Projections of quantum observables onto classical degrees of freedom in mixed quantum-classical simulations: understanding linear response failure for the photoexcited hydrated electron

We present a general analytic method for understanding how specific motions of a classical bath influence the dynamics of quantum-mechanical observables in mixed quantum-classical molecular dynamics simulations. We apply our method and develop expressions for the special case of quantum solvation, allowing us to examine how specific classical solvent motions couple to the equilibrium energy fluctuations and nonequilibrium energy relaxation of a quantum-mechanical solute. As a first application of our formalism, we investigate the motions of classical water underlying the equilibrium and nonequilibrium excited-state solvent response functions of the hydrated electron; the results allow us to explain why the linear response approximation fails for this system.

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.

A computationally efficient exact pseudopotential method. II. Application to the molecular pseudopotential of an excess electron interacting with tetrahydrofuran (THF)

In the preceding paper, we presented an analytic reformulation of the Phillips-Kleinman (PK) pseudopotential theory. In the PK theory, the number of explicitly treated electronic degrees of freedom in a multielectron problem is reduced by forcing the wave functions of the few electrons of interest (the valence electrons) to be orthogonal to those of the remaining electrons (the core electrons); this results in a new Schrodinger equation for the valence electrons in which the effects of the core electrons are treated implicitly via an extra term known as the pseudopotential. Although this pseudopotential must be evaluated iteratively, our reformulation of the theory allows the exact pseudopotential to be found without ever having to evaluate the potential energy operator, providing enormous computational savings. In this paper, we present a detailed computational procedure for implementing our reformulation of the PK theory, and we illustrate our procedure on the largest system for which an exact pseudopotential has been calculated, that of an excess electron interacting with a tetrahyrdrofuran (THF) molecule. We discuss the numerical stability of several approaches to the iterative solution for the pseudopotential, and find that once the core wave functions are available, the full e(-)-THF pseudopotential can be calculated in less than 3 s on a relatively modest single processor. We also comment on how the choice of basis set affects the calculated pseudopotential, and provide a prescription for correcting unphysical behavior that arises at long distances if a localized Gaussian basis set is used. Finally, we discuss the effective e(-)-THF potential in detail, and present a multisite analytic fit of the potential that is suitable for use in molecular simulation.

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.

Nonadiabatic Molecular Dynamics Simulations of Correlated Electrons in Solution. 1. Full Configuration Interaction (CI) Excited-State Relaxation Dynamics of Hydrated Dielectrons

The hydrated dielectron is composed of two excess electrons dissolved in liquid water that occupy a single cavity; in both its singlet and triplet spin states there is a significant exchange interaction so the two electrons cannot be considered to be independent. In this paper and the following paper,we present the results of mixed quantum/classical molecular dynamics simulations of the nonadiabatic relaxation dynamics of photoexcited hydrated dielectrons, where we use full configuration interaction (CI) to solve for the two-electron wave function at every simulation time step. To the best of our knowledge, this represents the first systematic treatment of excited-state solvation dynamics where the multiple-electron problem is solved exactly. The simulations show that the effects of exchange and correlation contribute significantly to the relaxation dynamics. For example, spin-singlet dielectrons relax to the ground state on a time scale similar to that of single electrons excited at the same energy, but spin-triplet dielectrons relax much faster. The difference in relaxation dynamics is caused by exchange and correlation: The Pauli exclusion principle imposes very different electronic structure when the electrons' spins are singlet paired than when they are triplet paired, altering the available nonadiabatic relaxation pathways. In addition, we monitor how electronic correlation changes dynamically during nonadiabatic relaxation and show that solvent dynamics cause electron correlation to evolve quite differently for singlet and triplet dielectrons. Despite such differences, our calculations show that both spin states are stable to excited-state dissociation, but that the excited-state stability has different origins for the two spin states. For singlet dielectrons, the stability depends on whether the solvent structure can rearrange to create a second cavity before the ground state is reached. For triplet dielectrons, in contrast, electronic correlation ensures that the two electrons do not dissociate, even if the dielectron is artificially kept from reaching the ground state. In addition, both singlet and triplet dielectrons change shape dramatically during relaxation, so that linear response fails to describe the solvation dynamics for either spin state. In the following paper (Larsen, R. E.; Schwartz, B. J. J. Phys. Chem. B 2006, 110, 9692), we use these simulations to calculate the pump-probe spectroscopic signal expected for photoexcited hydrated dielectrons and to predict an experiment to observe hydrated dielectrons directly.

