Structural information from two-dimensional fifth-order Raman spectroscopy

Full molecular dynamics simulations of molecular liquids for single-beam spectrally controlled two-dimensional Raman spectroscopy

Single-beam spectrally controlled (SBSC) two-dimensional (2D) Raman spectroscopy is a unique 2D vibrational measurement technique utilizing trains of short pulses that are generated from a single broadband pulse by pulse shaping. This approach overcomes the difficulty of 2D Raman spectroscopy in dealing with small-signal extraction and avoids complicated low-order cascading effects, thus providing a new possibility for measuring the intramolecular and intermolecular modes of molecular liquids using fifth-order 2D Raman spectroscopy. Recently, for quantitatively investigating the mode-mode coupling mechanism, Hurwitz et al. [Opt. Express 28, 3803 (2020)] have developed a new pulse design for this measurement to separate the contributions of the fifth- and third-order polarizations, which are often overlapped in the original single-beam measurements. Here, we describe a method for simulating these original measurements and the new 2D Raman measurements on the basis of a second-order response function approach. We carry out full molecular dynamics simulations for carbon tetrachloride and liquid water using an equilibrium-nonequilibrium hybrid algorithm, with the aim of explaining the key features of the SBSC 2D Raman spectroscopic method from a theoretical point of view. The predicted signal profiles and intensities provide valuable information that can be applied to 2D spectroscopy experiments, allowing them to be carried out more efficiently.

Single beam low frequency 2D Raman spectroscopy

Low frequency Raman spectroscopy resolves the slow vibrations resulting from collective motions of molecular structures. This frequency region is extremely challenging to access via other multidimensional methods such as 2D-IR. In this paper, we describe a new scheme which measures 2D Raman spectra in the low frequency regime. We separate the pulse into a spectrally shaped pump and a transform-limited probe, which can be distinguished by their polarization states. Low frequency 2D Raman spectra in liquid tetrabromoethane are presented, revealing coupling dynamics at frequencies as low as 115 cm^-1. The experimental results are supported by numerical simulations which replicate the key features of the measurement. This method opens the door for the deeper exploration of vibrational energy surfaces in complex molecular structures.

Notes on simulating two-dimensional Raman and terahertz-Raman signals with a full molecular dynamics simulation approach

Recent developments in two-dimensional (2D) THz-Raman and 2D Raman spectroscopies have created the possibility for quantitatively investigating the role of many dynamic and structural aspects of the molecular system. We explain the significant points for properly simulating 2D vibrational spectroscopic studies of intermolecular modes using the full molecular dynamics approach, in particular, regarding the system size, the treatment of the thermostat, and inclusion of an Ewald summation for the induced polarizability. Moreover, using the simulation results for water employing various polarization functions, we elucidate the roles of permanent and induced optical properties in determining the 2D profiles of the signal.

Biological and biomedical applications of two-dimensional vibrational spectroscopy: proteomics, imaging, and structural analysis

In the last 10 years, several forms of two-dimensional infrared (2DIR) spectroscopy have been developed, such as IR pump-probe spectroscopy and photon-echo techniques. In this Account, we describe a doubly vibrationally enhanced four-wave mixing method, in which a third-order nonlinear signal is generated from the interaction of two independently tunable IR beams and an electron-polarizing visible beam at 790 nm. When the IR beams are independently in resonance with coupled vibrational transitions, the signal is enhanced and cross-peaks appear in the spectrum. This method is known as either DOVE (doubly vibrationally enhanced) four-wave mixing or EVV (electron-vibration-vibration) 2DIR spectroscopy. We begin by discussing the basis and properties of EVV 2DIR. We then discuss several biological and potential biomedical applications. These include protein identification and quantification, as well as the potential of this label-free spectroscopy for protein and peptide structural analysis. In proteomics, we also show how post-translational modifications in peptides (tyrosine phosphorylation) can be detected by EVV 2DIR spectroscopy. The feasibility of EVV 2DIR spectroscopy for tissue imaging is also evaluated. Preliminary results were obtained on a mouse kidney histological section that was stained with hematoxylin (a small organic molecule). We obtained images by setting the IR frequencies to a specific cross-peak (the strongest for hematoxylin was obtained from its analysis in isolation; a general CH(3) cross-peak for proteins was also used) and then spatially mapping as a function of the beam position relative to the sample. Protein and hematoxylin distribution in the tissue were measured and show differential contrast, which can be entirely explained by the different tissue structures and their functions. The possibility of triply resonant EVV 2DIR spectroscopy was investigated on the retinal chromophore at the centre of the photosynthetic protein bacteriorhodopsin (bR). By putting the visible third beam in resonance with an electronic transition, we were able to enhance the signal and increase the sensitivity of the method by several orders of magnitude. This increase in sensitivity is of great importance for biological applications, in which the number of proteins, metabolites, or drug molecules to be detected is low (typically pico- to femtomoles). Finally, we present theoretical investigations for using EVV 2DIR spectroscopy as a structural analysis tool for inter- and intramolecular interaction geometries.

