Instantaneous normal mode theory of more complicated correlation functions: Third- and fifth-order optical response

Computational spectroscopy of complex systems

Numerous linear and non-linear spectroscopic techniques have been developed to elucidate structural and functional information of complex systems ranging from natural systems, such as proteins and light-harvesting systems, to synthetic systems, such as solar cell materials and light-emitting diodes. The obtained experimental data can be challenging to interpret due to the complexity and potential overlapping spectral signatures. Therefore, computational spectroscopy plays a crucial role in the interpretation and understanding of spectral observables of complex systems. Computational modeling of various spectroscopic techniques has seen significant developments in the past decade, when it comes to the systems that can be addressed, the size and complexity of the sample types, the accuracy of the methods, and the spectroscopic techniques that can be addressed. In this Perspective, I will review the computational spectroscopy methods that have been developed and applied for infrared and visible spectroscopies in the condensed phase. I will discuss some of the questions that this has allowed answering. Finally, I will discuss current and future challenges and how these may be addressed.

Pressure response of the THz spectrum of bulk liquid water revealed by intermolecular instantaneous normal mode analysis

The radial distribution functions of liquid water are known to change significantly their shape upon hydrostatic compression from ambient conditions deep into the kbar pressure regime. It has been shown that despite their eye-catching changes, the fundamental locally tetrahedral fourfold H-bonding pattern that characterizes ambient water is preserved up to about 10 kbar (1 GPa), which is the stability limit of liquid water at 300 K. The observed increase in coordination number comes from pushing water molecules into the first coordination sphere without establishing an H-bond, resulting in roughly two such additional interstitial molecules at 10 kbar. THz spectroscopy has been firmly established as a powerful experimental technique to analyze H-bonding in aqueous solutions given that it directly probes the far-infrared lineshape and thus the prominent H-bond network mode around 180 cm^-1. We, therefore, set out to assess pressure effects on the THz response of liquid water at 10 kbar in comparison to the 1 bar (0.1 MPa) reference, both at 300 K, with the aim to trace back the related lineshape changes to the structural level. To this end, we employ the instantaneous normal mode approximation to rigorously separate the H-bonding peak from the large background arising from the pronounced librational tail. By exactly decomposing the total molecular dynamics into hindered translations, hindered rotations, and intramolecular vibrations, we find that the H-bonding peak arises from translation-translation and translation-rotation correlations, which are successively decomposed down to the level of distinct local H-bond environments. Our utmost detailed analysis based on molecular pair classifications unveils that H-bonded double-donor water pairs contribute most to the THz response around 180 cm^-1, whereas interstitial waters are negligible. Moreover, short double-donor H-bonds have their peak maximum significantly shifted toward higher frequencies with respect to such long H-bonds. In conjunction with an increasing relative population of these short H-bonds versus the long ones (while the population of other water pair classes is essentially pressure insensitive), this explains not only the blue-shift of the H-bonding peak by about 20-30 cm^-1 in total from 1 bar to 10 kbar but also the filling of the shallow local minimum of the THz lineshape located in between the network peak and the red-wing of the librational band at 1 bar. Based on the changing populations as a function of pressure, we are also able to roughly estimate the pressure-dependence of the H-bond network mode and find that its pressure response and thus the blue-shifting are most pronounced at low kbar pressures.

Polarization Selectivity of Third-Order and Fifth-Order Raman Spectroscopies in Liquids and Solids

Polarization selectivity of third-order and fifth-order Raman spectroscopies is examined for both isotropic liquids and periodic lattices. Our approach directly applies the symmetry property of the probed system to decompose the polarization tensor elements into independent components. The polarization selectivity predicted by symmetry analysis is rigorous and applicable to higher-order Raman spectroscopy. The different polarization selectivities of isotropic systems and periodic lattices can be used as a signature of the liquid-solid phase transition.

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.

Fifth-Order Raman Spectroscopy of Liquid Benzene: Experiment and Theory

The heterodyned fifth-order Raman response of liquid benzene has been measured and characterized by exploiting the passive-phase stabilization of diffractive optics. This result builds on our previous work with liquid carbon disulfide and extends the spectroscopy to a new liquid for the first time. The all-parallel and Dutch Cross polarization tensor elements are presented for both the experimental results and a finite-field molecular dynamics simulation. The overall response characteristics are similar to those of liquid carbon disulfide: a complete lack of signal along the pump delay, an elongated signal along the probe delay, and a short-lived signal along the time diagonal. Of particular interest is the change in phase between the nuclear and electronic response along the probe delay and diagonal which is not seen in CS2. Good agreement is achieved between the experiment and the finite-field molecular dynamics simulation. The measurement of the low-frequency Raman two-time delay correlation function indicates the intermolecular modes of liquid benzene to be primarily homogeneously broadened and that the liquid loses its nuclear rephasing ability within 300 fs. This rapid loss of nuclear correlations indicates a lack of modal character in the low-frequency motions of liquid benzene. This result is a validation of the general nature of the technique and represents an important step forward with respect to the use of nonlinear spectroscopy to directly access information on the anharmonic motions of liquids.

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.

