Nuclear quantum effects on the nonadiabatic decay mechanism of an excited hydrated electron

We present a kinetic analysis of the nonadiabatic decay mechanism of an excited state hydrated electron to the ground state. The theoretical treatment is based on a quantized, gap dependent golden rule rate constant formula which describes the nonadiabatic transition rate between two quantum states. The rate formula is expressed in terms of quantum time correlation functions of the energy gap and of the nonadiabatic coupling. These gap dependent quantities are evaluated from three different sets of mixed quantum-classical molecular dynamics simulations of a hydrated electron equilibrated (a) in its ground state, (b) in its first excited state, and (c) on a hypothetical mixed potential energy surface which is the average of the ground and the first excited electronic states. The quantized, gap dependent rate results are applied in a phenomenological kinetic equation which provides the survival probability function of the excited state electron. Although the lifetime of the equilibrated excited state electron is computed to be very short (well under 100 fs), the survival probability function for the nonequilibrium process in pump-probe experiments yields an effective excited state lifetime of around 300 fs, a value that is consistent with the findings of several experimental groups and previous theoretical estimates.     

Semiclassical treatment of thermally activated electron transfer in the intermediate to strong electronic coupling regime under the fast dielectric relaxation

The generalized nonadiabatic transition-state theory (NA-TST) (Zhao, Y.; et al. J. Chem. Phys. 2004, 121, 8854) is used to study electron transfer with use of the Zhu-Nakamura (ZN) formulas of nonadiabatic transition in the case of fast dielectric relaxation. The rate constant is expressed as a product of the well-known Marcus formula and a coefficient which represents the correction due to the strong electronic coupling. In the case of general multidimensional systems, the Monte Carlo approach is utilized to evaluate the rate by taking into account the multidimensionality of the crossing seam surface. Numerical demonstration is made by using a model system of a collection of harmonic oscillators in the Marcus normal region. The results are naturally coincident with the perturbation theory in the weak electronic coupling limit; while in the intermediate to strong electronic coupling regime where the perturbation theory breaks down the present results are in good agreement with those from the quantum mechanical flux-flux correlation function within the model of effective one-dimensional mode.     

Quantum equilibrium structure for strong nonadiabatic coupling: Reaction rate enhancement

The quantum reactive flux correlation function is computed for a two-level system using an expression for the quantum equilibrium structure appropriate for strong nonadiabatic coupling, in conjunction with quantum–classical Liouville dynamics. The magnitude of the quantum mechanical enhancement of the reaction rate as a result of strong nonadiabatic coupling is studied. The reaction rate is found to increase strongly with an increase in the nonadiabatic coupling strength as well as with a decrease in the temperature. Equilibrium quantum effects increase the ground-state contribution to the rate constant but these effects decrease the excited-state contribution.     

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.     

Semiclassical treatment of thermally activated electron transfer in the inverted region under the fast dielectric relaxation

The previously formulated semiclassical theory (Zhao, Liang, and Nakamura, J. Phys. Chem. A 2006, 110, 8204) is used to study electron transfer in the Marcus inverted case by considering multidimensional potential energy surfaces of donor and acceptor. The Zhu-Nakamura formulas of nonadiabatic transition in the case of Landau-Zener type are incorporated into the approach. The theory properly takes into account the nonadiabatic transition coupled with the nuclear tunneling and can cover the whole range from weak to strong coupling regime uniformly under the assumption of fast solvent relaxation. The numerical calculations are performed for the 12-dimensional model of shifted harmonic oscillators and demonstrate that the reaction rate with respect to the electronic coupling shows a maximum, confirming the adiabatic suppression in the strong coupling limit. The adiabatic suppression is dramatically reduced by the effect of nuclear tunneling compared to the case that the Landau-Zener formula is used. The possible extension and applications to the case of the slow solvent dynamics are discussed.     

An unusually large nonadiabatic error in the BNB molecule

The vibronic coupling model of Ko?uppel, Domcke, and Cederbaum in one dimension is introduced as a means to estimate the effects of electronic nonadiabaticity on the vibrational energy levels of molecules that exhibit vibronic coupling. For the BNB molecule, the nonadiabatic contribution to the nominal fundamental vibrational energy of the antisymmetric stretching mode is approximately -80?cm(-1). The surprisingly large effect for this mode, which corresponds to an adiabatic potential that is essentially flat near the minimum due to the vibronic interaction, is contrasted with another model system that also exhibits a flat potential (precisely, a vanishing quadratic force constant) but has a significantly larger gap between interacting electronic states. For the latter case, the nonadiabatic contribution to the level energies is about two orders of magnitude smaller even though the effect on the potential is qualitatively identical. A simple analysis shows that significant nonadiabatic corrections to energy levels should occur only when the affected vibrational frequency is large enough to be of comparable magnitude to the energy gap involved in the coupling. The results provide evidence that nonadiabatic corrections should be given as much weight as issues such as high-level electron correlation, relativistic corrections, etc., in quantum chemical calculations of energy levels for radicals with close-lying and strongly coupled electronic states even in cases where conical intersections are not obviously involved. The same can be said for high-accuracy thermochemical studies, as the zero-point vibrational energy of the BNB example contains a nonadiabatic contribution of approximately -70?cm(-1) (-0.9?kJ mol(-1)).     

