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Electron transfer in strongly coupled systems, appropriate to mixed-valence compounds, is studied to explore the competition between electronic coherence and dissipation. A set of stochastic equations is derived for a spin-boson Hamiltonian with large tunneling coupling matrix element ~adiabatic regime! and strong system-bath-coupling. The bath dynamics is treated classically while the quantum character of the system is maintained. The bath dynamics is affected by the system dynamics, the effect being included by a mean-field description, valid for the adiabatic regime. Numerical solutions of the stochastic equations are presented and compared with exact quantum mechanical results. The numerical implementation of the method is straightforward and the long-time behavior of the system can be accessed. Analytic equilibrium solutions for the adiabatic regime are obtained, and we find good agreement between the long-time solution of the stochastic equations and these equilibrium solutions. We examine the dependence of the electronic population on the initial preparation of the bath and find that the proportion between oscillation ~coherence! and decay ~dissipation! is quite sensitive to this initial condition. Dirección General de Enseñanza Superior. España PB98-1120

A simulation method suited to characterize excited state proton transfer reaction kinetics in a polar solvent is developed, and applied to an intramolecular reaction of the form A–HB*→AH–B*. The model is applicable to an exothermic electronically excited proton potential energy surface (pes). The solvent modulates this surface but does not have enough coupling strength to symmetrize the proton pes with any significant probability. The proton transfer mechanism then is tunneling through an asymmetric proton pes. As the proton is a fast, quantum object relative to the solvent degrees of freedom, the tunneling is solvent configuration dependent. For each configuration, a rate constant is evaluated by a Wentzel–Kramers–Brillouin (WKB) method. Excitation to the excited reactant state initiates a coupled process of solvent relaxation to equilibrate to the new solute charge state and proton transfer. Hence, the kinetics of the reaction may be inhomogeneous. A survival time formalism is introduced to carry out th...

The effect of solvent and intramolecular dynamics on the rate of a bond breaking electron transfer reaction is investigated. The reaction takes place on a two‐dimensional potential energy surface with one coordinate the solvent’s polarization and the other the breaking bond’s displacement. The dynamics are governed by overdamped spatial diffusion along the polarization coordinate and by energy diffusion along the bond coordinate. A scheme is presented that treats the transition from rate control by the equilibrium rate constant kr (as evaluated by, e.g., a Golden Rule calculation) to dynamical control, where the rate is controlled by diffusion on the surface, with rate constant kd, that accounts for the different character of the dynamics in the two directions. The overall rate constant has the form appropriate to a consecutive reaction mechanism: k−1=k−1d+k−1r. The kd rate constant is analyzed numerically and the results compared with a number of approximation schemes. A method of analysis is developed f...

The effect of a solvent’s electronic polarization on the rate of a charge transfer reaction is studied in both continuum and discrete solvent models. An effective system Hamiltonian that contains the equilibrium solvation from the solvent electronic polarization is obtained, and leads to an effective matrix element Veff coupling the charge transfer states that is smaller than the gas phase value. Both the effective Hamiltonian and Veff are dependent on the solvent’s instantaneous nuclear configuration, and liquid state theory is used to carry out the configuration average. The solvent electronic polarization reduces the transition rate for both adiabatic and nonadiabatic reactions. A standard relation between the equilibrium solvation energy of the reactants and the solvent reorganization energy is established that permits evaluation of the effect of a molecular solvent (using a Drude model for the electronic degrees of freedom) on the rate by evaluating a partition function. This permits use of a path in...

An imaginary energy method is used to derive a rate constant expression that leads to the WKB tunneling rate at low temperature and to the quantum transition state rate at high temperature. For weak coupling, the imaginary energy method also leads to Fermi’s Golden Rule formula. The rate expression is analyzed for the model of one‐dimensional motion in a metastable potential profile with linear coupling to a bath of harmonic oscillators (Kramers’ problem). We recover the results of the Kramers, Grote‐Hynes, and Wolynes (KGHW) theory from the rate expression. Our rate expression is given, in part, in terms of partition functions and is suited to a path integral treatment. The path integral evaluation of the rate constant also yields the KGHW formula. We use the same expression to analyze the low‐temperature behavior of tunneling in a double‐well potential for a system linearly coupled to a bath, to provide a new expression for electron transfer in the case of strong coupling of the electron to its two sites.