Modeling, calculating, and analyzing multidimensional vibrational spectroscopies

Spectral line shapes in a condensed phase contain information from various dynamic processes that modulate the transition energy, such as microscopic dynamics, inter- and intramolecular couplings, and solvent dynamics. Because nonlinear response functions are sensitive to the complex dynamics of chemical processes, multidimensional vibrational spectroscopies can separate these processes. In multidimensional vibrational spectroscopy, the nonlinear response functions of a molecular dipole or polarizability are measured using ultrashort pulses to monitor inter- and intramolecular vibrational motions. Because a complex profile of such signals depends on the many dynamic and structural aspects of a molecular system, researchers would like to have a theoretical understanding of these phenomena. In this Account, we explore and describe the roles of different physical phenomena that arise from the peculiarities of the system-bath coupling in multidimensional spectra. We also present simple analytical expressions for a weakly coupled multimode Brownian system, which we use to analyze the results obtained by the experiments and simulations. To calculate the nonlinear optical response, researchers commonly use a particular form of a system Hamiltonian fit to the experimental results. The optical responses of molecular vibrational motions have been studied in either an oscillator model or a vibration energy state model. In principle, both models should give the same results as long as the energy states are chosen to be the eigenstates of the oscillator model. The energy state model can provide a simple description of nonlinear optical processes because the diagrammatic Liouville space theory that developed in the electronically resonant spectroscopies can easily handle three or four energy states involved in high-frequency vibrations. However, the energy state model breaks down if we include the thermal excitation and relaxation processes in the dynamics to put the system in a thermal equilibrium state. The roles of these excitation and relaxation processes are different and complicated compared with those in the resonant spectroscopy. Observing the effects of such thermal processes is more intuitive with the oscillator model because the bath modes, which cause the fluctuation and dissipation processes, are also described in the coordinate space. This coordinate space system-bath approach complements a realistic full molecular dynamics simulation approach. By comparing the calculated 2D spectra from the coordinate space model and the energy state model, we can examine the role of thermal processes and anharmonic mode-mode couplings in the energy state model. For this purpose, we employed the Brownian oscillator model with the nonlinear system-bath interaction. Using the hierarchy formalism, we could precisely calculate multidimensional spectra for a single and multimode anharmonic system for inter- and intramolecular vibrational modes.

Coherent multidimensional vibrational spectroscopy of biomolecules: concepts, simulations, and challenges

The response of complex molecules to sequences of femtosecond infrared pulses provides a unique window into their structure, dynamics, and fluctuating environments. Herein we survey the basic principles of modern two-dimensional infrared (2DIR) spectroscopy, which analogous to those of multidimensional NMR spectroscopy. The perturbative approach for computing the nonlinear optical response of coupled localized chromophores is introduced and applied to the amide backbone transitions of proteins, liquid water, membrane lipids, and amyloid fibrils. The signals are analyzed using classical molecular dynamics simulations combined with an effective fluctuating Hamiltonian for coupled localized anharmonic vibrations whose dependence on the local electrostatic environment is parameterized by an ab initio map. Several simulation methods, (cumulant expansion of Gaussian fluctuation, quasiparticle scattering, the stochastic Liouville equations, direct numerical propagation) are surveyed. Chirality-induced techniques which dramatically enhance the resolution are demonstrated. Signatures of conformational and hydrogen-bonding fluctuations, protein folding, and chemical-exchange processes are discussed.

