Dissipative mixed quantum-classical simulation of the aqueous solvated electron system

A size-consistent multi-state mapping approach to surface hopping

We develop a multi-state generalization of the recently proposed mapping approach to surface hopping (MASH) for the simulation of electronically nonadiabatic dynamics. This new approach extends the original MASH method to be able to treat systems with more than two electronic states. It differs from previous approaches in that it is size consistent and rigorously recovers the original two-state MASH in the appropriate limits. We demonstrate the accuracy of the method by applying it to a series of model systems for which exact benchmark results are available, and we find that the method is well suited to the simulation of photochemical relaxation processes.

Recovering Marcus Theory Rates and Beyond without the Need for Decoherence Corrections: The Mapping Approach to Surface Hopping

It is well-known that fewest-switches surface hopping (FSSH) fails to correctly capture the quadratic scaling of rate constants with diabatic coupling in the weak-coupling limit, as expected from Fermi's golden rule and Marcus theory. To address this deficiency, the most widely used approach is to introduce a "decoherence correction", which removes the inconsistency between the wave function coefficients and the active state. Here we investigate the behavior of a new nonadiabatic trajectory method, called the mapping approach to surface hopping (MASH), on systems that exhibit an incoherent rate behavior. Unlike FSSH, MASH hops between active surfaces deterministically and can never have an inconsistency between the wave function coefficients and the active state. We show that MASH not only can describe rates for intermediate and strong diabatic coupling but also can accurately reproduce the results of Marcus theory in the golden-rule limit, without the need for a decoherence correction. MASH is therefore a significant improvement over FSSH in the simulation of nonadiabatic reactions.

Nonadiabatic Dynamics in a Continuous Circularly Polarized Laser Field with Floquet Phase-Space Surface Hopping

Nonadiabatic chemical reactions involving continuous circularly polarized light (cw CPL) have not attracted as much attention as dynamics in unpolarized/linearly polarized light. However, including circularly (in contrast to linearly) polarized light allows one to effectively introduce a complex-valued time-dependent Hamiltonian, which offers a new path for control or exploration through the introduction of Berry forces. Here, we investigate several inexpensive semiclassical approaches for modeling such nonadiabatic dynamics in the presence of a time-dependent complex-valued Hamiltonian, beginning with a straightforward instantaneous adiabatic fewest-switches surface hopping (IA-FSSH) approach (where the electronic states depend on position and time), continuing to a standard Floquet fewest switches surface hopping (F-FSSH) approach (where the electronic states depend on position and frequency), and ending with an exotic Floquet phase-space surface hopping (F-PSSH) approach (where the electronic states depend on position, frequency, and momentum). Using a set of model systems with time-dependent complex-valued Hamiltonians, we show that the Floquet phase-space adiabats are the optimal choice of basis as far as accounting for Berry phase effects and delivering accuracy. Thus, the F-PSSH algorithm sets the stage for future modeling of nonadiabatic dynamics under strong externally pumped circular polarization.

Semiclassical description of nuclear dynamics moving through complex-valued single avoided crossings of two electronic states

The standard fewest-switches surface hopping (FSSH) approach fails to model nonadiabatic dynamics when the electronic Hamiltonian is complex-valued and there are multiple nuclear dimensions; FSSH does not include geometric magnetic effects and does not have access to a gauge independent direction for momentum rescaling. In this paper, for the case of a Hamiltonian with two electronic states, we propose an extension of Tully's FSSH algorithm, which includes geometric magnetic forces and, through diabatization, establishes a well-defined rescaling direction. When combined with a decoherence correction, our new algorithm shows satisfying results for a model set of two-dimensional single avoided crossings.

A "backtracking" correction for the fewest switches surface hopping algorithm

We propose a "backtracking" mechanism within Tully's fewest switches surface hopping (FSSH) algorithm, whereby whenever one detects consecutive (double) hops during a short period of time, one simply rewinds the dynamics backward in time. In doing so, one reduces the number of hopping events and comes closer to a truly fewest switches surface hopping approach with independent trajectories. With this algorithmic change, we demonstrate that surface hopping can be reasonably accurate for nuclear dynamics in a multidimensional configuration space with a complex-valued (i.e., not real-valued) electronic Hamiltonian; without this adjustment, surface hopping often fails. The added computational cost is marginal. Future research will be needed to assess whether or not this backtracking correction can improve the accuracy of a typical FSSH calculation with a real-valued electronic Hamiltonian (that ignores spin).

