Communication: Standard surface hopping predicts incorrect scaling for Marcus' golden-rule rate: the decoherence problem cannot be ignored

We evaluate the accuracy of Tully's surface hopping algorithm for the spin-boson model for the case of a small diabatic coupling parameter (V). We calculate the transition rates between diabatic surfaces, and we compare our results to the expected Marcus rates. We show that standard surface hopping yields an incorrect scaling with diabatic coupling (linear in V), which we demonstrate is due to an incorrect treatment of decoherence. By modifying standard surface hopping to include decoherence events, we recover the correct scaling (~V(2)).     

Numerical study of the accuracy and efficiency of various approaches for Monte Carlo surface hopping calculations

A one-dimensional, two-state model problem with two well-separated avoided crossing points is employed to test the efficiency and accuracy of a semiclassical surface hopping technique. The use of a one-dimensional model allows for the accurate numerical evaluation of both fully quantum-mechanical and semiclassical transition probabilities. The calculations demonstrate that the surface hopping procedure employed accounts for the interference between different hopping trajectories very well and provides highly accurate transition probabilities. It is, in general, not computationally feasible to completely sum over all hopping trajectories in the semiclassical calculations for multidimensional problems. In this case, a Monte Carlo procedure for selecting important trajectories can be employed. However, the cancellation due to the different phases associated with different trajectories limits the accuracy and efficiency of the Monte Carlo procedure. Various approaches for improving the accuracy and efficiency of Monte Carlo surface hopping procedures are investigated. These methods are found to significantly reduce the statistical sampling errors in the calculations, thereby increasing the accuracy of the transition probabilities obtained with a fixed number of trajectories sampled.     

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.     

Mixed quantum-classical equilibrium

We present an analysis of the equilibrium limits of the two most widely used approaches for simulating the dynamics of molecular systems that combine both quantum and classical degrees of freedom. For a two-level quantum system connected to an infinite number of classical particles, we derive a simple analytical expression for the equilibrium mean energy attained by the self-consistent-field (Ehrenfest) method and show that it deviates substantially from Boltzmann. By contrast, "fewest switches" surface hopping achieves Boltzmann quantum state populations. We verify these analytical results with simulations.     

Small polaron hopping in Li(x)FePO4 solid solutions: coupled lithium-ion and electron mobility

Transition metal phosphates such as LiFePO(4) have been recognized as very promising electrodes for lithium-ion batteries because of their energy storage capacity combined with electrochemical and thermal stability. A key issue in these materials is to unravel the factors governing electron and ion transport within the lattice. Lithium extraction from LiFePO(4) results in a two-phase mixture with FePO(4) that limits the power characteristics owing to the low mobility of the phase boundary. This boundary is a consequence of low solubility of the parent phases, and its mobility is impeded by slow migration of the charge carriers. In principle, these limitations could be diminished in a solid solution, Li(x)FePO(4). Here, we show that electron delocalization in the solid solution phases formed at elevated temperature is due to rapid small polaron hopping and is unrelated to consideration of the band gap. We give the first experimental evidence for a strong correlation between electron and lithium delocalization events that suggests they are coupled. Furthermore, the exquisite frequency sensitivity of M?ssbauer measurements provides direct insight into the electron hopping rate.     

Dynamic arrest in a liquid of symmetric dumbbells: reorientational hopping for small molecular elongations

We present extensive equilibrium and out-of-equilibrium molecular-dynamics simulations of a liquid of symmetric dumbbell molecules, for constant packing fraction, as a function of temperature and molecular elongation. We compute diffusion constants as well as odd and even orientational correlators. The notations odd and even refer to the parity of the order l of the corresponding Legendre l polynomial, evaluated for the orientation of the molecular axis relative to its initial position. Rotational degrees of freedom of order l are arrested if, in the long-time limit, the corresponding orientational l correlator does not decay to zero. It is found that for large elongations translational and rotational degrees of freedom freeze at the same temperature. For small elongations only the even rotational degrees of freedom remain coupled to translational motions and arrest at a finite common temperature. On the contrary, the odd rotational degrees of freedom remain ergodic at all investigated temperatures. Hence, in the translationally arrested state, each molecule remains trapped in the cage formed by its neighboring molecules, but is able to perform 180 degrees rotations, which lead to relaxation only for the odd orientational correlators. The temperature dependence of the characteristic time of these residual rotations is well described by an Arrhenius law. Finally, we discuss the evidence in favor of the presence of the type-A transition for the odd rotational degrees of freedom, as predicted by mode-coupling theory for small molecular elongations. This transition is distinct from the type-B transition, associated with the arrest of the translational and even rotational degrees of freedom for small elongations, and with all degrees of freedom for large elongations. Odd orientational correlators are computed for small elongations at very low temperatures in the translationally arrested state. The obtained results suggest that hopping events restore the ergodicity of the odd rotational degrees of freedom at temperatures far below the A transition.     

