Simulation algorithms for multidimensional nonlinear response of classical many-body systems

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.

Expected final online publication date for the Annual Review of Physical Chemistry, Volume 73 is April 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Ab Initio Transient Vibrational Spectral Analysis

Pump probe spectroscopy techniques have enabled the direct observation of a variety of transient molecular species in both ground and excited electronic states. Time-resolved vibrational spectroscopy is becoming an indispensable tool for investigating photoinduced nuclear dynamics of chemical systems of all kinds. On the other hand, a complete picture of the chemical dynamics encoded in these spectra cannot be achieved without a full temporal description of the structural relaxation, including the explicit time-dependence of vibrational coordinates that are substantially displaced from equilibrium by electronic excitation. Here we present a transient vibrational analysis protocol combining ab initio direct molecular dynamics and time-integrated normal modes introduced in this work, relying on the recent development of analytic time-dependent density functional theory (TDDFT) second derivatives for excited states. Prototypical molecules will be used as test cases, showing the evolution of the vibrational signatures that follow electronic excitation. This protocol provides a direct route to assigning the vibrations implicated in the (photo)dynamics of several (photoactive) systems.

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.

Multidimensional Measures of Response and Fluctuations in Stochastic Dynamical Systems

A new class of experiments is proposed which involve multiple measurements combined with multiple perturbations of a nonlinear classical complex system. A family of multipoint n + m - 1 dimensional measures R^(n,m) that provide complimentary information on complex systems is obtained by combining m non linear stimuli and n measurements. They represent the combined effect of causal response and noncausal spontaneous fluctuations. The proposed signals can be measured either in the frequency domain or in the time domain. Generalized fluctuation-dissipation relations which hold in the nonlinear regime restrict the number of independent techniques. Two dimensional correlation plots derived from such measurements can reveal various types of couplings among collective modes.

Phase-space algorithm for simulating quantum nonlinear response functions of bosons using stochastic classical trajectories

Using the positive P-representation of the density matrix, we develop an algorithm for calculating the quantum many-body nonlinear response functions of a system of bosons driven impulsively by external fields. The formalism maps the quantum time evolution of N boson degrees of freedom into a stochastic dynamics of 4N classical degrees of freedom. The first- and the third-order response functions are calculated by propagating the parameters of the P-representation using a set of coupled Langevin equations with multiplicative noise. These parameters serve as classical variables. Two classical ways for computing the response functions are presented. In the nonequilibrium method, an observable is calculated for weak impulsive pulses, and the response functions are obtained by taking its derivatives with respect to the pulse amplitudes. In the alternative, equilibrium simulation, the response functions are expressed in terms of time-correlation functions involving the P-representation parameters and stability matrices representing the perturbation of the trajectories. The stability matrices can be propagated simultaneously with the Langevin equations for the parameters. The formalism is generalized for a many-body boson system coupled to a harmonic bath.

Efficient calculation of time- and frequency-resolved four-wave-mixing signals

"Four-wave-mixing" is the generic name for a family of nonlinear electronic and vibrational spectroscopies. These techniques are widely used to explore dissipation, dephasing, solvation, and interstate coupling mechanisms in various material systems. Four-wave-mixing spectroscopy needs a firm theoretical support, because it delivers information on material systems indirectly, through certain transients, which are measured as functions of carrier frequencies, durations, and relative time delays of the laser pulses. The observed transients are uniquely determined by the three-pulse-induced third-order polarization. There exist two conceptually different approaches to the calculation of the nonlinear polarization. In the standard perturbative approach to nonlinear spectroscopy, the third-order polarization is expressed in terms of the nonlinear response functions. As the material systems become more complex, the evaluation of the response functions becomes cumbersome and the calculation of the signals necessitates a number of approximations. Herein, we review alternative methods for the calculation of four-wave-mixing signals, in which the relevant laser pulses are incorporated into the system Hamiltonian and the driven system dynamics is simulated numerically exactly. The emphasis is on the recently developed equation-of-motion phase-matching approach (EOM-PMA), which allows us to calculate the three-pulse-induced third-order polarization in any phase-matching direction by performing three (with the rotating wave approximation) or seven (without the rotating wave approximation) independent propagations of the density matrix. The EOM-PMA is limited to weak laser fields (its domain of validity is equivalent to the approach based on the third-order response functions) but allows for arbitrary pulse durations and automatically accounts for pulse-overlap effects. As an illustration, we apply the EOM-PMA to the calculation of optical three-pulse photon-echo two-dimensional (2D) signals for a generic model system, which represents a characteristic photophysical dynamics of large molecules or chromophores in condensed phases. The EOM-PMA is easy to implement and can straightforwardly be incorporated into any computational scheme, which provides the time-dependent density matrix or wave function of the material system of interest. In particular, EOM-PMA-based computer codes can efficiently be implemented on parallel computers. The EOM-PMA facilitates considerably the computation of four-wave-mixing signals and 2D spectra, in both vibrational and electronic spectroscopy. The EOM-PMA can be extended to higher order optical responses, e.g., heterodyned 3D IR, transient 2D IR, and other six-wave-mixing techniques.