Analyzing atomic liquids and solids by means of two-dimensional Raman spectra in frequency domain

A practical method to evaluate the contributions of the nonlinear polarizability and anharmonicity of potentials from the experimental and simulation data by using double Fourier transformation is presented. In a Lennard-Jones potential system, an approximated expression of the fifth-order response function using the ratio between nonlinear polarizability and anharmonicity exhibits a good agreement with the results of the molecular dynamics simulation. In a soft-core case, the fifth-order Raman signal indicates that the system consists of the delocalized and localized modes, and only the delocalized mode affects the dramatic change of the fifth-order Raman response functions between solid and liquid phases through nonlinear polarizability.

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.

Two-dimensional Raman spectra of atomic solids and liquids

We calculate third- and fifth-order Raman spectra of simple atoms interacting through a soft-core potential by means of molecular-dynamics (MD) simulations. The total polarizability of molecules is treated by the dipole-induced dipole model. Two- and three-body correlation functions of the polarizability at various temperatures are evaluated from equilibrium MD simulations based on a stability matrix formulation. To analyze the processes involved in the spectroscopic measurements, we divide the fifth-order response functions into symmetric and antisymmetric integrated response functions; the symmetric one is written as a simple three-body correlation function, while the antisymmetric one depends on a stability matrix. This analysis leads to a better understanding of the time scales and molecular motions that govern the two-dimensional (2D) signal. The 2D Raman spectra show novel differences between the solid and liquid phases, which are associated with the decay rates of coherent motions. On the other hand, these differences are not observed in the linear Raman spectra.

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.

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.

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.

Higher-order optical correlation spectroscopy in liquids

Linear optical spectroscopies have long been used to study the behavior of liquids. Laser technology has progressed to the point that it has become possible to perform nonlinear optical experiments that probe higher-order correlation functions in liquids, opening a new window into our understanding of the microscopic details of solution-phase processes. Here we review advances that have been made in recent years in employing higher-order electronic and vibrational spectroscopies to study liquid-state dynamics and structure.

Generalized Langevin equation approach to higher-order classical response: second-order-response time-resolved Raman experiment in CS2

A simple, systematic generalized Langevin equation approach for calculating classical nonlinear response functions is formulated and discussed. The two-time Poisson brackets appearing at second and higher order are rendered tractable by a physically motivated approximation. The method is used to calculate the fifth order (second order response) Raman response of liquid CS2. Agreement with simulation is good, but the simplicity of the theoretical expression suggests that the path to obtaining qualitatively new information about liquids with the fifth order experiment is uncertain. Further applications of the basic approach are suggested.

Multiple-point and multiple-time correlation functions in a hard-sphere fluid

A recent mode-coupling theory of higher-order correlation functions is tested on a simple hard-sphere fluid system at intermediate densities. Multiple-point and multiple-time correlation functions of the densities of conserved variables are calculated in the hydrodynamic limit and compared to results obtained from event-based molecular dynamics simulations. It is demonstrated that the mode-coupling theory results are in excellent agreement with the simulation results provided that dissipative couplings are included in the vertices appearing in the theory. In contrast, simplified mode-coupling theories in which the densities obey Gaussian statistics neglect important contributions to both the multiple point and multiple-time correlation functions on all time scales.

Mode-coupling theory of the fifth-order Raman spectrum of an atomic liquid

A fully microscopic molecular hydrodynamic theory for the two-dimensional (fifth-order) Raman spectrum of an atomic liquid (Xe) is presented. The spectrum is obtained from a simple mode-coupling theory by projecting the dynamics onto bilinear pairs of fluctuating density variables. Good agreement is obtained in comparison with recently reported molecular dynamics simulation results. The microscopic theory provides an understanding of the timescales and molecular motions that govern the two-dimensional signal. Predictions are made for the behavior of the spectrum as a function of temperature and density. The theory shows that novel signatures in the two-dimensional Raman spectrum of supercritical and supercooled liquids are expected.

Mean-atom-trajectory model for the velocity autocorrelation function of monatomic liquids

We present a model for the motion of an average atom in a liquid or supercooled liquid state and apply it to calculations of the velocity autocorrelation function Z(t) and diffusion coefficient D. The model trajectory consists of oscillations at a distribution of frequencies characteristic of the normal modes of a single potential valley, interspersed with position- and velocity-conserving transits to similar adjacent valleys. The resulting predictions for Z(t) and D agree remarkably well with molecular dynamics simulations of Na at up to almost three times its melting temperature. Two independent processes in the model relax velocity autocorrelations: (a) dephasing due to the presence of many frequency components, which operates at all temperatures but which produces no diffusion, and (b) the transit process, which increases with increasing temperature and which produces diffusion. Because the model provides a single-atom trajectory in real space and time, including transits, it may be used to calculate all single-atom correlation functions.

Fifth-order raman spectrum of an atomic liquid: simulation and instantaneous-normal-mode calculation

Experimental artifacts and technical difficulties in carrying out theoretical calculations have consistently frustrated attempts to obtain the two-dimensional (5th-order) Raman spectrum of a liquid. We report here a new theoretical development: the first microscopic numerical simulation of the 5th-order Raman signal in a liquid. Comparison with an instantaneous-normal-mode treatment, a fully microscopic model which interprets liquid dynamics as arising from coherent harmonic modes, shows that the 5th-order spectrum reveals profound effects stemming from dynamical anharmonicity.