On the proper derivation of the Floquet-based quantum classical Liouville equation and surface hopping describing a molecule or material subject to an external field

We investigate different approaches to derive the proper Floquet-based quantum-classical Liouville equation (F-QCLE) for laser-driven electron-nuclear dynamics. The first approach projects the operator form of the standard QCLE onto the diabatic Floquet basis and then transforms to the adiabatic representation. The second approach directly projects the QCLE onto the Floquet adiabatic basis. Both approaches yield a form that is similar to the usual QCLE with two modifications: (1) The electronic degrees of freedom are expanded to infinite dimension and (2) the nuclear motion follows Floquet quasi-energy surfaces. However, the second approach includes an additional cross derivative force due to the dual dependence on time and nuclear motion of the Floquet adiabatic states. Our analysis and numerical tests indicate that this cross derivative force is a fictitious artifact, suggesting that one cannot safely exchange the order of Floquet state projection with adiabatic transformation. Our results are in accord with similar findings by Izmaylov et al., [J. Chem. Phys. 140, 084104 (2014)] who found that transforming to the adiabatic representation must always be the last operation applied, although now we have extended this result to a time-dependent Hamiltonian. This paper and the proper derivation of the F-QCLE should lay the basis for further improvements of Floquet surface hopping.

Nonadiabatic Dynamics in a Laser Field: Using Floquet Fewest Switches Surface Hopping To Calculate Electronic Populations for Slow Nuclear Velocities

We investigate two well-known approaches for extending the fewest switches surface hopping (FSSH) algorithm to periodic time-dependent couplings. The first formalism acts as if the instantaneous adiabatic electronic states were standard adiabatic states, which just happen to evolve in time. The second formalism replaces the role of the usual adiabatic states by the time-independent adiabatic Floquet states. For a set of modified Tully model problems, the Floquet FSSH (F-FSSH) formalism gives a better estimate for both transmission and reflection probabilities than the instantaneous adiabatic FSSH (IA-FSSH) formalism, especially for slow nuclear velocities. More importantly, only F-FSSH predicts the correct final scattering momentum. Finally, in order to use Floquet theory accurately, we find that it is crucial to account for the interference between wavepackets on different Floquet states. Our results should be of interest to all those interested in laser-induced molecular dynamics.

Nonadiabatic Molecular Dynamics at Metal Surfaces

Dynamics at molecule-metal interfaces are a subject of intense current interest and come in many different flavors of experiments: gas-phase scattering, chemisorption, electrochemistry, nanojunction transport, and heterogeneous catalysis, to name a few. These dynamics involve nuclear degrees of freedom entangled with many electronic degrees of freedom (in the metal), and as such there is always the possibility for nonadiabatic phenomena to appear: the nuclei do not necessarily need to move slower than the electrons to break the Born-Oppenheimer (BO) approximation. In this Feature Article, we review a set of dynamical methods developed recently to deal with such nonadiabatic phenomena at a metal surface, methods that serve as alternatives to Tully's independent electron surface hopping (IESH) model. In the weak molecule-metal coupling regime, a classical master equation (CME) can be derived and a simple surface hopping approach is proposed to propagate nuclear and electronic dynamics stochastically. In the strong molecule-metal interaction regime, a Fokker-Planck equation can be derived for the nuclear dynamics, with electronic DoFs incorporated into the overall friction and random force. Lastly, a broadened classical master equation (BCME) can interpolate between the weak and strong molecule-metal interactions. Here, we briefly review these methods and the relevant benchmarking data, showing in particular how the methods can be used to calculate nonequilibrium transport properties. We highlight several open questions and pose several avenues for future study.

Ultrafast Reactive Quantum Dynamics Coupled to Classical Solvent Dynamics Using an Ehrenfest Approach

The inclusion of solvent effects in the theoretical analysis of molecular processes becomes increasingly important. Currently, it is not feasible to directly include the solvent on the quantum level. We use an Ehrenfest approach to study the coupled time evolution of quantum dynamically treated solutes and classical solvents system. The classical dynamics of the solvent is coupled to the wavepacket dynamics of the solute and rotational and translational degrees of freedom of the solute are included classically. This allows quantum dynamics simulations for ultrafast processes that are decided by environment interactions without explicit separation of time scales. We show the application to the dissociation of ICN in liquid Ar as a proof of principal system and to the more applied example of uracil in water.