Computational vibrational spectroscopy of peptides and proteins in one and two dimensions

Vibrational spectroscopy provides direct information on molecular environment and motions but, its interpretation is often hampered by band broadening. Over the past decade, two-dimensional (2D) vibrational spectroscopy has emerged as a promising technique to overcome a number of difficulties associated with linear spectroscopy and provided significantly detailed information on the structure and dynamics of complex molecules in condensed phases. This Account reviews recently developed computational methods used to simulate 1D and 2D vibrational spectra. The central quantity to calculate in computational spectroscopy is the spectroscopic response function, which is the product of many contributing factors such as vibrational transition energies, transition moments, and their modulations by fluctuating local environment around a solute. Accurate calculations of such linear and nonlinear responses thus require a concerted effort employing a wide range of methods including electronic structure calculation (ESC) and molecular dynamics (MD) simulation. The electronic structure calculation can provide fundamental quantities such as normal-mode frequencies and transition multipole strengths. However, since the treatable system size is limited with this method, classical MD simulation has also been used to account for the dynamics of the solvent environment. To achieve chemical accuracy, these two results are combined to generate time series of fluctuating transition frequencies and transition moments with the distributed multipole analysis, and this particular approach has been known as the hybrid ESC/MD method. For coupled multichromophore systems, vibrational properties of each chromophore such as a peptide are individually calculated by electronic structure methods and the Hessian matrix reconstruction scheme was used to obtain local mode frequencies and couplings of constituting anharmonic oscillators. The spectra thus obtained, especially for biomolecules including polypeptides and proteins, have proven to be reliable and in good agreement with experimental spectra. An alternative to the hybrid method has also been developed, where the classical limit of the vibrational response function was considered. Its main attraction is the capability to obtain the spectra directly from a set of MD trajectories. A novel development along this direction has been achieved by using quantum mechanical/molecular mechanical (QM/MM) force fields for the accurate description of vibrational anharmonicity and chromophore polarization effects. The latter aspects are critical in the 2D case because classical force fields employing harmonic intramolecular potential cannot produce reliable 2D signal. We anticipate that the computational methods presented here will continue to evolve along with experimental advancements and will be of use to further elucidate ultrafast dynamics of chemical and biological systems.

Classical nonlinear response of a chaotic system. I. Collective resonances

We develop a general semiquantitative picture of nonlinear classical response in strongly chaotic systems. In contrast to behavior in integrable or almost integrable systems, the nonlinear classical response in chaotic systems vanishes at long times. The exponential decay of the response functions in the case of strong chaos is attributed to both exponentially decaying and growing elements in the stability matrices. We calculate the linear and second-order response in one of the simplest chaotic systems: free classical motion on a compact surface of constant negative curvature. The response reveals certain features of collective resonances which do not correspond to any periodic classical trajectories. We demonstrate the relevance of the model for the interpretation of spectroscopic experiments.

Classical nonlinear response of a chaotic system. II. Langevin dynamics and spectral decomposition

The spectrum of a strongly chaotic system consists of discrete complex Ruelle-Pollicott (RP) resonances. We interpret the RP resonances as eigenstates and eigenvalues of the Fokker-Planck operator obtained by adding an infinitesimal diffusion term to the first-order Liouville operator. We demonstrate how the deterministic expression for the linear response is reproduced in the limit of vanishing noise. For the second-order response function we establish an equivalence of the spectral decomposition in the limit of vanishing noise and the long-time asymptotic expansion in the deterministic case.

Collective oscillations in the classical nonlinear response of a chaotic system

We establish a general semiquantitative phase-space picture of the classical nonlinear response in a strongly chaotic system. As opposed to the case of stable dynamics, the response functions decay exponentially at long times. Damped oscillations in response functions are attributed to collective resonances which do not correspond to any periodic classical motions. We calculate analytically the second-order response in a simple chaotic system and demonstrate the relevance of the concept for the interpretation of spectroscopic data.

Classical and quantum mechanical infrared echoes from resonantly coupled molecular vibrations

The nonlinear response function associated with the infrared vibrational echo is calculated for a quantum mechanical model of resonantly coupled, anharmonic oscillators at zero temperature. The classical mechanical response function is determined from the quantum response function by setting variant Planck's over 2pi-->0, permitting the comparison of the effects of resonant vibrational coupling among an arbitrary number of anharmonic oscillators on quantum and classical vibrational echoes. The quantum response function displays a time dependence that reflects both anharmonicity and resonant coupling, while the classical response function depends on anharmonicity only through a time-independent amplitude, and shows a time dependence controlled only by the resonant coupling. In addition, the classical response function grows without bound in time, a phenomenon associated with the nonlinearity of classical mechanics, and absent in quantum mechanics. This unbounded growth was previously identified in the response function for a system without resonant vibrational energy transfer, and is observed to persist in the presence of resonant coupling among vibrations. Quantitative agreement between classical and quantum response functions is limited to a time scale of duration inversely proportional to the anharmonicity.

