The molecular origins of the two-dimensional Raman spectrum of an atomic liquid. I. Molecular dynamics simulation

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 Raman spectroscopy of Lennard-Jones liquids via ring-polymer molecular dynamics

The simulation of multidimensional vibrational spectroscopy of condensed-phase systems including nuclear quantum effects is challenging since full quantum-mechanical calculations are still intractable for large systems comprising many degrees of freedom. Here, we apply the recently developed double Kubo transform (DKT) methodology in combination with ring-polymer molecular dynamics (RPMD) for evaluating multi-time correlation functions [K. A. Jung et al., J. Chem. Phys. 148, 244105 (2018)], providing a practical method for incorporating nuclear quantum effects in nonlinear spectroscopy of condensed-phase systems. We showcase the DKT approach in the simulation of the fifth-order two-dimensional (2D) Raman spectroscopy of Lennard-Jones liquids as a prototypical example, which involves nontrivial nonlinear spectroscopic observables of systems described by anharmonic potentials. Our results show that the DKT can faithfully reproduce the 2D Raman response of liquid xenon at high temperatures, where the system behaves classically. In contrast, liquid neon at low temperatures exhibits moderate but discernible nuclear quantum effects in the 2D Raman response compared to the responses obtained with classical molecular dynamics approaches. Thus, the DKT formalism in combination with RPMD simulations enables simulations of multidimensional optical spectroscopy of condensed-phase systems that partially account for nuclear quantum effects.

Time correlation function and finite field approaches to the calculation of the fifth order Raman response in liquid xenon

The fifth order, two-dimensional Raman response in liquid xenon is calculated via a time correlation function (TCF) theory and the numerically exact finite field method. Both employ classical molecular dynamics simulations. The results are shown to be in excellent agreement, suggesting the efficacy of the TCF approach, in which the response function is written approximately in terms of a single classical multitime TCF.

Fifth-order two-dimensional Raman spectroscopy of liquid water, crystalline ice Ih and amorphous ices: Sensitivity to anharmonic dynamics and local hydrogen bond network structure

The theoretical study of off-resonant fifth-order two-dimensional (2D)-Raman spectroscopy is made to analyze the intermolecular dynamics of liquid and solid water. The 2D-Raman spectroscopy is susceptible to the nonlinear anharmonic dynamics and local hydrogen bond structure in water. It is found that the distinct 2D-Raman response appears as the negative signal near the t(2) axis. The origin of this negative signal for t(2)<15 fs is from the nonlinear polarizability in the librational motions, whereas that for 30 fs<t(2)<150 fs is attributed to the anharmonic translational motions. It is found that the mechanical anharmonicity and nonlinear polarizability couplings among modes clearly can be observed as the sum- and difference-frequency peaks in the 2D-Raman spectrum (i.e., Fourier transforms of the response). The 2D-Raman spectroscopies of ice Ih and amorphous ices, i.e., low density, high density, and very high density amorphous ices, are also investigated. It is found that the 2D-Raman spectroscopy is very sensitive to the anisotropy of the structure of ice Ih. The strong hydrogen bond stretching band is seen in the 2D-Raman spectroscopy of the polarization directions parallel to the c axis, whereas the contributions of the librational motion can be also seen in the spectrum with the polarization directions parallel to the a axis. The 2D-Raman spectroscopy is also found to be also very sensitive to the differences in local hydrogen bond network structures in various amorphous phases.

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.

Applications of a time correlation function theory for the fifth-order Raman response function I: Atomic liquids

Multidimensional spectroscopy has the ability to provide great insight into the complex dynamics and time-resolved structure of liquids. Theoretically describing these experiments requires calculating the nonlinear-response function, which is a combination of quantum-mechanical time correlation functions R5(t1,t2) was expressed with a two-time, computationally tractable, classical TCF. Writing the response function in terms of classical TCFs brings the full power of atomistically detailed molecular dynamics to the problem. In this paper, the new TCF theory is employed to calculate the fifth-order Raman response function for liquid xenon and investigate several of the polarization conditions for which experiments can be performed on an isotropic system. The theory is shown to reproduce line-shape characteristics predicted by earlier theoretical work.

