The Ehrenfest method with quantum corrections to simulate the relaxation of molecules in solution: Equilibrium and dynamics

On detailed balance in nonadiabatic dynamics: From spin spheres to equilibrium ellipsoids

Trajectory-based methods that propagate classical nuclei on multiple quantum electronic states are often used to simulate nonadiabatic processes in the condensed phase. A long-standing problem of these methods is their lack of detailed balance, meaning that they do not conserve the equilibrium distribution. In this article, we investigate ideas for restoring detailed balance in mixed quantum-classical systems by tailoring the previously proposed spin-mapping approach to thermal equilibrium. We find that adapting the spin magnitude can recover the correct long-time populations but is insufficient to conserve the full equilibrium distribution. The latter can however be achieved by a more flexible mapping of the spin onto an ellipsoid, which is constructed to fulfill detailed balance for arbitrary potentials. This ellipsoid approach solves the problem of negative populations that has plagued previous mapping approaches and can therefore be applied also to strongly asymmetric and anharmonic systems. Because it conserves the thermal distribution, the method can also exploit efficient sampling schemes used in standard molecular dynamics, which drastically reduces the number of trajectories needed for convergence. The dynamics does however still have mean-field character, as is observed most clearly by evaluating reaction rates in the golden-rule limit. This implies that although the ellipsoid mapping provides a rigorous framework, further work is required to find an accurate classical-trajectory approximation that captures more properties of the true quantum dynamics.

Efficient Surface Hopping Approach for Modeling Charge Transport in Organic Semiconductors

The trajectory surface hopping (TSH) method is nowadays widely applied to study the charge/exciton transport process in organic semiconductors (OSCs). In the present study, we systematically examine the performance of two approximations in the fewest switched surface hopping (FSSH) simulations for charge transport (CT) in several representative OSCs. These approximations include (i) the substitution of the nuclear velocity scaling along the nonadiabatic coupling vector (NCV) by rescaling the hopping probability with the Boltzmann factor (Boltzmann correction (BC)) and (ii) a phenomenological approach to treat the quantum feedback from the electronic system to the nuclear system (implicit charge relaxation (IR)) in the OSCs. We find that charge mobilities computed by FSSH-BC-IR are in very good agreement with the mobilities obtained by standard FSSH simulations with explicit charge relaxation (FSSH-ER), however, at reduced computational cost. A key parameter determining the charge carrier mobility is the reorganization energy, which is sensitively dependent on DFT functionals applied. By employing the IR approximation, the FSSH method allows systematic investigation of the effect of the reorganization energies obtained by different DFT functionals like B3LYP or ωB97XD on CT in OSCs. In comparison to the experiments, FSSH-BC-IR using ωB97XD reorganization energy underestimates mobilities in the low-coupling regime, which may indicate the lack of nuclear quantum effects (e.g., zero point energy (ZPE)) in the simulations. The mobilities obtained by FSSH-BC-IR using the B3LYP reorganization energy agree well with experimental values in 3 orders of magnitude. The accidental agreement may be the consequence of the underestimation of the reorganization energy by the B3LYP functional, which compensates for the neglect of nuclear ZPE in the simulations.

Performance of Mixed Quantum-Classical Approaches on Modeling the Crossover from Hopping to Bandlike Charge Transport in Organic Semiconductors

In the present study, several mixed quantum-classical (MQC) methods are applied to on-the-fly nonadiabatic molecular dynamics simulations of hole transport in molecular organic semiconductors (OSCs). The tested MQC methods contain the mean-field Ehrenfest (MFE), trajectory surface hopping (TSH) approaches based on Tully's fewest switches surface hopping (FSSH) and the global flux surface hopping (GFSH), the latter in the diabatic/adiabatic representation, and a Landau-Zener type trajectory surface hopping (LZSH). We also tested several correction schemes which were proposed to identify trivial crossings and to remove unphysical long-range charge transfers due to decoherence corrections. In addition, several cost-effective approaches for the nuclear velocity adjustment after an energy-allowed/energy-forbidden hop are investigated with respect to detailed balance and internal consistency conditions. To model a broad spectrum of OSCs with different charge transport characteristics, we derived from the anthracene structural model the construction of two additional models by uniformly scaling down the electronic couplings by the factors of 0.1 and 0.5. Anthracene shows a bandlike charge transport mechanism, characterized by slightly delocalized charge carriers 'diffusing' through the crystal. For smaller couplings, the mechanism changes to a hopping type, characteristically differing in the charge delocalization and temperature dependence. The MFE and corrected adiabatic TSH approaches are able to quantitatively reproduce the expected behavior, while the diabatic LZSH method fails for large couplings, as do approaches which are based on the hopping of localized charge between neighboring sites. Moreover, we find that while the hole mobility of the anthracene crystal simulated using the celebrated Marcus theory is in good agreement with the experimental value, its agreement has to be regarded as an accident due to the overestimation of the prefactor in the Marcus rate equation.

