Calculating Multidimensional Optical Spectra from Classical Trajectories.

Multidimensional optical spectra are measured from the response of a material system to a sequence of laser pulses and have the capacity to elucidate specific molecular interactions and dynamics whose influences are absent or obscured in a conventional linear absorption spectrum. Interpretation of complex spectra is supported by theoretical modeling of the spectroscopic observable, requiring implementation of quantum dynamics for coupled electrons and nuclei. Performing numerically correct quantum dynamics in this context may pose computational challenges, particularly in the condensed phase. Semiclassical methods based on calculating classical trajectories offer a practical alternative. Here I review the recent application of some semiclassical, trajectory-based methods to nonlinear molecular vibrational and electronic spectra.

Voltage fluctuations and probe frequency jitter in electric force microscopy of a conductor.

Electric force microscopy probes the statistics of electric field fluctuations from a sample surface, both through measurement of the noncontact friction exerted on the oscillating charged probe and by determination of the power spectrum of stochastic probe frequency fluctuations, referred to as "jitter." Here we calculate the frequency jitter power spectrum determined over a conducting sample of finite thickness, whose response is characterized by a dielectric function that is wavevector-dependent. These calculations complement previous predictions of the coefficient of noncontact friction in an electric force microscopy measurement for the same model, and also previous predictions of the jitter power spectrum for a dielectric continuum. The inclusion both of a finite sample thickness and a wavevector-dependent dielectric response can significantly enhance the magnitude of the predicted jitter spectrum for a conductor, relative to a simpler model of an infinitely thick dielectric continuum. These calculations provide a baseline prediction of the jitter power spectrum generated by the dynamics of conduction electrons in a metal sample.

Noncontact Friction in Electric Force Microscopy over a Conductor with Nonlocal Dielectric Response.

Electric force microscopy, in which a charged probe moves above a surface, can measure thermally generated electric field fluctuations from the sample. Noncontact friction measurements of energy loss from the probe have been performed over insulators, semiconductors, and conductors and have been interpreted in terms of the dielectric response of the sample. Noncontact friction over metal surfaces has recently been ascribed to dielectric relaxation of adsorbed molecules, motivating the examination of the baseline noncontact friction over a bare metal surface. The noncontact friction for a thin conducting film is calculated for a wavevector-dependent dielectric function, complementing previous calculations for insulators and semiconductors employing a dielectric continuum representation. Inclusion of the wavevector dependence in the dielectric response of a conductor enhances the friction and alters its dependence on tip-sample separation, relative to the continuum treatment.

2D electronic-vibrational spectroscopy with classical trajectories.

Two-dimensional electronic-vibrational (2DEV) spectra have the capacity to probe electron-nuclear interactions in molecules by measuring correlations between initial electronic excitations and vibrational transitions at a later time. The trajectory-based semiclassical optimized mean trajectory approach is applied to compute 2DEV spectra for a system with excitonically coupled electronic excited states vibronically coupled to a chromophore vibration. The chromophore mode is in turn coupled to a bath, inducing redistribution of vibrational populations. The lineshapes and delay-time dynamics of the resulting spectra compare well with benchmark calculations, both at the level of the observable and with respect to contributions from distinct spectroscopic processes.

Two-dimensional vibronic spectroscopy with semiclassical thermofield dynamics.

Thermofield dynamics is an exactly correct formulation of quantum mechanics at finite temperature in which a wavefunction is governed by an effective temperature-dependent quantum Hamiltonian. The optimized mean trajectory (OMT) approximation allows the calculation of spectroscopic response functions from trajectories produced by the classical limit of a mapping Hamiltonian that includes physical nuclear degrees of freedom and other effective degrees of freedom representing discrete vibronic states. Here, we develop a thermofield OMT (TF-OMT) approach in which the OMT procedure is applied to a temperature-dependent classical Hamiltonian determined from the thermofield-transformed quantum mapping Hamiltonian. Initial conditions for bath nuclear degrees of freedom are sampled from a zero-temperature distribution. Calculations of two-dimensional electronic spectra and two-dimensional vibrational-electronic spectra are performed for models that include excitonically coupled electronic states. The TF-OMT calculations agree very closely with the corresponding OMT results, which, in turn, represent well benchmark calculations with the hierarchical equations of motion method.