On the definition of a Monte Carlo model for binary crystal growth

We show that consistency of the transition probabilities in a lattice Monte Carlo (MC) model for binary crystal growth with the thermodynamic properties of a system does not guarantee the MC simulations near equilibrium to be in agreement with the thermodynamic equilibrium phase diagram for that system. The deviations remain small for systems with small bond energies, but they can increase significantly for systems with large melting entropy, typical for molecular systems. These deviations are attributed to the surface kinetics, which is responsible for a metastable zone below the liquidus line where no growth occurs, even in the absence of a 2D nucleation barrier. Here we propose an extension of the MC model that introduces a freedom of choice in the transition probabilities while staying within the thermodynamic constraints. This freedom can be used to eliminate the discrepancy between the MC simulations and the thermodynamic equilibrium phase diagram. Agreement is achieved for that choice of the transition probabilities yielding the fastest decrease of the free energy (i.e., largest growth rate) of the system at a temperature slightly below the equilibrium temperature. An analytical model is developed, which reproduces quite well the MC results, enabling a straightforward determination of the optimal set of transition probabilities. Application of both the MC and analytical model to conditions well away from equilibrium, giving rise to kinetic phase diagrams, shows that the effect of kinetics on segregation is even stronger than that predicted by previous models.     

Exploring the role of decoherence in condensed-phase nonadiabatic dynamics: a comparison of different mixed quantum/classical simulation algorithms for the excited hydrated electron

Mixed quantum/classical (MQC) molecular dynamics simulation has become the method of choice for simulating the dynamics of quantum mechanical objects that interact with condensed-phase systems. There are many MQC algorithms available, however, and in cases where nonadiabatic coupling is important, different algorithms may lead to different results. Thus, it has been difficult to reach definitive conclusions about relaxation dynamics using nonadiabatic MQC methods because one is never certain whether any given algorithm includes enough of the necessary physics. In this paper, we explore the physics underlying different nonadiabatic MQC algorithms by comparing and contrasting the excited-state relaxation dynamics of the prototypical condensed-phase MQC system, the hydrated electron, calculated using different algorithms, including: fewest-switches surface hopping, stationary-phase surface hopping, and mean-field dynamics with surface hopping. We also describe in detail how a new nonadiabatic algorithm, mean-field dynamics with stochastic decoherence (MF-SD), is to be implemented for condensed-phase problems, and we apply MF-SD to the excited-state relaxation of the hydrated electron. Our discussion emphasizes the different ways quantum decoherence is treated in each algorithm and the resulting implications for hydrated-electron relaxation dynamics. We find that for three MQC methods that use Tully's fewest-switches criterion to determine surface hopping probabilities, the excited-state lifetime of the electron is the same. Moreover, the nonequilibrium solvent response function of the excited hydrated electron is the same with all of the nonadiabatic MQC algorithms discussed here, so that all of the algorithms would produce similar agreement with experiment. Despite the identical solvent response predicted by each MQC algorithm, we find that MF-SD allows much more mixing of multiple basis states into the quantum wave function than do other methods. This leads to an excited-state lifetime that is longer with MF-SD than with any method that incorporates nonadiabatic effects with the fewest-switches surface hopping criterion.     

Coherent switching with decay of mixing: an improved treatment of electronic coherence for non-Born-Oppenheimer trajectories

The self-consistent decay-of-mixing (SCDM) semiclassical trajectory method for electronically nonadiabatic dynamics is improved by modifying the switching probability that determines the instantaneous electronic state toward which the system decoheres. This method is called coherent switching with decay of mixing (CSDM), and it differs from the previously presented SCDM method in that the electronic amplitudes controlling the switching of the decoherent state are treated fully coherently in the electronic equations of motion for each complete passage through a strong interaction region. It is tested against accurate quantum mechanical calculations for 12 atom-diatom scattering test cases. Also tested are the SCDM method and the trajectory surface hopping method of Parlant and Gislason that requires coherent passages through each strong interaction region, and which we call the "exact complete passage" trajectory surface hopping (ECP-TSH) method. The results are compared with previously presented results for the fewest switches with time uncertainty and Tully's fewest switches (TFS) surface hopping methods and the semiclassical Ehrenfest method. We find that the CSDM method is the most accurate of the semiclassical trajectory methods tested. Including coherent passages improves the accuracy of the SCDM method (i.e., the CSDM method is more accurate than the SCDM method) but not of the trajectory surface hopping method (i.e., the ECP-TSH method is not more accurate on average than the TFS method).     