Trajectory-based solution of the nonadiabatic quantum dynamics equations: an on-the-fly approach for molecular dynamics simulations

The non-relativistic quantum dynamics of nuclei and electrons is solved within the framework of quantum hydrodynamics using the adiabatic representation of the electronic states. An on-the-fly trajectory-based nonadiabatic molecular dynamics algorithm is derived, which is also able to capture nuclear quantum effects that are missing in the traditional trajectory surface hopping approach based on the independent trajectory approximation. The use of correlated trajectories produces quantum dynamics, which is in principle exact and computationally very efficient. The method is first tested on a series of model potentials and then applied to study the molecular collision of H with H(2) using on-the-fly TDDFT potential energy surfaces and nonadiabatic coupling vectors.     

Nonadiabatic molecular dynamics with solvent effects: A LR-TDDFT QM/MM study of ruthenium (II) tris (bipyridine) in water

The previously derived trajectory-based nonadiabatic molecular dynamics scheme [E. Tapavicza, I. Tavernelli, U. Rothlisberger, Phys. Rev. Lett. 98 (2007) 023001] is extended to include the coupling of the quantum system with a classically described environment. The dynamics is performed using LR-TDDFT energies and forces computed on-the-fly together with the nonadiabatic coupling vectors needed for the propagation of the nuclear coefficients according to Tully’s fewest-switches surface hopping algorithm. The resulting LR-TDDFT-QM/MM approach is applied to the study of the ultrafast relaxation of the photoexcited singlet metal-to-ligand-charge-transfer state (¹MLCT) of [Ru(bpy)₃]²⁺ (bpy = 2,2′-bipyridine) in water. The observed intersystem crossing dynamics with the triplet MLCT is in good agreement with available experimental results.     

Nonadiabatic relaxation dynamics of water anion cluster and its isotope effects by ring‐polymer molecular dynamics simulation

We have applied a recently developed hybrid quantum ring‐polymer molecular dynamics method to the nonadiabatic p → s relaxation dynamics in water anion clusters to understand the isotope effects observed in previous experiments. The average relaxation times for (H₂O)₅₀^? and (D₂O)₅₀^? were calculated at 120 and 207 fs, respectively, and are comparable to the experimental results. Therefore, we conclude that nuclear quantum effects play an essential role in understanding the observed isotope effects for water anion cluster nonadiabatic dynamics. The nonadiabatic relaxation mechanisms are also discussed in detail. © 2014 Wiley Periodicals, Inc.     

Semiclassical calculation of nonadiabatic thermal rate constants: application to condensed phase reactions

The nonadiabatic transition state theory proposed recently by Zhao et al. [J. Chem. Phys. 121, 8854 (2004)] is extended to calculate rate constants of complex systems by using the Monte Carlo and umbrella sampling methods. Surface hopping molecular dynamics technique is incorporated to take into account the dynamic recrossing effect. A nontrivial benchmark model of the nonadiabatic reaction in the condensed phase is used for the numerical test. It is found that our semiclassical results agree well with those produced by the rigorous quantum mechanical method. Comparing with available analytical approaches, we find that the simple statistical theory proposed by Straub and Berne [J. Chem. Phys. 87, 6111 (1987)] is applicable for a wide friction region although their formula is obtained using Landau-Zener [Phys. Z. Sowjetunion 2, 46 (1932); Proc. R. Soc. London, Ser. A 137, 696 (1932)] nonadiabatic transition probability along a one-dimensional diffusive coordinate. We also investigate how the nuclear tunneling events affect the dependence of the rate constant on the friction.     

Reaction-rate approach to some low temperature processes in alkali halides

The general reaction-rate theory is applied to some processes in alkali halide crystals that occur by lattice tunneling at low temperature. The theoretical equations used refer to the strong coupling case comprising both adiabatic and nonadiabatic transitions. The thermal ionization of excited F centers is considered from this point of view. The theory reproduces rather well the experimental temperature dependence of the ionization efficiency. Similarly the reorientation of OH^? impurity dipoles is treated and a good agreement is again found between theoretical and experimental relaxation rates in several host crystals.     

Dynamics of nonadiabatic chemical reactions

New methods are proposed to treat nonadiabatic chemical dynamics in realistic large molecular systems by using the Zhu-Nakamura (ZN) theory of curve-crossing problems. They include the incorporation of the ZN formulas into the Herman-Kluk type semiclassical wave packet propagation method and the trajectory surface hopping (TSH) method, formulation of the nonadiabatic transition state theory, and its application to the electron transfer problem. Because the nonadiabatic coupling is a vector in multidimensional space, the one-dimensional ZN theory works all right. Even the classically forbidden transitions can be correctly treated by the ZN formulas. In the case of electron transfer, a new formula that can improve the celebrated Marcus theory in the case of normal regime is obtained so that it can work nicely in the intermediate and strong electronic coupling regimes. All these formulations mentioned above are demonstrated to work well in comparison with the exact quantum mechanical numerical solutions and are expected to be applicable to large systems that cannot be treated quantum mechanically numerically exactly. To take into account another quantum mechanical effect, namely, the tunneling effect, an efficient method to detect caustics from which tunneling trajectories emanate is proposed. All the works reported here are the results of recent activities carried out in the author's research group. Finally, the whole set of ZN formulas is presented in Appendix.     