Dynamics of a Multimode System Coupled to Multiple Heat Baths Probed by Two-Dimensional Infrared Spectroscopy

Reduced equation of motion for a multimode system coupled to multiple heat baths is constructed by extending the quantum Fokker-Planck equation with low-temperature correction terms (J. Phys. Soc. Jpn. 2005, 74, 3131). Unlike such common approaches used to describe intramolecular multimode vibration as a Bloch-Redfield theory and a stochastic theory, the present formalism is defined by the molecular coordinates. To explore the correlation among different modes through baths, we consider two cases of system-bath couplings. One is a correlated case in which two modes are coupled to a single bath, and the other is an uncorrelated case in which each mode is coupled to a different bath. We further classify the correlated case into two cases, the plus- and minus-correlated cases, according to distinct correlation manners. For these, one-dimensional and two-dimensional infrared (2D-IR) spectra are calculated numerically by solving the equation of motion. It is demonstrated that 2D-IR spectroscopy has the ability to analyze the correlation of fluctuation-dissipation processes among different modes.

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.

Modeling vibrational dephasing and energy relaxation of intramolecular anharmonic modes for multidimensional infrared spectroscopies

Starting from a system-bath Hamiltonian in a molecular coordinate representation, we examine an applicability of a stochastic multilevel model for vibrational dephasing and energy relaxation in multidimensional infrared spectroscopy. We consider an intramolecular anharmonic mode nonlinearly coupled to a colored noise bath at finite temperature. The system-bath interaction is assumed linear plus square in the system coordinate, but linear in the bath coordinates. The square-linear system-bath interaction leads to dephasing due to the frequency fluctuation of system vibration, while the linear-linear interaction contributes to energy relaxation and a part of dephasing arises from anharmonicity. To clarify the role and origin of vibrational dephasing and energy relaxation in the stochastic model, the system part is then transformed into an energy eigenstate representation without using the rotating wave approximation. Two-dimensional (2D) infrared spectra are then calculated by solving a low-temperature corrected quantum Fokker-Planck (LTC-QFP) equation for a colored noise bath and by the stochastic theory. In motional narrowing regime, the spectra from the stochastic model are quite different from those from the LTC-QFP. In spectral diffusion regime, however, the 2D line shapes from the stochastic model resemble those from the LTC-QFP besides the blueshifts caused by the dissipation from the colored noise bath. The preconditions for validity of the stochastic theory for molecular vibrational motion are also discussed.

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.

Time-Domain Calculations of the Polarized Raman Spectra, the Transient Infrared Absorption Anisotropy, and the Extent of Delocalization of the OH Stretching Mode of Liquid Water

The polarized Raman spectrum and the time dependence of the transient infrared (TRIR) absorption anisotropy are calculated for the OH stretching mode of liquid water (neat liquid H2O) by using time-domain formulations, which include the effects of both the diagonal frequency modulations (of individual oscillators) induced by the interactions between the dipole derivatives and the intermolecular electric field, and the off-diagonal (intermolecular) vibrational coupling described by the transition dipole coupling (TDC) mechanism. The IR spectrum of neat liquid H2O and the TRIR anisotropy of a liquid mixture of H2O/HDO/D2O are also calculated. It is shown that the calculated features of these optical signals, including the temperature dependence of the polarized Raman and IR spectra, are in reasonable agreement with the experimental results, indicating that the frequency separation between the isotropic and anisotropic components of the polarized Raman spectrum and the rapid decay (approximately 0.1 ps) of the TRIR anisotropy of the OH stretching mode of neat liquid H2O are mainly controlled by the resonant intermolecular vibrational coupling described by the TDC mechanism. Comparing with the time evolution of vibrational excitations, it is suggested that the TRIR anisotropy decays in the time needed for the initially localized vibrational excitations to delocalize over a few oscillators. It is also shown that the enhancement of the dipole derivatives by the interactions with surrounding molecules is an important factor in generating the spectral profiles of the OH stretching Raman band. The time-domain behavior of the molecular motions that affect the spectroscopic features is discussed.

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.