Charge Carrier Transport Mechanism in Ta2O5, TaON,and Ta3N5 Studied by Applying Polaron Hopping and Bandlike Models

TaON and Ta₃N₅ are considered promising materials for photocatalytic and photoelectrochemical water splitting. In contrast, their counterpart Ta₂O₅ does not exhibit good photocatalytic performance. This may be explained with the different charge carrier transport mechanisms in these materials, which are not well understood yet. Herein, we investigate the charge transport properties in Ta₂O₅, TaON, and Ta₃N₅ by polaron hopping and bandlike models. First, the polaron binding energies were calculated to evaluate whether the small polaron occurs in these materials. Then we performed calculations to localize the excess carriers as small polarons using a hybrid density functional. We find that the small polaron hopping is the charge transfer mechanism in Ta₂O_5, whereas our calculations indicate that this mechanism may not occur in TaON and Ta₃N₅. We also investigated the bandlike model mechanism by calculating the charge carrier mobility of these materials using the effective mass approximation, but the calculated mobility is not consistent with experimental results. This study is a first step towards understanding charge transport in oxynitrides and nitrides and furthermore establishes a simple rule to determine whether a small polaron occurs in a material.     

Single switch surface hopping for molecular quantum dynamics

The aim of this text is to present a surface hopping approximation for molecular quantum dynamics obeying a Schr?dinger equation with crossing eigenvalue surfaces. After motivating Schr?dinger equations with matrix valued potentials, we describe the single switch algorithm and present some numerical results. Then we discuss the algorithm’s mathematical justification and describe extensions to more general situations, where three eigenvalue surfaces intersect or the eigenvalues are of multiplicity two. We emphasize the generality of this surface hopping approximation for non-adiabatic transitions.     

A unified theory for charge-carrier transport in organic crystals

To characterize the crossover from bandlike transport to hopping transport in molecular crystals, we study a microscopic model that treats electron-phonon interactions explicitly. A finite-temperature variational method combining Merrifield's transformation with Bogoliubov's theorem is developed to obtain the optimal basis for an interacting electron-phonon system, which is then used to calculate the bandlike and hopping mobilities for charge carriers. Our calculations on the one dimensional (1D) Holstein model at T=0 K and finite temperatures show that the variational basis gives results that compared favorably to other analytical methods. We also study the structures of polaron states at a broad range of parameters including different temperatures. Furthermore, we calculate the bandlike and hopping mobilities of the 1D Holstein model in different parameters and show that our theory predicts universal power-law decay at low temperatures and an almost temperature independent behavior at higher temperatures, in agreement with experimental observations. In addition, we show that as the temperature increases, hopping transport can become dominant even before the polaron state changes its character. Thus, our result indicates that the self-trapping transition studied in conventional polaron theories does not necessarily correspond to the bandlike to hopping transition in the transport properties in organic molecular crystals. Finally, a comparison of our 1D results with experiments on ultrapure naphthalene crystals suggests that the theory can describe the charge-carrier mobilities quantitatively across the whole experimental temperature range.     

Critical importance of length-scale dependence in implicit modeling of hydrophobic interactions

The existence of length-scale dependence of hydrophobic solvation has important implications in the equilibrium of disordered, partially folded, and folded protein conformations. Neglecting this dependence, such as in popular solute surface-area based implicit solvent models with fixed surface tension coefficients, severely limits the ability to accurately model protein conformational equilibrium. We illustrate such fundamental limitations by examining the potentials of mean force of forming dimeric and trimeric nonpolar clusters and propose a new empirical model that effectively captures the context dependence of the local effective surface tension. Further optimization of the new model with other components of the implicit solvent force fields provides promise to significantly improve one's ability to simulate protein folding and conformational transitions. The existence of length-scale dependence of hydrophobic solvation has important implications in the equilibrium of disordered, partially folded, and folded protein conformations. Neglecting this dependence, such as in popular solute surface-area based implicit solvent models with fixed surface tension coefficients, severely limits the ability to accurately model protein conformational equilibrium. We illustrate such fundamental limitations by examining the potentials of mean force of forming dimeric and trimeric nonpolar clusters and propose a new empirical model that effectively captures the context dependence of the local effective surface tension. Further optimization of the new model with other components of the implicit solvent force fields provides promise to significantly improve one's ability to simulate protein folding and conformational transitions.     

An analysis of the accuracy of an initial value representation surface hopping wave function in the interaction and asymptotic regions

The behavior of an initial value representation surface hopping wave function is examined. Since this method is an initial value representation for the semiclassical solution of the time independent Schrodinger equation for nonadiabatic problems, it has computational advantages over the primitive surface hopping wave function. The primitive wave function has been shown to provide transition probabilities that accurately compare with quantum results for model problems. The analysis presented in this work shows that the multistate initial value representation surface hopping wave function should approach the primitive result in asymptotic regions and provide transition probabilities with the same level of accuracy for scattering problems as the primitive method.     