Two-dimensional vibrational-electronic spectra with semiclassical mechanics.

Two-dimensional vibrational-electronic (2DVE) spectra probe the effects on vibronic spectra of initial vibrational excitation in an electronic ground state. The optimized mean trajectory (OMT) approximation is a semiclassical method for computing nonlinear spectra from response functions. Ensembles of classical trajectories are subject to semiclassical quantization conditions, with the radiation-matter interaction inducing discontinuous transitions. This approach has been previously applied to two-dimensional infrared and electronic spectra and is extended here to 2DVE spectra. For a system including excitonic coupling, vibronic coupling, and interaction of a chromophore vibration with a resonant environment, the OMT method is shown to well approximate exact quantum dynamics.

Spectroscopic response theory with classical mapping Hamiltonians.

Exact quantum dynamics with a time-independent Hamiltonian in a discrete state space can be computed using classical mechanics through the classical Meyer-Miller-Stock-Thoss mapping Hamiltonian. In order to compute quantum response functions from classical dynamics, we extend this mapping to a quantum Hamiltonian with time-dependence arising from a classical field. This generalization requires attention to time-ordering in quantum and classical propagators. Quantum response theory with the original quantum Hamiltonian is equivalent to classical response theory with the classical mapping Hamiltonian. We elucidate the structure of classical response theory with the mapping Hamiltonian, thereby generating classical versions of the two-sided quantum density operator diagrams conventionally used to describe spectroscopic processes. This formal development can provide a foundation for new semiclassical approximations to spectroscopic observables for models in which classical nuclear degrees of freedom are introduced into a mapping Hamiltonian describing electronic states. Calculations of the temperature-dependence of two-dimensional electronic spectra for an exciton dimer using two semiclassical approaches are compared with benchmark calculations using the hierarchical equations of motion method.

Two-dimensional vibronic spectra from classical trajectories.

We present a semiclassical procedure for calculating nonlinear optical spectra from a quantum Hamiltonian with discrete electronic states. The purely electronic Hamiltonian for N states is first mapped to the associated Meyer-Miller Hamiltonian for N quantum harmonic oscillators. The classical limit is then taken, and classical nuclear degrees of freedom are introduced. Spectra are calculated by propagating the classical analogs of transition dipole operators subject to semiclassical quantization conditions on action variables. This method generalizes the optimized-mean-trajectory approach, originally developed for nonlinear vibrational spectroscopy and subsequently extended to vibronic spectroscopy, to models with multiple interacting electronic states. Calculations for two electronic excited states with displaced harmonic nuclear potentials illustrate the implementation of this approach.

Mean-trajectory approximation for electronic and vibrational-electronic nonlinear spectroscopy.

Mean-trajectory approximations permit the calculation of nonlinear vibrational spectra from semiclassically quantized trajectories on a single electronically adiabatic potential surface. By describing electronic degrees of freedom with classical phase-space variables and subjecting these to semiclassical quantization, mean-trajectory approximations may be extended to compute both nonlinear electronic spectra and vibrational-electronic spectra. A general mean-trajectory approximation for both electronic and nuclear degrees of freedom is presented, and the results for purely electronic and for vibrational-electronic four-wave mixing experiments are quantitatively assessed for harmonic surfaces with linear electronic-nuclear coupling.

Lattice model of spatial correlations in catalysis.

Optically detected single-turnover measurements of biological and inorganic catalysts provide a detailed picture of structural and dynamical influences on catalytic activity. Measurement at the single-molecule level of catalysis of a fluorogenic reaction (or its reverse) yields a stochastic fluorescence trajectory reflecting the statistics of individual reaction and product dissociation events. Analysis of time correlations displayed by this trajectory reveals reaction details inaccessible in a bulk measurement of averaged dynamics. Superresolution optical detection techniques can provide a spatial resolution over which correlations could be observed in space as well as time. A model is constructed here for spatial correlations in catalytic activity produced by an entity transported among multiple active sites. An approximation strategy based on perturbation theory in the coupling between transport and reaction dynamics is applied to calculate the mean dwell time of a reactant on an active site and the correlation between dwell times of reactants at different locations.

Vibrational coherence and energy transfer in two-dimensional spectra with the optimized mean-trajectory approximation.