The effects of solvent and intramolecular dynamics on the rates of bond‐breaking electron transfer (BBET) reactions is investigated. In the model we adopt, suggested by Saveant [J. Am. Chem. Soc. 109, 6788 (1987)], electron transfer and bond breaking are considered to occur as a concerted process. Thermal equilibrium rate constants kie [i=1(2) denoting the forward (reverse) reaction] are derived and exhibit a characteristic Marcus form, with the reorganization energy equal to the sum of contributions from the solvent, intramolecular vibrational and bond‐breaking coordinates. The effect of dynamics on the BBET rate constants is studied by using diffusion‐reaction equations. We assume that the intramolecular vibrational coordinate is in equilibrium and the solvent and the bond‐breaking coordinates can be out of equilibrium. The survival probabilities are derived analytically with the use of a decoupling approximation. The single exponential decay of the survival probabilities leads to nonthermal‐equilibrium...

A mean‐field theory of a localized, excess electron in a classical, nonpolar fluid, presented in an earlier paper [J. Zhu and R. I. Cukier, J. Chem. Phys. 99, 1288 (1993)], is extended to polar fluids. The mean‐field potential, and the effective potential, characterizing the electron–solvent interaction, are both modified by the addition of a long‐ranged, attractive term arising from the charge–dipole interaction between the electron density and the solvent dipoles. The attractive part of this effective interaction is similar to that of an anion–dipole interaction, which makes possible the closure of the Ornstein–Zernike equations, characterizing the solvent–solvent and electron–solvent structure, by a suitably modified form of the mean spherical approximation, familiar from ion–dipole theories. The theory is compared with simulations of an excess electron in water, carried out with a new electron–solvent pseudopotential, designed to mimic the potentials of the mean‐field theory. Agreement between the theory and simulation is good. Both theory and simulation predict that the solvation structure around the electron is weak. We find that the repulsive part of the force is dominant in the electron’s localization, and the long‐ranged force serves only to contract the electron. We explore the contrast between the electron–solvation structure in water and a polar liquid with a more ideal dipole, to confirm that the deficiencies of the mean‐field theory, constructed on the basis of an ideal‐dipole fluid, are associated with the nonideality of the water dipole.

A mean‐field, density‐functional theory for a ground‐state, localized excess electron in a classical solvent is presented. We obtain a Schrodinger equation for the electron’s wave function, with a mean‐field potential dependent on the local density of the solvent, and an integral equation for the electron–solvent correlation function, with an effective (averaged over the electron density) electron–solvent interaction potential. We show that this effective interaction is weak and use this feature to suggest closures of the integral equation characterizing the electron–solvent correlations. The coupled system of the Schrodinger and integral equations are solved self‐consistently, using an iterative method. The results are in good agreement with path‐integral and time‐dependent self‐consistent‐field simulations of an excess electron in supercritical helium. We show that these two simulation methods should agree when the electron is essentially always in its ground electronic state, as is the case for an electron in sufficiently dense helium.

The structure, energetics, and dynamics of a ground‐state, excess electron in the polar solvent methanol are simulated. Two pseudopotentials describing the interaction of the excess electron and the methanol molecules are developed. An adiabatic simulation method is used whereby the Schrodinger equation for the electron is solved in the presence of a fixed solvent configuration and the solvent configuration is advanced with the forces arising from the methanol interactions and the expectation value of the electron–methanol interaction. We find that the electron is localized with average radii of 3.1 and 2.6 A, depending on which pseudopotential is used, and both show a fairly strong solvation structure. The methanols are on average methoxyl bond‐dipole oriented toward the electron in one model and hydroxyl bond‐dipole ordered in the other. The binding energy (kinetic plus potential) of the electron fluctuates about the value −2.2 eV. The electron solvates on about a 400 fs time scale with a fast decay component of ∼30 fs. The dynamics of the electron’s center of mass can be described by a diffusion process with a diffusion constant ≂1.9×10−5 cm2 s−1.

