Amber Jain

Understanding the Surface Hopping View of Electronic Transitions and Decoherence

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.

Surface hopping, transition state theory, and decoherence. II. Thermal rate constants and detailed balance.

We investigate a simple approach to compute a non-adiabatic thermal rate constant using the fewest switches surface hopping (FSSH) dynamics. We study the effects of both decoherence (using our augmented-FSSH (A-FSSH) algorithm) and forbidden hops over a large range of parameters, including high and low friction regimes, and weak and strong electronic coupling regimes. Furthermore, when possible, we benchmark our results against exact hierarchy equations of motion results, where we usually find a maximum error of roughly a factor of two (at reasonably large temperatures). In agreement with Hammes-Schiffer and Tully, we find that a merger of transition state theory and surface hopping can be both accurate and efficient when performed correctly. We further show that detailed balance is followed approximately by A-FSSH dynamics.

An assessment of mean-field mixed semiclassical approaches: Equilibrium populations and algorithm stability

We study several recent mean-field semiclassical dynamics methods, focusing on the ability to recover detailed balance for long time (equilibrium) populations. We focus especially on Miller and Cottons [J. Phys. Chem. A 117, 7190 (2013)] suggestion to include both zero point electronic energy and windowing on top of Ehrenfest dynamics. We investigate three regimes: harmonic surfaces with weak electronic coupling, harmonic surfaces with strong electronic coupling, and anharmonic surfaces with weak electronic coupling. In most cases, recent additions to Ehrenfest dynamics are a strong improvement upon mean-fieldtheory. However, for methods that include zero point electronic energy, we show that anharmonic potential energy surfaces often lead to numerical instabilities, as caused by negative populations and forces. We also show that, though the effect of negative forces can appear hidden in harmonic systems, the resulting equilibrium limits do remain dependent on any windowing and zero point energy parameters.

Tunneling splittings in formic acid dimer: An adiabatic approximation to the Herring formula

Small symmetric molecules and low-dimensional model Hamiltonians are excellent systems for benchmarking theories to compute tunneling splittings. In this work, we investigate a three dimensional model Hamiltonian coupled to a harmonic bath that describes concerted proton transfer in the formic acid dimer. The three modes include the symmetric proton stretch, the symmetric dimer rock, and the dimer stretch. These modes provide a paradigm for the symmetric and anti-symmetric coupled tunneling pathways, these being recognized in the literature as two of the more important classes of coupling. The effects of selective vibrational excitation and coupling to a bath on the tunneling splittings are presented. The splittings for highly excited states are computed using a novel method that makes an adiabatic approximation to the Herring estimate. Results, which are in excellent agreement with the exact splittings, are compared with those obtained using the Makri-Miller approach. This latter method has been shown to provide quality results for tunneling splittings including highly excited vibrational states.

Surface hopping, transition state theory and decoherence. I. Scattering theory and time-reversibility.

We provide an in-depth investigation of transmission coefficients as computed using the augmented-fewest switches surface hopping algorithm in the low energy regime. Empirically, microscopic reversibility is shown to hold approximately. Furthermore, we show that, in some circumstances, including decoherence on top of surface hopping calculations can help recover (as opposed to destroy) oscillations in the transmission coefficient as a function of energy; these oscillations can be studied analytically with semiclassical scattering theory. Finally, in the spirit of transition state theory, we also show that transmission coefficients can be calculated rather accurately starting from the curve crossing point and running trajectories forwards and backwards.

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.

Dynamics of Barrier Crossings for the Generalized Anderson–Holstein Model: Beyond Electronic Friction and Conventional Surface Hopping

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 Marcuss nonadiabatic theory of electron transfer in the limit of small molecule-metal couplings. The latter conclusion establishes a hidden connection between Marcuss nonadiabatic theory and Kramers adiabatic theory of rate constants.

Vibrational Energy Relaxation: A Benchmark for Mixed Quantum–Classical Methods

We investigate the ability of mixed quantum-classical methods to capture the dynamics of vibrational energy relaxation. Several methods, including surface hopping, and Ehrenfest and symmetrical quasiclassical (SQC) dynamics, are benchmarked for the exactly solvable model problem of a harmonic oscillator bilinearly coupled to a bath of harmonic oscillators. Results show that, very often, one can recover accurate vibrational relaxation rates and detailed balance using simple mixed quantum-classical approaches. A few anomalous results do appear, however, especially regarding Ehrenfest and SQC dynamics.

Ultrafast Electronic Relaxation through a Conical Intersection: Nonadiabatic Dynamics Disentangled through an Oscillator Strength-Based Diabatization Framework

We employ surface hopping trajectories to model the short-time dynamics of gas-phase and partially solvated 4-(N,N-dimethylamino)benzonitrile (DMABN), a dual fluorescent molecule that is known to undergo a nonadiabatic transition through a conical intersection. To compare theory vs time-resolved fluorescence measurements, we calculate the mixed quantum-classical density matrix and the ensemble averaged transition dipole moment. We introduce a diabatization scheme based on the oscillator strength to convert the TDDFT adiabatic states into diabatic states of La and Lb character. Somewhat surprisingly, we find that the rate of relaxation reported by emission to the ground state is almost 50% slower than the adiabatic population relaxation. Although our calculated adiabatic rates are largely consistent with previous theoretical calculations and no obvious effects of decoherence are seen, the diabatization procedure introduced here enables an explicit picture of dynamics in the branching plane, raising tantalizing questions about geometric phase effects in systems with dozens of atoms.

Vibrational relaxation of chloroiodomethane in cold argon.

Electronically exciting the C-I stretch in the molecule chloroiodomethane CH2ClI embedded in a matrix of argon at 12 K can lead to an isomer, iso-chloroiodomethane CH2Cl-I, that features a chlorine iodine bond. By temporally probing the isomer at two different frequencies of 435 nm and 485 nm, multiple timescales for isomerization and vibrational energy relaxation were inferred [T. J. Preston, et al., J. Chem. Phys. 135, 114503 (2011)]. This relaxation is studied theoretically using molecular dynamics by considering 2 and 3 dimensional models. Multiple decay rate constants of the same order of magnitude as the experiment are observed. These decay rate constants are interpreted within the context of the Landau-Teller theory. Sensitivity of the decay rate constants on the bath and system parameters shed more light into the mechanism of vibrational energy relaxation.