Does Nonadiabatic Transition State Theory Make Sense Without Decoherence?

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.

Surface hopping with a manifold of electronic states. I. Incorporating surface-leaking to capture lifetimes

We investigate the incorporation of the surface-leaking (SL) algorithm into Tully's fewest-switches surface hopping (FSSH) algorithm to simulate some electronic relaxation induced by an electronic bath in conjunction with some electronic transitions between discrete states. The resulting SL-FSSH algorithm is benchmarked against exact quantum scattering calculations for three one-dimensional model problems. The results show excellent agreement between SL-FSSH and exact quantum dynamics in the wide band limit, suggesting the potential for a SL-FSSH algorithm. Discrepancies and failures are investigated in detail to understand the factors that will limit the reliability of SL-FSSH, especially the wide band approximation. Considering the easiness of implementation and the low computational cost, we expect this method to be useful in studying processes involving both a continuum of electronic states (where electronic dynamics are probabilistic) and processes involving only a few electronic states (where non-adiabatic processes cannot ignore short-time coherence).

Surface hopping outperforms secular Redfield theory when reorganization energies range from small to moderate (and nuclei are classical)

We evaluate the accuracy of Tully's surface hopping algorithm for the spin-boson model in the limit of small to moderate reorganization energy. We calculate transition rates between diabatic surfaces in the exciton basis and compare against exact results from the hierarchical equations of motion; we also compare against approximate rates from the secular Redfield equation and Ehrenfest dynamics. We show that decoherence-corrected surface hopping performs very well in this regime, agreeing with secular Redfield theory for very weak system-bath coupling and outperforming secular Redfield theory for moderate system-bath coupling. Surface hopping can also be extended beyond the Markovian limits of standard Redfield theory. Given previous work [B. R. Landry and J. E. Subotnik, J. Chem. Phys. 137, 22A513 (2012)] that establishes the accuracy of decoherence-corrected surface-hopping in the Marcus regime, this work suggests that surface hopping may well have a very wide range of applicability.

On surface hopping and time-reversal

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.

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

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.

A new approach to decoherence and momentum rescaling in the surface hopping algorithm

As originally proposed, the fewest switches surface hopping (FSSH) algorithm does not allow for decoherence between wavefunction amplitudes on different adiabatic surfaces. In this paper, we propose an inexpensive correction to standard FSSH dynamics wherein we explicitly model the decoherence of nuclear wave packets on distinct electronic surfaces. Our augmented fewest switches surface hopping approach is conceptually simple and, thus far, it has allowed us to capture several key features of the exact quantum results. Two points in particular merit attention. First, we obtain the correct branching ratios when a quantum particle passes through more than one region of nonadiabatic coupling. Second, our formalism provides a new and natural approach for rescaling nuclear momenta after a surface hop. Both of these features should become increasingly important as surface hopping schemes are applied to higher-dimensional problems.

Exploring dynamical electron theory beyond the Born-Oppenheimer framework: from chemical reactivity to non-adiabatically coupled electronic and nuclear wavepackets on-the-fly under laser field

Chemical theory and its application to dynamical electrons in molecules under intense electromagnetic fields is explored, in which we take an explicit account of nuclear nonadiabatic (kinematic) interactions along with simultaneous coupling with intense optical interactions. All the electronic wavefunctions studied here are necessarily time-dependent, and thereby beyond stationary state quantum chemistry based on the Born-Oppenheimer framework. As a general and tractable alternative framework with which to track the electronic and nuclear simultaneous dynamics, we propose an on-the-fly method to calculate the electron and nuclear wavepackets coupled along the branching non-Born-Oppenheimer paths, through which their bifurcations, strong quantum entanglement between nuclear electronic motions, and coherence and decoherence among the phases associated with them are properly represented. Some illustrative numerical examples are also reported, which are aimed at our final goals; real time tracking of nonadiabatic electronic states, chemical dynamics in densely degenerate electronic states coupled with nuclear motions and manipulation and/or creation of new electronic states in terms of intense lasers, and so on. Other examples are also presented as to how the electron wavepacket dynamics can be used to analyze chemical reactions, shedding a new light on some typical and conventional chemical reactions such as proton transfer followed by tautomerization.