The possibility of controlling the tunneling of a proton in a condensed phase with the use of static or time varying external fields, which couple to the transition dipole moment of the tunneling proton, is investigated. Starting from a Hamiltonian, an equation of motion describing the tunnel dynamics of the proton as a stochastically modulated, externally driven, two‐level system is derived under suitable restrictions. For external fields that satisfy a precise connection between frequency and amplitude, whereby the resulting Floquet eigenvalues (quasienergies) are degenerate, tunneling can be suppressed in the absence of the medium. With the medium present, we examine the consequences to this tunnel suppression. Static fields, if sufficiently strong, can also suppress tunneling. Expressions are derived for the effect of a static external field on the medium‐influenced, tunnel‐rate constant. The rate constant can be enhanced or decreased, depending on the sizes of the medium‐reorganization energy and ext...

Recent studies of electron transfer (ET) reactions in polar solvents have shown that solvent dynamics, in addition to solvent statics, can have a dramatic effect on the rate. ET reactions were analyzed with the aim of suggesting methods that permit a separation of solvent static and dynamic effects on the rate. Two schemes that can separate these effects are (1) the measurement of ET charge separation (typically a normal regime reaction) and charge recombination (typically an inverted regime reaction) rates in the same solvent and (2) the variation of the static (transition state) rate in a given solvent, as induced by altering the electronic coupling via a series of donor–acceptor complexes at different fixed distances. To emphasize dynamical effects, the static rate contribution should be large, and therefore in the adiabatic, strong electronic coupling limit of Marcus theory. We obtain an expression for the solvent dynamically influenced rate, for the inverted ET regime, which can describe this limit a...

Serious doubt has been cast on the validity of the independent binary collision (IBC) theory for describing nonlinear solvent density effects on the rate of vibrational to translational and rotational energy transfer [P. S. Dardi and R. I. Cukier, J. Chem. Phys. 89, 4145 (1988)]. Here, we compare the IBC theory predictions with experiment. Inconsistencies in previous applications of the IBC theory are discussed. Correcting these inconsistencies, we find that the agreement between the IBC theory and experiment is not good. Furthermore, we find that the collision diameter needed to obtain the best fit with experiment is significantly temperature dependent. We conclude that the IBC theory is not a valid theory to describe solvent density effects on vibrational relaxation.

The equilibrium, dynamic, and trapping properties of an excess electron in dense helium are simulated. An adiabatic simulation method is used whereby the Schrodinger equation for the electron in the presence of a fixed, classical solvent configuration is solved. The solvent configuration is advanced by molecular dynamics with the force on a particular helium atom arising from the classical helium–helium potential and the expectation value of the electron‐helium potential. The equilibrium properties of the electron are contrasted with those obtained by Coker and Berne [D. F. Coker and B. F. Berne, J. Chem. Phys. 89, 2128 (1988)] using a different procedure for generating helium configurations. The diffusion coefficient of the electron is obtained and, for ρ*=ρσ 3=0.9, is De=5.0×10−3 cm2 s−1. This is an order of magnitude greater than the diffusion coefficient of the helium atoms and corresponds to a very mobile electron. The distribution of times for an electron to move between donor and acceptor sites ins...