Decoherence and surface hopping: when can averaging over initial conditions help capture the effects of wave packet separation?

Fewest-switches surface hopping (FSSH) is a popular nonadiabatic dynamics method which treats nuclei with classical mechanics and electrons with quantum mechanics. In order to simulate the motion of a wave packet as accurately as possible, standard FSSH requires a stochastic sampling of the trajectories over a distribution of initial conditions corresponding, e.g., to the Wigner distribution of the initial quantum wave packet. Although it is well-known that FSSH does not properly account for decoherence effects, there is some confusion in the literature about whether or not this averaging over a distribution of initial conditions can approximate some of the effects of decoherence. In this paper, we not only show that averaging over initial conditions does not generally account for decoherence, but also why it fails to do so. We also show how an apparent improvement in accuracy can be obtained for a fortuitous choice of model problems, even though this improvement is not possible, in general. For a basic set of one-dimensional and two-dimensional examples, we find significantly improved results using our recently introduced augmented FSSH algorithm.     

Charge hopping in DNA.

The efficiency of charge migration through stacked Watson-Crick base pairs is analyzed for coherent hole motion interrupted by localization on guanine (G) bases. Our analysis rests on recent experiments, which demonstrate the competition of hole hopping transitions between nearest neighbor G bases and a chemical reaction of the cation G(+) with water. In addition, it has been assumed that the presence of units with several adjacent stacked G bases on the same strand leads to the additional vibronic relaxation process (G(+)G...G) --> (GG...G)(+). The latter may also compete with the hole transfer from (G(+)G...G) to a single G site, depending on the relative positions of energy levels for G(+) and (G(+)G...G). A hopping model is proposed to take the competition of these three rate steps into account. It is shown that the model includes two important limits. One corresponds to the situation where the charge relaxation inside a multiple guanine unit is faster than hopping. In this case hopping is terminated by several adjacent G bases located on the same strand, as has been observed for the GGG triple. In the opposite, slow relaxation limit the GG...G unit allows a hole to migrate further in accord with experiments on strand cleavage exploiting GG pairs. We demonstrate that for base pair sequences with only the GGG triple, the fast relaxation limit of our model yields practically the same sequence- and distance dependencies as measurements, without invoking adjustable parameters. For sequences with a certain number of repeating adenine:thymine pairs between neighboring G bases, our analysis predicts that the hole transfer efficiency varies in inverse proportion to the sequence length for short sequences, with change to slow exponential decay for longer sequences. Calculations performed within the slow relaxation limit enable us to specify parameters that provide a reasonable fit of our numerical results to the hole migration efficiency deduced from experiments with sequences containing GG pairs. The relation of the results obtained to other theoretical and experimental studies of charge transfer in DNA is discussed. We propose experiments to gain a deeper insight into complicated kinetics of charge-transfer hopping in DNA.     

Using semiclassical surface hopping for coupled partial wave calculations on systems with non-spherically symmetric potentials

It is shown that a semiclassical surface hopping (SH) approach provides a simple and efficient method for scattering calculations with non-spherically symmetric potentials. The calculations are performed by expanding the wave function in an angular momentum state basis. Since the potential is not spherically symmetric, the different angular states are coupled. The semiclassical SH method, which is typically used for problems with coupled electronic states, can, in principle, be employed for any coupled state problem. The particular SH method employed is known to provide highly accurate results for coupled electronic state problems. The method is tested on model two angular state problems using potential surfaces and couplings arising from a non-spherically symmetric scattering problem. The results for these model problems are in excellent agreement with exact quantum calculations. Full calculations, which are converged with regard to the number of angular basis states, are also performed for the non-spherically symmetric problem. It is shown that an approximation to the surface hopping amplitudes that simplifies the numerical implementation of the method provides results in excellent agreement with the full surface hopping calculation.     

Photofoams: remote control of foam destabilization by exposure to light using an azobenzene surfactant

We report evidence for photocontrolled stability and breakage of aqueous foams made from solutions of a cationic azobenzene-containing surfactant over a wide range of concentrations. Exposure to UV or visible lights results in shape and polarity switches in the surfactant molecule, which in turn affects several properties including critical micelle concentration, equilibrium surface tension, and the air-water interfacial composition (cis isomers are displaced by trans ones). We demonstrate that the trans isomer stabilizes foams, whereas the cis isomer forms unstable foams, a property that does not correlate with effects of light on surface tension, nor with total surfactant concentration. Achieving in situ breakage of foam is accordingly ascribed to the remote control of the dynamics of adsorption/desorption of the surfactant, accompanied by gradients of concentrations out of equilibrium. Photomodulation of adsorption kinetics and/or diffusion dynamics on interfaces is reached here by a noninvasive clean trigger, bringing a new tool for the study of foams.