Nonadiabatic Molecular Dynamics Simulations of Correlated Electrons in Solution. 2. A Prediction for the Observation of Hydrated Dielectrons with Pump−Probe Spectroscopy

The hydrated dielectron is a highly correlated, two-electron, solvent-supported state consisting of two spin-paired electrons confined to a single cavity in liquid water. Although dielectrons have been predicted to exist theoretically and have been used to explain the lack of ionic strength effect in the bimolecular reaction kinetics of hydrated electrons, they have not yet been observed directly. In this paper, we use the extensive nonadiabatic mixed quantum/classical excited-state molecular dynamics simulations from the previous paper to calculate the transient spectroscopy of hydrated dielectrons. Because our simulations use full configuration interaction (CI) to determine the ground and excited state two-electron wave functions at every instant, our nonequilibrium simulations allow us to compute the absorption, stimulated emission (SE), and bleach spectroscopic signals of both singlet and triplet dielectrons following excitation by ultraviolet light. Excited singlet dielectrons are predicted to display strong SE in the mid infrared and a transient absorption in the near-infrared. The near-infrared transient absorption of the singlet dielectron, which occurs near the peak of the (single) hydrated electron's equilibrium absorption, arises because the two electrons tend to separate in the excited state. In contrast, excitation of the hydrated electron gives a bleach signal in this wavelength region. Thus, our calculations suggest a clear pump-probe spectroscopic signature that may be used in the laboratory to distinguish hydrated singlet dielectrons from hydrated electrons: By choosing an excitation energy that is to the blue of the peak of the hydrated electron's absorption spectrum and probing near the maximum of the single electron's absorption, the single electron's transient bleach signal should shrink or even turn into a net absorption as sample conditions are varied to produce more dielectrons.

Full Configuration Interaction Computer Simulation Study of the Thermodynamic and Kinetic Stability of Hydrated Dielectrons

The hydrated electron is a unique solvent-supported state comprised of an excess electron that is confined to a cavity by the surrounding water. Theoretical studies have suggested that two-electron solvent-supported states also can be formed; in particular, simulations indicate that two excess electrons could pair up and occupy a single cavity, forming a so-called hydrated dielectron. Although hydrated dielectrons have not been observed directly by experiment, their existence has been posited to explain the lack of an ionic strength effect in hydrated electron bimolecular annihilation [Schmidt, K. H.; Bartels, D. M. Chem. Phys. 1995, 190, 145]. To determine whether dielectrons may be created in the laboratory, we use thermodynamic integration (TI), combined with mixed quantum/classical molecular dynamics simulation, to examine the thermodynamic stability of hydrated electrons and dielectrons. For the dielectron calculations, we solve the two-electron quantum problem using full configuration interaction. Our results suggest that hydrated dielectrons are thermodynamically unstable relative to separated (single) hydrated electrons, although we also show that increasing the pressure could drive the equilibrium toward the formation of dielectrons. Because the simulations suggest that hydrated dielectrons are kinetically stable, we also examine a scenario for creating metstable, nonequilibrium populations of dielectrons, which involves the capture of a newly injected electron by a preexisting, equilibrated hydrated electron. These calculations, which allow for the full nonadiabatic relaxation of the injected electron, show that hydrated electrons may indeed act as trapping sites for unequilibrated electrons, so that capture may be a viable mechanism for creating dielectrons. We suggest possible experimental procedures to create such nonequilibrium hydrated dielectrons using either pulse radiolysis or ultrafast spectroscopic techniques.