A theory of proton transfer reactions oriented toward transfer in solids is presented. This theory is used to interpret the experimental data of Nagaoka et al. [S. Nagaoka, T. Terao, F. Imashiro, A. Saika, and N. Hirota, J. Chem. Phys. 79, 4694 (1983)], who measured proton transfer rates in dicarboxylic acids by the use of NMR T1 measurements. The experiments, carried out over a large range of temperature, imply that the proton transfer rate is activated at high temperature and falls to a temperature‐independent rate at low temperature. It is assumed that the proton motion occurs by tunneling between equivalent sites and is modulated by coupling to the acoustic phonons of the solid. The rate of transition is evaluated by the ‘‘golden rule,’’ the perturbation being the splitting between the localized states. It is shown that for the potential surfaces of interest here, another pair of localized states exist in the potential well and, for the higher temperatures of the experiment, excitation to this higher ...

We present a theory of proton transfer reactions which incorporate the modulation of the proton’s potential surface by intermolecular vibrations and the effect of coupling to solvent degrees of freedom. The proton tunnels between states corresponding to it being localized in the wells of a double minimum potential. The resulting tunnel splitting depends on the intermolecular separation. The solvent response to the proton’s charge is modeled as that of a damped oscillator, allowing the introduction of friction effects in the solvent dynamics. The rate of transfer is evaluated by perturbation theory in the level splitting. We find that typically the intermolecular and solvent contributions enhance the rate relative to the values obtained in their absence. This effect is evident at low temperature where friction can enhance the rate by increasing opportunity for solvent tunneling. At high temperature the intermolecular motion enhances the rate by sampling over a distribution of tunnel splittings.

Time scales available to biomolecular simulations are limited by barriers among states in a high-dimensional configuration space. If equilibrium averages are to be computed, methods that accelerate barrier passage can be carried out by non-Boltzmann sampling. Barriers can be reduced by modifying the potential-energy function and running dynamics on the modified surface. The Boltzmann average can be restored by reweighting each point along the trajectory. We introduce a targeted reweighting scheme where some barriers are reduced, while others are not modified. If only equilibrium properties are desired, trajectories in configuration space can be generated by Langevin dynamics. Once past a transient time, these trajectories guarantee equilibrium sampling when reweighted. A relatively high-order stochastic integration method can be used to generate trajectories. The targeted reweighting scheme is illustrated by a series of double-well models with varying degrees of freedom and shown to be a very efficient method to provide the correct equilibrium distributions, in comparison with analytic results. The scheme is applied to a protein model consisting of a chain of connected beads characterized by dihedral angles and the van der Waals interactions among the beads. We investigate the sampling of configuration space for a model of a helix-turn-helix motif. The targeted reweighting is found to be essential to permit the original all-helical conformation to bend and generate turn structures while still maintaining the alpha-helical segments. Institutos Nacionales de Salud de Estados Unidos GM47274 Ministerio de Educación y Ciencia de España BFM2002-03822

We show that a recent theory of strong field spectroscopy (SFS) [R. I. Cukier and M. Morillo, Phys. Rev. B 57, 6972 (1998), M. Morillo and R. I. Cukier, J. Chem. Phys. 110, 7966 (1999)] can be used to circumvent the effects of inhomogeneous broadening on this spectroscopy. In SFS, a strong external field is used to connect, with the transition dipole, two electronic states of a solute immersed in a medium. The electronic dephasing due to the medium is characterized via the power absorbed by the solute. The average absorbed power ���̄(t) for resonant, strong fields exhibits an oscillatory decay in time, reflecting the finite change in the population difference of the electronic states and the dephasing arising from the coupling to the medium. The decay rate is characterized by d≡Δ2τc, where Δ and τc are, respectively, the strength and time constant of the correlation function characterizing the solute–medium coupling. The decay can be very rapid, on a 10–100 fs time scale, and this necessitates an indirect procedure to experimentally probe ���̄(t) that we develop. For strong, off-resonance fields, ���̄(t) returns to an exponential decay regime. The contrasting behavior of resonant and nonresonant strong fields can be used to avoid the loss of information about the homogeneous properties due to inhomogeneous broadening of the optical transition, when this broadening arises from inhomogeneity in the optical transition frequency. Ministerio de Educación y Ciencia. España PB98-1120

