We investigate barrier crossings within the context of the Anderson–Holstein model, as relevant to coupled nuclear–electronic dynamics near a metal surface. Beyond standard electronic friction or conventional surface-hopping dynamics, we show that a broadened classical master equation can recover both the correct nonadiabatic and the correct adiabatic dynamics for a general escape problem (even with possibly multiple escape channels). In the case of a large barrier with only a single escape channel, we also find a surprising conclusion: electronic friction can recover Marcus’s nonadiabatic theory of electron transfer in the limit of small molecule–metal couplings. The latter conclusion establishes a hidden connection between Marcus’s nonadiabatic theory and Kramer’s adiabatic theory of rate constants.

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.

Whereas surface hopping is usually used to study populations and mean-field dynamics to study coherences, in two recent papers, we described a procedure for calculating dipole–dipole correlation functions (and therefore absorption spectra) directly from ensembles of surface hopping trajectories. We previously applied this method to a handful of one-dimensional model problems intended to mimic the gas phase. In this article, we now benchmark this new procedure on a set of multidimensional model problems intended to mimic the condensed phase and compare our results against other standard semiclassical methods. By comparison, we demonstrate that methods that include only dynamical information from one PES (the standard Kubo approaches) exhibit large discrepancies with the results of exact quantum dynamics. Furthermore, for model problems with nonadiabatic excited state dynamics but no quantized vibrational structure in the spectra, our surface hopping approach performs comparably to using Ehrenfest dynamics to calculate the electronic coherences. That being said, however, when quantized vibrational structures are present in the spectra but the electronic states are uncoupled, performing the dynamics on the mean PES still outperforms our present method. These benchmark results should influence future studies that use ensembles of independent semiclassical trajectories to model linear as well as multidimensional spectra in the condensed phase.

We analyze thermal rate constants as computed with surface hopping dynamics and resolve certain inconsistencies that have permeated the literature. On one hand, according to Landry and Subotnik (J. Chem. Phys. 2012, 137, 22A513), without decoherence, direct dynamics with surface hopping overestimates the rate of relaxation for the spin-boson Hamiltonian. On the other hand, according to Jain and Subotnik (J. Chem. Phys. 2015, 143, 134107), without decoherence, a transition state theory with surface hopping underestimates spin-boson rate constants. In this Letter, we resolve this apparent contradiction. We show that, without decoherence, direct dynamics and transition state theory should not agree; agreement is guaranteed only with decoherence. We also show that, even though the effects of decoherence may be hidden for isoenergetic reactions, these decoherence failures are exposed for exothermic reactions. We believe these lessons are essential when interpreting surface hopping papers published in the literature without any decoherence corrections.

A pseudo-wavefunction description of time-dependent Hartree–Fock (TDHF) states is proposed and used to develop an analytic expression for derivative couplings between TDHF excited states based on the Hellmann–Feynman theorem. The resulting expression includes Pulay terms associated with using an atom-centered basis as well as a correction to ensure translational invariance. We demonstrate that our formalism recovers the well-known Chernyak–Mukamel expression near a crossing and in the limit of a complete basis, and thus our approach is consistent with time-dependent response theory. In a companion paper (DOI 10.1021/jp5057682), we investigate these derivative couplings near conical intersections and show that they behave correctly.

We use a recent flavor of surface hopping dynamics to investigate the nonequilibrium transport of electrons as carried by a few ions through a solution of Lennard-Jones spheres across a finite voltage. We analyze our nonequilibrium results through a combination of equilibrium simulations and steady state rate equations. While the nonequilibrium dynamics approach in this paper is computationally demanding and likely limited to reasonably small simulation sizes (or short time scales), the present study does provide a simple means for us to understand the interplay between nuclear motion near a metal surface and charge injection into/from the metal surface. This interplay is especially interesting when the system of interest is far from the linear response regime (which might well be very common in electrochemistry).

In this paper, we present a formalism for derivative couplings between time-dependent density functional theory (TD-DFT) excited states within the random-phase approximation (RPA) using analytic gradient theory. Our formalism is based on a pseudo-wavefunction approach in a companion paper (DOI 10.1021/jp505767b), and can be checked against finite-difference overlaps. Our approach recovers the correct properties of derivative couplings around a conical intersection (CI), which is a crucial prerequisite for any derivative coupling expression. As an example, we study the test case of protonated formaldimine (CH2NH2+).

While individual Tully-style fewest switches surface hopping (FSSH) trajectories are stochastic and cannot be inverted in time, it is possible to reverse in time the dynamics of a swarm of FSSH trajectories. Here we show exactly how to invert such dynamics, and we investigate the stability of such time-reversed surface hopping dynamics. We demonstrate that FSSH trajectories can be inverted successfully for short time periods, but the time-reversed dynamics become unstable for long times with multiple hopping events. We argue that this instability of FSSH going backward in time can be correlated with the stability of the FSSH algorithm going forward in time.