Generalization of classical mechanics for nuclear motions on nonadiabatically coupled potential energy surfaces in chemical reactions

Classical trajectory study of nuclear motion on the Born-Oppenheimer potential energy surfaces is now one of the standard methods of chemical dynamics. In particular, this approach is inevitable in the studies of large molecular systems. However, as soon as more than a single potential energy surface is involved due to nonadiabatic coupling, such a naive application of classical mechanics loses its theoretical foundation. This is a classic and fundamental issue in the foundation of chemistry. To cope with this problem, we propose a generalization of classical mechanics that provides a path even in cases where multiple potential energy surfaces are involved in a single event and the Born-Oppenheimer approximation breaks down. This generalization is made by diagonalization of the matrix representation of nuclear forces in nonadiabatic dynamics, which is derived from a mixed quantum-classical representation of the electron-nucleus entangled Hamiltonian [Takatsuka, K. J. Chem. Phys. 2006, 124, 064111]. A manifestation of quantum fluctuation on a classical subsystem that directly contacts with a quantum subsystem is discussed. We also show that the Hamiltonian thus represented gives a theoretical foundation to examine the validity of the so-called semiclassical Ehrenfest theory (or mean-field theory) for electron quantum wavepacket dynamics, and indeed, it is pointed out that the electronic Hamiltonian to be used in this theory should be slightly modified.

Decoherence and quantum-classical master equation dynamics

The conditions under which quantum-classical Liouville dynamics may be reduced to a master equation are investigated. Systems that can be partitioned into a quantum-classical subsystem interacting with a classical bath are considered. Starting with an exact non-Markovian equation for the diagonal elements of the density matrix, an evolution equation for the subsystem density matrix is derived. One contribution to this equation contains the bath average of a memory kernel that accounts for all coherences in the system. It is shown to be a rapidly decaying function, motivating a Markovian approximation on this term in the evolution equation. The resulting subsystem density matrix equation is still non-Markovian due to the fact that bath degrees of freedom have been projected out of the dynamics. Provided the computation of nonequilibrium average values or correlation functions is considered, the non-Markovian character of this equation can be removed by lifting the equation into the full phase space of the system. This leads to a trajectory description of the dynamics where each fictitious trajectory accounts for decoherence due to the bath degrees of freedom. The results are illustrated by computations of the rate constant of a model nonadiabatic chemical reaction.

A computationally efficient exact pseudopotential method. I. Analytic reformulation of the Phillips-Kleinman theory

Even with modern computers, it is still not possible to solve the Schrodinger equation exactly for systems with more than a handful of electrons. For many systems, the deeply bound core electrons serve merely as placeholders and only a few valence electrons participate in the chemical process of interest. Pseudopotential theory takes advantage of this fact to reduce the dimensionality of a multielectron chemical problem: the Schrodinger equation is solved only for the valence electrons, and the effects of the core electrons are included implicitly via an extra term in the Hamiltonian known as the pseudopotential. Phillips and Kleinman (PK) [Phys. Rev. 116, 287 (1959)]. demonstrated that it is possible to derive a pseudopotential that guarantees that the valence electron wave function is orthogonal to the (implicitly included) core electron wave functions. The PK theory, however, is expensive to implement since the pseudopotential is nonlocal and its computation involves iterative evaluation of the full Hamiltonian. In this paper, we present an analytically exact reformulation of the PK pseudopotential theory. Our reformulation has the advantage that it greatly simplifies the expressions that need to be evaluated during the iterative determination of the pseudopotential, greatly increasing the computational efficiency. We demonstrate our new formalism by calculating the pseudopotential for the 3s valence electron of the Na atom, and in the subsequent paper, we show that pseudopotentials for molecules as complex as tetrahydrofuran can be calculated with our formalism in only a few seconds. Our reformulation also provides a clear geometric interpretation of how the constraint equations in the PK theory, which are required to obtain a unique solution, are themselves sufficient to calculate the pseudopotential.