We discuss the effects of a strong external field on the optical transition between two electronic states of a solute immersed in a medium. The solute states may be weakly or quite strongly coupled to the medium. The electronic dephasing process is characterized via the power absorbed by the solute. The average absorbed power P¯(t) for resonant, strong fields exhibits an oscillatory decay in time, reflecting the finite change in the population difference of the electronic states, and the dephasing arising from the coupling to the medium. The coefficients of P¯(t) depend on the detuning from resonance as well as the coupling strength between the external field and the solute’s transition dipole. Our method is nonperturbative in the external field strength and shows that the spectral line shapes can be systematically altered by the application of a strong external field. We also show that for strong but off-resonance fields, P¯(t) returns to the linear response regime. Center for Fundamental Materials Research at Michigan State University Dirección General de Enseñanza Superior de España PB95-0536 Junta de Andalucía

Thermally driven materials characterized by complex energy landscapes, such as proteins, exhibit motions on a broad range of space and time scales. Principal component analysis (PCA) is often used to extract modes of motion from protein trajectory data that correspond to coherent, functional motions. In this work, two other methods, maximum covariance analysis (MCA) and canonical correlation analysis (CCA) are formulated in a way appropriate to analyze protein trajectory data. Both methods partition the coordinates used to describe the system into two sets (two measurement domains) and inquire as to the correlations that may exist between them. MCA and CCA provide rotations of the original coordinate system that successively maximize the covariance (MCA) or correlation (CCA) between modes of each measurement domain under suitable constraint conditions. We provide a common framework based on the singular value decomposition of appropriate matrices to derive MCA and CCA. The differences between and strengths and weaknesses of MCA and CCA are discussed and illustrated. The application presented here examines the correlation between the backbone and side chain of the peptide met-enkephalin as it fluctuates between open conformations, found in solution, to closed conformations appropriate to when it is bound to its receptor. Difficulties with PCA carried out in Cartesian coordinates are found and motivate a formulation in terms of dihedral angles for the backbone atoms and selected atom distances for the side chains. These internal coordinates are a more reliable basis for all the methods explored here. MCA uncovers a correlation between combinations of several backbone dihedral angles and selected side chain atom distances of met-enkephalin. It could be used to suggest residues and dihedral angles to focus on to favor specific side chain conformers. These methods could be applied to proteins with domains that, when they rearrange upon ligand binding, may have correlated functional motions or, for multi-subunit proteins, may exhibit correlated subunit motions.

The large contact distance of electron bridging dihydrogen bond (EBDB), which is over 2.4 A, is the most prominent characteristic for the imidazole-contained anion derivatives. The elongation of N-H bond and the shortening of H...H distance can be observed upon hydration and hydrogenation. Transformation from EBDB to dissociative H2 is convenient upon sequential hydrogenation. The H...H distance decreases with the enhancement of the electronegativity of the heavy atom which contacts directly with one of these two hydrogen atoms. NMR shielding of the bonding N varies significantly upon hydration and hydrogenation. The spin-spin coupling constants, 1J(H-H), is dominated predominantly by the paramagnetic spin-orbit and diamagnetic spin-orbit contributions instead of the Fermi-contact term. Enhancement of electronegativity of the heavy atom leads to the increase of 1J(H-H) coupling constants. The stabilization is enhanced upon hydration predominantly for the formation of O-H...N H bond, while it is reversed upon hydrogenation for the cleavage of big pi bond, Pi5(6). Enhancement of the stability is demonstrated by the increase of stabilization energy and vertical electron detachment energy with the electronegativity of the heavy atom. The dominant contributions for the formation of such electron bridging dihydrogen bond are the high polarity of each fragment, large electron density between two fragments, and strong bonding interaction of the bridging electron with H(N) atoms. The H...H interaction can be formed by X-Hdelta+ and Hdelta- -Y polar molecules in Hdelta+...Hdelta- and Hdelta+...e...Hdelta+ of two forms.