The optimized mean-trajectory (OMT) approximation is a semiclassical method for computing vibrational response functions from action-quantized classical trajectories connected by discrete transitions that represent radiation-matter interactions. Here, we extend the OMT to include additional vibrational coherence and energy transfer processes. This generalized approximation is applied to a pair of anharmonic chromophores coupled to a bath. The resulting 2D spectra are shown to reflect coherence transfer between normal modes.

Thermal weights for semiclassical vibrational response functions.

Semiclassical approximations to response functions can allow the calculation of linear and nonlinear spectroscopic observables from classical dynamics. Evaluating a canonical response function requires the related tasks of determining thermal weights for initial states and computing the dynamics of these states. A class of approximations for vibrational response functions employs classical trajectories at quantized values of action variables and represents the effects of the radiation-matter interaction by discontinuous transitions. Here, we evaluate choices for a thermal weight function which are consistent with this dynamical approximation. Weight functions associated with different semiclassical approximations are compared, and two forms are constructed which yield the correct linear response function for a harmonic potential at any temperature and are also correct for anharmonic potentials in the classical mechanical limit of high temperature. Approximations to the vibrational linear response function with quantized classical trajectories and proposed thermal weight functions are assessed for ensembles of one-dimensional anharmonic oscillators. This approach is shown to perform well for an anharmonic potential that is not locally harmonic over a temperature range encompassing the quantum limit of a two-level system and the limit of classical dynamics.

An optimized semiclassical approximation for vibrational response functions.

The observables of multidimensional infrared spectroscopy may be calculated from nonlinear vibrational response functions. Fully quantum dynamical calculations of vibrational response functions are generally impractical, while completely classical calculations are qualitatively incorrect at long times. These challenges motivate the development of semiclassical approximations to quantum mechanics, which use classical mechanical information to reconstruct quantum effects. The mean-trajectory (MT) approximation is a semiclassical approach to quantum vibrational response functions employing classical trajectories linked by deterministic transitions representing the effects of the radiation-matter interaction. Previous application of the MT approximation to the third-order response function R(3)(t3, t2, t1) demonstrated that the method quantitatively describes the coherence dynamics of the t3 and t1 evolution times, but is qualitatively incorrect for the waiting-time t2 period. Here we develop an optimized version of the MT approximation by elucidating the connection between this semiclassical approach and the double-sided Feynman diagrams (2FD) that represent the quantum response. Establishing the direct connection between 2FD and semiclassical paths motivates a systematic derivation of an optimized MT approximation (OMT). The OMT uses classical mechanical inputs to accurately reproduce quantum dynamics associated with all three propagation times of the third-order vibrational response function.

Electric force microscopy of semiconductors: theory of cantilever frequency fluctuations and noncontact friction.

An electric force microscope employs a charged atomic force microscope probe in vacuum to measure fluctuating electric forces above the sample surface generated by dynamics of molecules and charge carriers. We present a theoretical description of two observables in electric force microscopy of a semiconductor: the spectral density of cantilever frequency fluctuations (jitter), which are associated with low-frequency dynamics in the sample, and the coefficient of noncontact friction, induced by higher-frequency motions. The treatment is classical-mechanical, based on linear response theory and classical electrodynamics of diffusing charges in a dielectric continuum. Calculations of frequency jitter explain the absence of contributions from carrier dynamics to previous measurements of an organic field effect transistor. Calculations of noncontact friction predict decreasing friction with increasing carrier density through the suppression of carrier density fluctuations by intercarrier Coulomb interactions. The predicted carrier density dependence of the friction coefficient is consistent with measurements of the dopant density dependence of noncontact friction over Si. Our calculations predict that in contrast to the measurement of cantilever frequency jitter, a noncontact friction measurement over an organic semiconductor could show appreciable contributions from charge carriers.

Charge carrier dynamics and interactions in electric force microscopy.

In electric force microscopy, a charged atomic force microscope tip in vacuum senses a fluctuating electrical force generated by the sample. Such measurements can in principle probe electrical noise generated by moving charge carriers in an organic semiconductor. We present a theory of cantilever frequency fluctuations in electric force microscopy, driven by coupled charge carrier dynamics and dielectric fluctuations. The connection between observable frequency fluctuations in electric force microscopy and the Casimir-Lifshitz force is described. This classical electrodynamic calculation is based on Maxwell's equations coupled to diffusive carrier transport. The effects of carrier transport and inter-carrier interactions on the spectrum of cantilever frequency noise are elucidated. We find that a simplified model of freely diffusing carriers can overestimate cantilever frequency noise by several orders of magnitude because of the neglect of interactions. Electric force microscopy measurements on an organic field effect transistor are reported and qualitatively interpreted in terms of the suppression of electrical noise from charge carriers by Coulomb interactions.