Calculating fifth-order Raman signals for various molecular liquids by equilibrium and nonequilibrium hybrid molecular dynamics simulation algorithms

The fifth-order two-dimensional (2D) Raman signals have been calculated from the equilibrium and nonequilibrium (finite field) molecular dynamics simulations. The equilibrium method evaluates response functions with equilibrium trajectories, while the nonequilibrium method calculates a molecular polarizability from nonequilibrium trajectories for different pulse configurations and sequences. In this paper, we introduce an efficient algorithm which hybridizes the existing two methods to avoid the time-consuming calculations of the stability matrices which are inherent in the equilibrium method. Using nonequilibrium trajectories for a single laser excitation, we are able to dramatically simplify the sampling process. With this approach, the 2D Raman signals for liquid xenon, carbon disulfide, water, acetonitrile, and formamide are calculated and discussed. Intensities of 2D Raman signals are also estimated and the peak strength of formamide is found to be only five times smaller than that of carbon disulfide.

Effect of noise on the classical and quantum mechanical nonlinear response of resonantly coupled anharmonic oscillators

Multidimensional infrared spectroscopy probes coupled molecular vibrations in complex, condensed phase systems. Recent theoretical studies have focused on the analytic structure of the nonlinear response functions required to calculate experimental observables in a perturbative treatment of the radiation-matter interaction. Classical mechanical nonlinear response functions have been shown to exhibit unbounded growth for anharmonic, integrable systems, as a consequence of the nonlinearity of classical mechanics, a feature that is absent in a quantum mechanical treatment. We explore the analytic structure of the third-order vibrational response function for an exactly solvable quantum mechanical model that includes some of the important and theoretically challenging aspects of realistic models of condensed phase systems: anharmonicity, resonant coupling, fluctuations, and a well-defined classical mechanical limit.

Time- and frequency-resolved fluorescence spectra of nonadiabatic dissipative systems: What photons can tell us

The monitoring of the excited-state dynamics by time- and frequency-resolved spontaneous emission spectroscopy has been studied in detail for a model exhibiting an excited-state curve crossing. The model represents characteristic aspects of the photoinduced ultrafast dynamics in large molecules in the gas or condensed phases and accounts for strong nonadiabatic and electron-vibrational coupling effects, as well as for vibrational relaxation and optical dephasing. A comprehensive overview of the dependence of spontaneous emission spectra on the characteristics of the excitation and detection processes (such as carrier frequencies, pump/gate pulse durations, as well as optical dephasing) is presented. A systematic comparison of ideal spectra, which provide simultaneously perfect time and frequency resolution and thus contain maximal information on the system dynamics, with actually measurable time- and frequency-gated spectra has been carried out. The calculations of real time- and frequency-gated spectra demonstrate that complementary information on the excited-state dynamics can be extracted when the duration of the gate pulse is varied.

Interpreting nonlinear vibrational spectroscopy with the classical mechanical analogs of double-sided Feynman diagrams

Observables in coherent, multiple-pulse infrared spectroscopy may be computed from a vibrational nonlinear response function. This response function is conventionally calculated quantum-mechanically, but the challenges in applying quantum mechanics to large, anharmonic systems motivate the examination of classical mechanical vibrational nonlinear response functions. We present an approximate formulation of the classical mechanical third-order vibrational response function for an anharmonic solute oscillator interacting with a harmonic solvent, which establishes a clear connection between classical and quantum mechanical treatments. This formalism permits the identification of the classical mechanical analog of the pure dephasing of a quantum mechanical degree of freedom, and suggests the construction of classical mechanical analogs of the double-sided Feynman diagrams of quantum mechanics, which are widely applied to nonlinear spectroscopy. Application of a rotating wave approximation permits the analytic extraction of signals obeying particular spatial phase matching conditions from a classical-mechanical response function. Calculations of the third-order response function for an anharmonic oscillator coupled to a harmonic solvent are compared to numerically correct classical mechanical results.

All-forward semiclassical simulations of nonlinear response functions

We propose a quantum trajectory algorithm for computing nonlinear response functions of condensed phase molecular systems based on a time-ordered expansion of the density matrix. The nth-order response function is expressed as a sum of 2(n) impulsive response pathways representing trajectories involving zero, one, and up to n interactions with short external pulses. These are evaluated using a forward propagation algorithm based upon a Liouville space extension of the Bohmian propagation method.