Mean-field dynamics with stochastic decoherence (MF-SD): A new algorithm for nonadiabatic mixed quantum/classical molecular-dynamics simulations with nuclear-induced decoherence

The key factors that distinguish algorithms for nonadiabatic mixed quantum/classical (MQC) simulations from each other are how they incorporate quantum decoherence-the fact that classical nuclei must eventually cause a quantum superposition state to collapse into a pure state-and how they model the effects of decoherence on the quantum and classical subsystems. Most algorithms use distinct mechanisms for modeling nonadiabatic transitions between pure quantum basis states ("surface hops") and for calculating the loss of quantum-mechanical phase information (e.g., the decay of the off-diagonal elements of the density matrix). In our view, however, both processes should be unified in a single description of decoherence. In this paper, we start from the density matrix of the total system and use the frozen Gaussian approximation for the nuclear wave function to derive a nuclear-induced decoherence rate for the electronic degrees of freedom. We then use this decoherence rate as the basis for a new nonadiabatic MQC molecular-dynamics (MD) algorithm, which we call mean-field dynamics with stochastic decoherence (MF-SD). MF-SD begins by evolving the quantum subsystem according to the time-dependent Schrodinger equation, leading to mean-field dynamics. MF-SD then uses the nuclear-induced decoherence rate to determine stochastically at each time step whether the system remains in a coherent mixed state or decoheres. Once it is determined that the system should decohere, the quantum subsystem undergoes an instantaneous total wave-function collapse onto one of the adiabatic basis states and the classical velocities are adjusted to conserve energy. Thus, MF-SD combines surface hops and decoherence into a single idea: decoherence in MF-SD does not require the artificial introduction of reference states, auxiliary trajectories, or trajectory swarms, which also makes MF-SD much more computationally efficient than other nonadiabatic MQC MD algorithms. The unified definition of decoherence in MF-SD requires only a single ad hoc parameter, which is not adjustable but instead is determined by the spatial extent of the nonadiabatic coupling. We use MF-SD to solve a series of one-dimensional scattering problems and find that MF-SD is as quantitatively accurate as several existing nonadiabatic MQC MD algorithms and significantly more accurate for some problems.

The role of solvent structure in the absorption spectrum of solvated electrons: Mixed quantum/classical simulations in tetrahydrofuran

In polar fluids such as water and methanol, the peak of the solvated electron's absorption spectrum in the red has been assigned as a sum of transitions between an s-like ground state and three nearly degenerate p-like excited states bound in a quasispherical cavity. In contrast, in weakly polar solvents such as tetrahydrofuran (THF), the solvated electron has an absorption spectrum that peaks in the mid-infrared, but no definitive assignment has been offered about the origins of the spectrum or the underlying structure. In this paper, we present the results of adiabatic mixed quantum/classical molecular dynamic simulations of the solvated electron in THF, and provide a detailed explanation of the THF-solvated electron's absorption spectrum and electronic structure. Using a classical solvent model and a fully quantum mechanical excess electron, our simulations show that although the ground and first excited states are bound in a quasispherical cavity, a multitude of other, nearby solvent cavities support numerous, nearly degenerate, bound excited states that have little Franck-Condon overlap with the ground state. We show that these solvent cavities, which are partially polarized so that they act as electron trapping sites, are an inherent property of the way THF molecules pack in the liquid. The absorption spectrum is thus assigned to a sum of bound-to-bound transitions between a localized ground state and multiple disjoint excited states scattered throughout the fluid. Furthermore, we find that the usual spherical harmonic labels (e.g., s-like, p-like) are not good descriptors of the excited-state wave functions of the solvated electron in THF. Our observation of multiple disjoint excited states is consistent with femtosecond pump-probe experiments in the literature that suggest that photoexcitation of solvated electrons in THF causes them to relocalize into solvent cavities far from where they originated.