Interpreting single turnover catalysis measurements with constrained mean dwell times.

Observation of a chemical transformation at the single-molecule level yields a detailed view of kinetic pathways contributing to the averaged results obtained in a bulk measurement. Studies of a fluorogenic reaction catalyzed by gold nanoparticles have revealed heterogeneous reaction dynamics for these catalysts. Measurements on single nanoparticles yield binary trajectories with stochastic transitions between a dark state in which no product molecules are adsorbed and a fluorescent state in which one product molecule is present. The mean dwell time in either state gives information corresponding to a bulk measurement. Quantifying fluctuations from mean kinetics requires identifying properties of the fluorescence trajectory that are selective in emphasizing certain dynamic processes according to their time scales. We propose the use of constrained mean dwell times, defined as the mean dwell time in a state with the constraint that the immediately preceding dwell time in the other state is, for example, less than a variable time. Calculations of constrained mean dwell times for a kinetic model with dynamic disorder demonstrate that these quantities reveal correlations among dynamic fluctuations at different active sites on a multisite catalyst. Constrained mean dwell times are determined from measurements of single nanoparticle catalysis. The results indicate that dynamical fluctuations at different active sites are correlated, and that especially rapid reaction events produce particularly slowly desorbing product molecules.

Dynamics of a myoglobin mutant enzyme: 2D IR vibrational echo experiments and simulations.

Myoglobin (Mb) double mutant T67R/S92D displays peroxidase enzymatic activity in contrast to the wild type protein. The CO adduct of T67R/S92D shows two CO absorption bands corresponding to the A(1) and A(3) substates. The equilibrium protein dynamics for the two distinct substates of the Mb double mutant are investigated by using two-dimensional infrared (2D IR) vibrational echo spectroscopy and molecular dynamics (MD) simulations. The time-dependent changes in the 2D IR vibrational echo line shapes for both of the substates are analyzed using the center line slope (CLS) method to obtain the frequency-frequency correlation function (FFCF). The results for the double mutant are compared to those from the wild type Mb. The experimentally determined FFCF is compared to the FFCF obtained from molecular dynamics simulations, thereby testing the capacity of a force field to determine the amplitudes and time scales of protein structural fluctuations on fast time scales. The results provide insights into the nature of the energy landscape around the free energy minimum of the folded protein structure.

Quantifying electric field gradient fluctuations over polymers using ultrasensitive cantilevers.

An ultrasensitive cantilever, oscillating parallel to a surface in vacuum, is used to probe weak thermal electric field gradient fluctuations over thin polymer films. We measure the power spectrum of cantilever frequency fluctuations as a function of cantilever height and voltage over polymers of various compositions and thicknesses. The data are well described by a linear-response theory that calculates stochastic electric fields arising from thermally driven dielectric fluctuations.

One and Two Dimensional Vibronic Spectra for an Exciton Dimer from Classical Trajectories.

We extend the semiclassical optimized mean trajectory (OMT) procedure to calculate electronic spectra for a dimer with excitonic and vibronic interactions. The electronic part of the quantum Hamiltonian is expressed in the Miller-Meyer-Stock-Thoss form with one fictitious harmonic oscillator per electronic state, and the classical limit is taken, transforming a quantum Hamiltonian governing discrete states to an equivalent classical form. The ad hoc addition of classical nuclear degrees of freedom and electron-nuclear coupling yields a classical Hamiltonian with one degree of freedom per each electronic state and also per each nuclear motion. Semiclassical quantization is applied to this Hamiltonian through the OMT, originally developed to describe nuclear dynamics on a single potential surface and subsequently generalized to include electronic transitions. The accuracy and practicality of this trajectory-based method is assessed for an excitonically coupled dimer. The semiclassical one- and two-dimensional spectra are shown to compare well with quantum dynamical calculations performed with the hierarchical equations of motion method.