Qualitative features of the two-dimensional Raman spectrum in liquids

The theory presented earlier [J. Kim and T. Keyes, Phys. Rev. E 66, 051110 (2002)] is analyzed to determine the information available from the two-dimensional Raman spectrum R((5))(t(2),t(1)) in liquids. The known spectra are well represented by the sum of two products of ordinary time correlations predicted by the theory. The shape of R((5)) is related in general to the values of simple same-time averages and concepts amenable to physical intuition. Using standard models for the time correlations entering the theory, specific analytic expressions for the spectrum are obtained depending on two parameters and a time scale, and the behavior of the spectrum is mapped out in the parameter space.

Tractable theory of nonlinear response and multidimensional nonlinear spectroscopy

Nonlinear spectroscopy provides insights into dynamics, but the response functions required for its interpretation pose a challenge to theorists. We proposed an approach in which the fifth-order response function [R5( t1, t2)] was expressed as a two-time classical time correlation function (TCF). Here, we present TCF theory results for R5( t1, t2) in liquid xenon. Using a first-order dipole-induced dipole polarizability model, the result is compared to an exact numerical calculation showing remarkable agreement. In addition, R5( t1, t2) is calculated using the exactly solved polarizability model, yielding different results and predicting an echo signal.

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.

A time correlation function theory of two-dimensional infrared spectroscopy with applications to liquid water

A theory describing the third-order response function R((3))(t(1),t(2),t(3)), which is associated with two-dimensional infrared (2DIR) spectroscopy, has been developed. R((3)) can be written as sums and differences of four distinct quantum mechanical dipole (multi)time correlation functions (TCF's), each with the same classical limit; the combination of TCF's has a leading contribution of order variant Planck's over 2pi (3) and thus there is no obvious classical limit that can be written in terms of a TCF. In order to calculate the response function in a form amenable to classical mechanical simulation techniques, it is rewritten approximately in terms of a single classical TCF, B(R)(t(1),t(2),t(3))=micro(j)(t(2)+t(1))micro(i)(t(3)+t(2)+t(1))micro(k)(t(1))micro(l)(0), where the subscripts denote the Cartesian dipole directions. The response function is then given, in the frequency domain, as the Fourier transform of a classical TCF multiplied by frequency factors. This classical expression can then further be quantum corrected to approximate the true response function, although for low frequency spectroscopy no correction is needed. In the classical limit, R((3)) becomes the sum of multidimensional time derivatives of B(R)(t(1),t(2),t(3)). To construct the theory, the response function's four TCF's are rewritten in terms of a single TCF: first, two TCF's are eliminated from R((3)) using frequency domain detailed balance relationships, and next, two more are removed by relating the remaining TCF's to each other within a harmonic oscillator approximation; the theory invokes a harmonic approximation only in relating the TCF's and applications of theory involve fully anharmonic, atomistically detailed molecular dynamics (MD). Writing the response function as a single TCF thus yields a form amenable to calculation using classical MD methods along with a suitable spectroscopic model. To demonstrate the theory, the response function is obtained for liquid water with emphasis on the OH stretching portion of the spectrum. This approach to evaluating R((3)) can easily be applied to chemically interesting systems currently being explored experimentally by 2DIR and to help understand the information content of the emerging multidimensional spectroscopy.

Semiclassical calculation of the vibrational echo

The infrared echo measurement probes the time scales of the molecular motions that couple to a vibrational transition. Computation of the echo observable within rigorous quantum mechanics is problematic for systems with many degrees of freedom, motivating the development of semiclassical approximations to the nonlinear optical response. We present a semiclassical approximation to the echo observable, based on the Herman-Kluk propagator. This calculation requires averaging over a quantity generated by two pairs of classical trajectories and associated stability matrices, connected by a pair of phase-space jumps. Quantum, classical, and semiclassical echo calculations are compared for a thermal ensemble of noninteracting anharmonic oscillators. The semiclassical approach uses input from classical mechanics to reproduce the significant features of a complete, quantum mechanical calculation of the nonlinear response.

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.