For chemically accurate excited state energies, one is forced to include electron–electron correlation at a level of theory significantly higher than configuration interaction singles (CIS). Post-CIS corrections do exist, but most often, if they are computationally inexpensive, these methods rely on perturbation theory. At the same time, inexpensive variational post-CIS methods would be ideal since modeling electronic relaxation usually requires globally smooth potential energy surfaces (PESs) and there will inevitably be regions of near electronic degeneracy. With that goal in mind, we now present a new method entitled variationally orbital adapted CIS (VOA-CIS). On the one hand, we show that in the ground-state geometry, VOA-CIS performs comparably to CIS(D) at predicting relative excited state energies. On the other hand, far beyond CIS(D) or any other perturbative method, VOA-CIS correctly rebalances the energy of charge-transfer (CT) states versus non-CT states, while simultaneously producing smooth PESs—including the important case of avoided crossings. In fact, through localized diabatization of VOA-CIS excited states, one can find a set of reasonable diabatic states modeling CT chemical dynamics. After significant benchmarking, we are now confident VOA-CIS and VOA-CIS-like methods should play a major role in future excited state calculations.

In a previous paper [Fatehi, S.; et al. J. Chem. Phys. 2013, 139, 124112], we demonstrated a practical method by which analytic derivative couplings of Boys-localized CIS states can be obtained. In this paper, we now apply that same method to the analysis of triplet–triplet energy transfer systems studied by Closs and collaborators [Closs, G. L.; et al. J. Am. Chem. Soc. 1988, 110, 2652]. For the systems examined, we are able to conclude that (i) the derivative coupling in the BoysOV basis is negligible, and (ii) the diabatic coupling will likely change little over the configuration space explored at room temperature. Furthermore, we propose and evaluate an approximation that allows for the inexpensive calculation of accurate diabatic energy gradients, called the “strictly diabatic” approximation. This work highlights the effectiveness of diabatic state analytic gradient theory in realistic systems and demonstrates that localized diabatic states can serve as an acceptable approximation to strictly diabatic states.

The photophysics of benzaldehyde are analyzed through the lens of TD-DFT adiabatic excited states and Boys or Edmiston–Ruedenberg localized diabatic states. We predict rate constants for two processes in excited benzaldehyde: (i) the intersystem crossing from S1 → T2 and (ii) the phosphorescence from T1 → S0. We also study (iii) the conical intersection between T2 and T1 that is putatively responsible for an ultrafast internal conversion process, T2 → T1. In agreement with Ohmori et al. (J. Phys. Chem. 1988, 92 (5), 1086–1093), our results suggest that the S1 → T2 intersystem crossing in benzaldehyde is rapid not only because of a large spin–orbit matrix element (i.e., El-Sayed’s rule) but also because of a fortuitously small energy barrier. Furthermore, when studying the T2 → T1 internal conversion, we find that both Boys and Edmiston–Ruedenberg localization give remarkably stable and accurate diabatic states which will be useful for ongoing studies of dynamics near conical intersections. To our knowledge, this is the first example whereby localized diabatization techniques have been tested and have successfully recovered the topology of a conical intersection.

In a recent article (Subotnik, J. E.; Shenvi, N. J. Chem. Phys.2011, 134, 24105), we introduced a new approach for incorporating decoherence into the fewest-switches surface-hopping (FSSH) algorithm, titled augmented FSSH (A-FSSH). The A-FSSH algorithm was designed to correct the well-known overcoherence problem in traditional FSSH, and thus allow wave packets on different surfaces to separate naturally subject to different forces. As presented earlier, however, the A-FSSH algorithm was restricted to two electronic states. We now extend the method to more than two electronic states and present several new model problems with multiple electronic and nuclear dimensions. Lastly, starting with the quantum Liouville equation, we rederive and implement the new phase correction suggested by Shenvi (Shenvi, N.; Subotnik, J. E.; Yang, W. J. Chem. Phys.2011, 135, 24101) and co-workers for propagating the electronic amplitude along a specified nuclear trajectory and find much improved results in certain cases.

We model the triplet−triplet energy-transfer experiments from the Closs group [Closs, G. L.; et al. J. Am. Chem. Soc. 1988, 110, 2652.] using a combination of Marcus theory and either Boys or Edmiston−Ruedenberg localized diabatization, and we show that relative and absolute rates of electronic excitation transfer may be computed successfully. For the case where both the donor and acceptor occupy equatorial positions on a rigid cyclohexane bridge, we find βcalc = 2.8 per C−C bond, compared with the experimental value βexp = 2.6. This work highlights the power of using localized diabatization methods as a tool for modeling nonequilibrium processes.

A common strategy to calculate electronic coupling matrix elements for charge or energy transfer is to take the adiabatic states generated by electronic structure computations and rotate them to form localized diabatic states. In this paper, we show that, for intermolecular transfer of singlet electronic excitation, usually we cannot fully localize the electronic excitations in this way. Instead, we calculate putative initial and final states with small excitation tails caused by weak interactions with high energy excited states in the electronic manifold. These tails do not lead to substantial changes in the total diabatic coupling between states, but they do lead to a different partitioning of the total coupling between Coulomb (Förster), exchange (Dexter), and one-electron components. The tails may be reduced by using a multistate diabatic model or eliminated entirely by truncation (denoted as “chopping”). Without more information, we are unable to conclude with certainty whether the observed diabatic tails are a physical reality or a computational artifact. This research suggests that decomposition of the diabatic coupling between chromophores into Coulomb, exchange, and one-electron components may depend strongly on the number of states considered, and such results should be treated with caution.