Ultrafast electron-transfer and solvent adiabaticity effects in viologen charge-transfer complexes

We report ultrafast electron transfer (ET) in charge-transfer complexes that shows solvent relaxation effects consistent with adiabatic crossover models of nonadiabatic ET. The complexes of either dimethyl viologen (MV) or diheptyl viologen (HV) with 4,4'-biphenol (BP) (MVBP or HVBP complexes) have identical charge-transfer spectra and kinetics in ethylene glycol with approximately 900 fs ET decay. We assign this decay time as largely due to adiabatic control of a predicted approximately 40 fs nonadiabatic ET. The MVBP decay in methanol of 470 fs is reduced in mixtures having low (2-20%) concentrations of acetonitrile to as short as 330 fs; these effects are associated with faster relaxation time in methanol and its mixtures. In contrast, HVBP has much longer ET decay in methanol (730 fs) and mixture effects that only reduce its decay to 550 fs. We identify the heptyl substituent as creating major perturbations to solvent relaxation times in the methanol solvation shell of HVBP. These charge-transfer systems have reasonably well-defined geometry with weak electronic coupling where the electronic transitions are not dependent on intramolecular motions. We used a nonadiabatic ET model with several models for adiabatic crossover predictions to discuss the small variation of energy gap with solvent and the ET rates derived from adiabatic solvent control. A time correlation model of solvent relaxation was used to define the solvent relaxation times for this case of approximately zero-barrier ET.     

Nonadiabatic surface hopping Herman-Kluk semiclassical initial value representation method revisited: applications to Tully's three model systems

The nonadiabatic surface hopping Herman-Kluk (HK) semiclassical initial value representation (SC-IVR) method for nonadiabatic problems is reformulated. The method has the same spirit as Tully's surface hopping technique [J. Chem. Phys. 93, 1061 (1990)] and almost keeps the same structure as the original single-surface HK SC-IVR method except that trajectories can hop to other surfaces according to the hopping probabilities and phases, which can be easily integrated along the paths. The method is based on a rather general nonadiabatic semiclassical surface hopping theory developed by Herman [J. Chem. Phys. 103, 8081 (1995)], which has been shown to be accurate to the first order in h and through all the orders of the nonadiabatic coupling amplitude. Our simulation studies on the three model systems suggested by Tully demonstrate that this method is practical and capable of describing nonadiabatic quantum dynamics for various coupling situations in very good agreement with benchmark calculations.     

On the properties of a primitive semiclassical surface hopping propagator for nonadiabatic quantum dynamics

A previously developed nonadiabatic semiclassical surface hopping propagator [M. F. Herman J. Chem. Phys. 103, 8081 (1995)] is further studied. The propagator has been shown to satisfy the time-dependent Schrodinger equation (TDSE) through order h, and the O(h2) terms are treated as small errors, consistent with standard semiclassical analysis. Energy is conserved at each hopping point and the change in momentum accompanying each hop is parallel to the direction of the nonadiabatic coupling vector resulting in both transmission and reflection types of hops. Quantum mechanical analysis and numerical calculations presented in this paper show that the h2 terms involving the interstate coupling functions have significant effects on the quantum transition probabilities. Motivated by these data, the h2 terms are analyzed for the nonadiabatic semiclassical propagator. It is shown that the propagator can satisfy the TDSE for multidimensional systems by including another type of nonclassical trajectories that reflect on the same surfaces. This h2 analysis gives three conditions for these three types of trajectories so that their coefficients are uniquely determined. Besides the nonadiabatic semiclassical propagator, a numerically useful quantum propagator in the adiabatic representation is developed to describe nonadiabatic transitions.     

Computing charge-exchange cross sections for Ca+ collisions with Rb at low and ultralow energies

Collisions between Ca cations and Rb atoms are computed within a quantum approach that generates the most relevant potential energy curves from accurate ab initio methods and carries out the low-energy scattering calculations by including nonadiabatic and spin-orbit coupling terms. The cross sections are obtained at relative energies typical for the likely arrangements of Rb atoms in a Magneto-Optical Trap overlapped with a Coulomb Crystal of Ca cations. The dominant nonadiabatic process is clearly identified and the efficiency of the nonadiabatic coupling terms which lead to the charge-exchange process is discussed.     

Molecular aspects of superconductivity: The role of the one-particle term at the gap formation

An exact solution for the electron-vibrational problem of the nonadiabatic molecular system has been obtained. By the quasi-particle transformation technique, the fermionic Hamiltonian has been derived and solved at the ab initio level. Results clearly and unambiguously show that the gap formation due to nonadiabatic electron–phonon coupling is mediated by the one-particle electron–phonon interaction term, whereas the two-particle one represents just a correction to the correlation energy. The temperature dependence of the gap and electronic specific heat connected with the electron–phonon coupling have also been derived.