Elucidating the initial dynamics of electron photodetachment from atoms in liquids using variably-time-delayed resonant multiphoton ionization

We study the photodetachment of electrons from sodium anions in room temperature liquid tetrahydrofuran (THF) using a new type of three-pulse pump-probe spectroscopy. Our experiments use two variably-time-delayed pulses for excitation in what is essentially a resonant 1+1 two-photon ionization: By varying the arrival time of the second excitation pulse, we can directly observe how solvent motions stabilize and trap the excited electron prior to electron detachment. Moreover, by varying the arrival times of the ionization (excitation) and probe pulses, we also can determine the fate of the photoionized electrons and the distance they are ejected from their parent Na atoms. We find that as solvent reorganization proceeds, the second excitation pulse becomes less effective at achieving photoionization, and that the solvent motions that stabilize the excited electron following the first excitation pulse occur over a time of approximately 450 fs. We also find that there is no spectroscopic evidence for significant solvent relaxation after detachment of the electron is complete. In combination with the results of previous experiments and molecular dynamics simulations, the data provide new insight into the role of the solvent in solution-phase electron detachment and charge-transfer-to-solvent reactions.

Conjugated polymers as molecular materials: how chain conformation and film morphology influence energy transfer and interchain interactions

The electronic structure of conjugated polymers is of current interest because of the wide range of potential applications for such materials in optoelectronic devices. It is increasingly clear that the electronic properties of conjugated polymers depend sensitively on the physical conformation of the polymer chains and the way the chains pack together in films. This article reviews the evidence that interchain electronic species do form in conjugated polymer films, and that their number and chemical nature depend on processing conditions; the chain conformation, degree of interchain contact, and rate of energy transfer can be controlled by factors such as choice of solvent, polymer concentration, thermal annealing, presence of electrically charged side groups, and encapsulation of the polymer chains in mesoporous silica. Taken together, the results reconcile many contradictions in the literature and provide a prescription for the optimization of conjugated polymer film morphology for device applications.

Manipulating the production and recombination of electrons during electron transfer: Femtosecond control of the charge-transfer-to-solvent (CTTS) dynamics of the sodium anion

The scavenging of a solvated electron represents the simplest possible electron-transfer (ET) reaction. In this work, we show how a sequence of femtosecond laser pulses can be used to manipulate an ET reaction that has only electronic degrees of freedom: the scavenging of a solvated electron by a single atom in solution. Solvated electrons in tetrahydrofuran are created via photodetachment using the charge-transfer-to-solvent (CTTS) transition of sodide (Na(-)). The CTTS process ejects electrons to well-defined distances, leading to three possible initial geometries for the back ET reaction between the solvated electrons and their geminate sodium atom partners (Na(0)). Electrons that are ejected within the same solvent cavity as the sodium atom (immediate contact pairs) undergo back ET in approximately 1 ps. Electrons ejected one solvent shell away from the Na(0) (solvent-separated contact pairs) take hundreds of picoseconds to undergo back ET. Electrons ejected more than one solvent shell from the sodium atom (free solvated electrons) do not recombine on subnanosecond time scales. We manipulate the back ET reaction for each of these geometries by applying a "re-excitation" pulse to promote the localized solvated electron ground state into a highly delocalized excited-state wave function in the fluid's conduction band. We find that re-excitation of electrons in immediate contact pairs suppresses the back ET reaction. The kinetics at different probe wavelengths and in different solvents suggest that the recombination is suppressed because the excited electrons can relocalize into different solvent cavities upon relaxation to the ground state. Roughly one-third of the re-excited electrons do not collapse back into their original solvent cavities, and of these, the majority relocalize into a cavity one solvent shell away. In contrast to the behavior of the immediate pair electrons, re-excitation of electrons in solvent-separated contact pairs leads to an early time enhancement of the back ET reaction, followed by a longer-time recombination suppression. The recombination enhancement results from the improved overlap between the electron and the Na(0) one solvent shell away due to the delocalization of the wave function upon re-excitation. Once the excited state decays, however, the enhanced back ET is shut off, and some of the re-excited electrons relocalize even farther from their geminate partners, leading to a long-time suppression of the recombination; the rates for recombination enhancement and relocalization are comparable. Enhanced recombination is still observed even when the re-excitation pulse is applied hundreds of picoseconds after the initial CTTS photodetachment, verifying that solvent-separated contact pairs are long-lived, metastable entities. Taken together, all these results, combined with the simplicity and convenient spectroscopy of the sodide CTTS system, allow for an unprecedented degree of control that is a significant step toward building a full molecular-level picture of condensed-phase ET reactions.