrate that Tully's fewest-switches surface hopping (FSSH) algorithm approximately obeys the mixed quantum-classical Liouville equation (QCLE), provided that several conditions are satisfied--some major conditions, and some minor. The major conditions are: (1) nuclei must be moving quickly with large momenta; (2) there cannot be explicit recoherences or interference effects between nuclear wave packets; (3) force-based decoherence must be added to the FSSH algorithm, and the trajectories can no longer rigorously be independent (though approximations for independent trajectories are possible). We furthermore expect that FSSH (with decoherence) will be most robust when nonadiabatic transitions in an adiabatic basis are dictated primarily by derivative couplings that are presumably localized to crossing regions, rather than by small but pervasive off-diagonal force matrix elements. In the end, our results emphasize the strengths of and possibilities for the FSSH algorithm when decoherence is included, while also demonstrating the limitations of the FSSH algorithm and its inherent inability to follow the QCLE exactly.

We present a slightly improved version of our augmented fewest switches surface hopping (A-FSSH) algorithm and apply it to the calculation of transition rates between diabatic electronic states within the spin-boson model. We compare A-FSSH rates with (i) Marcus rates from the golden rule, (ii) Tully-style FSSH rates, and (iii) FSSH rates using a simple, intuitive decoherence criterion. We show that unlike FSSH, A-FSSH recovers the correct scaling with diabatic coupling (quadratic in V) as well as the lack of dependence on harmonic frequency ω for small enough values of ω and large enough temperatures.

Can we derive Tully's surface-hopping algorithm from the semiclassical quantum Liouville equation? Almost, but only with decoherence

In a recent paper, we presented a road map for how Tully's fewest switches surface hopping (FSSH) algorithm can be derived, under certain circumstances, from the mixed quantum-classical Liouville equation. In this communication, we now demonstrate how this new interpretation of surface hopping can yield significantly enhanced results for electronic properties in nonadiabatic calculations. Specifically, we calculate diabatic populations for the spin-boson problem using FSSH trajectories. We show that, for some Hamiltonians, without changing the FSSH algorithm at all but rather simply reinterpreting the ensemble of surface hopping trajectories, we recover excellent results and remove any and all ambiguity about the initial condition problem.

Electronically photoexcited dynamics are complicated because there are so many different relaxation pathways: fluorescence, phosphorescence, radiationless decay, electon transfer, etc. In practice, to model photoexcited systems is a very difficult enterprise, requiring accurate and very efficient tools in both electronic structure theory and nonadiabatic chemical dynamics. Moreover, these theoretical tools are not traditional tools. On the one hand, the electronic structure tools involve couplings between electonic states (rather than typical single state energies and gradients). On the other hand, the dynamics tools involve propagating nuclei on multiple potential energy surfaces (rather than the usual ground state dynamics). In this Account, we review recent developments in electronic structure theory as directly applicable for modeling photoexcited systems. In particular, we focus on how one may evaluate the couplings between two different electronic states. These couplings come in two flavors. If we order states energetically, the resulting adiabatic states are coupled via derivative couplings. Derivative couplings capture how electronic wave functions change as a function of nuclear geometry and can usually be calculated with straightforward tools from analytic gradient theory. One nuance arises, however, in the context of time-dependent density functional theory (TD-DFT): how do we evaluate derivative couplings between TD-DFT excited states (which are tricky, because no wave function is available)? This conundrum was recently solved, and we review the solution below. We also discuss the solution to a second, pesky problem of origin dependence, whereby the derivative couplings do not (strictly) satisfy translation variance, which can lead to a lack of momentum conservation. Apart from adiabatic states, if we order states according to their electronic character, the resulting diabatic states are coupled via electronic or diabatic couplings. The couplings between diabatic states |ΞA⟩ and |ΞB⟩ are just the simple matrix elements, ⟨ΞA|H|ΞB⟩. A difficulty arises, however, because constructing exactly diabatic states is formally impossible and constructing quasi-diabatic states is not unique. To that end, we review recent advances in localized diabatization, which is one approach for generating adiabatic-to-diabatic (ATD) transformations. We also highlight outstanding questions in the arena of diabatization, especially how to generate multiple globally stable diabatic surfaces.

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We compare the dynamics of Fewest Switches Surface Hopping (FSSH) in different parameter regimes of the spin-boson model. We show that for exceptional regions of the spin-boson parameter space, FSSH dynamics are in fact time-reversible. In these rare instances, FSSH does recover the correct Marcus rate scaling (as a function of diabatic coupling) without the addition of decoherence. In regions where dynamics are irreversible, however, FSSH does not recover the correct Marcus rate scaling. Finally, by comparing the friction dependence of rates predicted by various decoherence schemes to an analytic result by Zusman, we provide yet more evidence that the method of introducing decoherence has a qualitative effect on the accuracy of results and this effect must be treated carefully.