Multidimensional infrared spectroscopy for molecular vibrational modes with dipolar interactions, anharmonicity, and nonlinearity of dipole moments and polarizability

We present an analytical expression for the linear and nonlinear infrared spectra of interacting molecular vibrational motions. Each of the molecular modes is explicitly represented by a classical damped oscillator on an anharmonic multidimensional potential-energy surface. The two essential interactions, the dipole-dipole (DD) and the dipole-induced-dipole (DID) interactions, are taken into account, and each dipole moment and polarizability are expanded to nonlinear order with respect to the nuclear vibrational coordinate. Our analytical treatment leads to expressions for the contributions of anharmonicity, DD and DID interactions, and the nonlinearity of dipole moments and polarizability elements to the one-, two-, and three-dimensional spectra as separated terms, which allows us to discuss the relative importance of these respective contributions. We can calculate multidimensional signals for various configurations of molecules interacting through DD and DID interactions for different material parameters over the whole range of frequencies. We demonstrate that contributions from the DD and DID interactions and anharmonicity are separately detectable through the third-order three-dimensional IR spectroscopy, whereas they cannot be distinguished from each other in either the linear or the second-order IR spectroscopies. The possibility of obtaining the intra- or intermolecular structural information from multidimensional spectra is also discussed.

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.

Multidimensional vibrational spectroscopy for tunneling processes in a dissipative environment

Simulating tunneling processes as well as their observation are challenging problems for many areas. In this study, we consider a double-well potential system coupled to a heat bath with a linear-linear (LL) and square-linear (SL) system-bath interactions. The LL interaction leads to longitudinal (T1) and transversal (T2) homogeneous relaxations, whereas the SL interaction leads to the inhomogeneous dephasing (T2*) relaxation in the white noise limit with a rotating wave approximation. We discuss the dynamics of the double-well system under infrared (IR) laser excitations from a Gaussian-Markovian quantum Fokker-Planck equation approach, which was developed by generalizing Kubo's stochastic Liouville equation. Analytical expression of the Green function is obtained for a case of two-state-jump modulation by performing the Fourier-Laplace transformation. We then calculate a two-dimensional infrared signal, which is defined by the four-body correlation function of optical dipole, for various noise correlation time, system-bath coupling parameters, and temperatures. It is shown that the bath-induced vibrational excitation and relaxation dynamics between the tunneling splitting levels can be detected as the isolated off-diagonal peaks in the third-order two-dimensional infrared (2D-IR) spectroscopy for a specific phase matching condition. Furthermore, this spectroscopy also allows us to directly evaluate the rate constants for tunneling reactions, which relates to the coherence between the splitting levels; it can be regarded as a novel technique for measuring chemical reaction rates. We depict the change of reaction rates as a function of system-bath coupling strength and a temperature through the 2D-IR signal.

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.

Two-dimensional Raman and infrared vibrational spectroscopy for a harmonic oscillator system nonlinearly coupled with a colored noise bath

Multidimensional vibrational response functions of a harmonic oscillator are reconsidered by assuming nonlinear system-bath couplings. In addition to a standard linear-linear (LL) system-bath interaction, we consider a square-linear (SL) interaction. The LL interaction causes the vibrational energy relaxation, while the SL interaction is mainly responsible for the vibrational phase relaxation. The dynamics of the relevant system are investigated by the numerical integration of the Gaussian-Markovian Fokker-Planck equation under the condition of strong couplings with a colored noise bath, where the conventional perturbative approach cannot be applied. The response functions for the fifth-order nonresonant Raman and the third-order infrared (or equivalently the second-order infrared and the seventh-order nonresonant Raman) spectra are calculated under the various combinations of the LL and the SL coupling strengths. Calculated two-dimensional response functions demonstrate that those spectroscopic techniques are very sensitive to the mechanism of the system-bath couplings and the correlation time of the bath fluctuation. We discuss the primary optical transition pathways involved to elucidate the corresponding spectroscopic features and to relate them to the microscopic sources of the vibrational nonlinearity induced by the system-bath interactions. Optical pathways for the fifth-order Raman spectroscopies from an "anisotropic" medium were newly found in this study, which were not predicted by the weak system-bath coupling theory or the standard Brownian harmonic oscillator model.

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.

Signatures of beta-peptide unfolding in two-dimensional vibrational echo spectroscopy: a simulation study

An ensemble of exciton Hamiltonians for the amide-I band of the folded and unfolded states of a helical beta-heptapeptide is generated using a molecular dynamics (MD) simulation. The correlated fluctuations of its parameters and their signatures in two-dimensional (2D) vibrational echo spectroscopy are computed. This technique uses infrared pulse sequences to provide ultrafast snapshots of molecular structural fluctuations, in analogy with multidimensional NMR. The present study demonstrates that, by combining a method of calculating the vibrational Hamiltonian from MD snapshots and the nonlinear exciton equations (NEE), it may be possible to simulate realistic multidimensional IR spectra of chemically and biologically interesting systems.