Modeling nonadiabatic dynamics in condensed matter materials: some recent advances and applications

This review focuses on recent developments in the field of nonadiabatic molecular dynamics (NA-MD), with particular attention given to condensed-matter systems. NA-MD simulations for small molecular systems can be performed using high-level electronic structure (ES) calculations, methods accounting for the quantization of nuclear motion, and using fewer approximations in the dynamical methodology itself. Modeling condensed-matter systems imposes many limitations on various aspects of NA-MD computations, requiring approximations at various levels of theory-from the ES, to the ways in which the coupling of electrons and nuclei are accounted for. Nonetheless, the approximate treatment of NA-MD in condensed-phase materials has gained a spin lately in many applied studies. A number of advancements of the methodology and computational tools have been undertaken, including general-purpose methods, as well as those tailored to nanoscale and condensed matter systems. This review summarizes such methodological and software developments, puts them into the broader context of existing approaches, and highlights some of the challenges that remain to be solved.

Pressure effects on the vibrational and rotational relaxation of vibrationally excited OH (ν, J) in an argon bath

Quasi-classical molecular dynamics simulations were used to study the energy relaxation of an initially non-rotating, vibrationally excited (ν = 4) hydroxyl radical (OH) in an Ar bath at 300 K and at high pressures from 50 atm to 400 atm. A Morse oscillator potential represented the OH, and two sets of interaction potentials were used based on whether the Ar-H potential was a Buckingham (Exp6) or a Lennard-Jones (LJ) potential. The vibrational and rotational energies were monitored for 25 000-90 000 ps for Exp6 trajectories and 5000 ps for LJ trajectories. Comparisons to measured vibrational relaxation rates show that Exp6 rates are superior. Simulated initial vibrational relaxation rates are linearly proportional to pressure, implying no effect of high-pressure breakdown in the isolated binary collision approximation. The vibrational decay curves upward from single-exponential decay. A model based on transition rates that exponentially depend on the anharmonic energy gap between vibrational levels fits the vibrational decay well at all pressures, suggesting that anharmonicity is a major cause of the curvature. Due to the competition of vibration-to-rotation energy transfer and bath gas relaxation, the rotational energy overshoots and then relaxes to its thermal value. Approximate models with adjustable rates for this competition successfully reproduced the rotational results. These models show that a large fraction of the vibrational energy loss is initially converted to rotational energy but that fraction decreases rapidly as the vibrational energy content of OH decreases. While simulated rates change dramatically between Exp6 and LJ potentials, the mechanisms remain the same.

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.

A new trajectory branching approximation to propagate the mixed quantum-classical Liouville equation

Starting from the mixed quantum-classical Liouville (MQCL) equation, we derive a new trajectory branching method as a modification to the conventional mean field approximation. In the new method, the mean field approximation is used to propagate the mixed quantum-classical dynamics for short times. When the mean field description becomes invalid, new trajectories are added in the simulation by branching the single trajectory into multiple ones. To achieve this, a new set of variables are defined to monitor the deviations of the dynamics on different potential energy surfaces from the reference mean field trajectory, and their equations of motion are derived from the MQCL equation based on the method of first moment expansion. The new method is tested on several one-dimensional two surface problems and is shown to correctly solve the problem of the mean field approximation in several cases.

Juxtaposing density matrix and classical path-based wave packet dynamics

In many physical, chemical, and biological systems energy and charge transfer processes are of utmost importance. To determine the influence of the environment on these transport processes, equilibrium molecular dynamics simulations become more and more popular. From these simulations, one usually determines the thermal fluctuations of certain energy gaps, which are then either used to perform ensemble-averaged wave packet simulations, also called Ehrenfest dynamics, or to employ a density matrix approach via spectral densities. These two approaches are analyzed through energy gap fluctuations that are generated to correspond to a predetermined spectral density. Subsequently, density matrix and wave packet simulations are compared through population dynamics and absorption spectra for different parameter regimes. Furthermore, a previously proposed approach to enforce the correct long-time behavior in the wave packet simulations is probed and an improvement is proposed.

Hybrid quantum/classical simulations of the vibrational relaxation of the amide I mode of N-methylacetamide in D2O solution

Hybrid quantum/classical molecular dynamics (MD) is applied to simulate the vibrational relaxation (VR) of the amide I mode of deuterated N-methylacetamide (NMAD) in aqueous (D(2)O) solution. A novel version of the vibrational molecular dynamics with quantum transitions (MDQT) treatment is developed in which the amide I mode is treated quantum mechanically while the remaining degrees of freedom are treated classically. The instantaneous normal modes of the initially excited NMAD molecule (INM(0)) are used as internal coordinates since they provide a proper initial partition of the system in quantum and classical subsystems. The evolution in time of the energy stored in each individual normal mode is subsequently quantified using the hybrid quantum-classical instantaneous normal modes (INM(t)). The identities of both the INM(0)s and the INM(t)s are tracked using the equilibrium normal modes (ENMs) as templates. The results extracted from the hybrid MDQT simulations show that the quantum treatment of the amide I mode accelerates the whole VR process versus pure classical simulations and gives better agreement with experiments. The relaxation of the amide I mode is found to be essentially an intramolecular vibrational redistribution (IVR) process with little contribution from the solvent, in agreement with previous theoretical and experimental studies. Two well-defined relaxation mechanisms are identified. The faster one accounts for ≈40% of the total vibrational energy that flows through the NMAD molecule and involves the participation of the lowest frequency vibrations as short-life intermediate modes. The second and slower mechanism accounts for the remaining ≈60% of the energy released and is associated to the energy flow through specific mid-range and high-frequency modes.