Two-Dimensional Raman Echoes: Femtosecond View of Molecular Structure and Vibrational Coherence

Time-resolved enantiomer-exchange probed by using the orbital angular momentum of X-ray light

Molecular chirality, a geometric property of utmost importance in biochemistry, is now being investigated in the time-domain. Ultrafast chiral techniques can probe the formation or disappearance of stereogenic centers in molecules. The element-sensitivity of X-rays adds the capability to probe chiral nuclear dynamics locally within the molecular system. However, the implementation of ultrafast techniques for measuring transient chirality remains a challenge because of the intrinsic weakness of chiral-sensitive signals based on circularly polarized light. We propose a novel approach for probing the enantiomeric dynamics by using the orbital angular momentum (OAM) of X-ray light, which can directly monitor the real-time chirality of molecules. Our simulations probe the oscillations in excited chiral formamide on different potential energy surfaces and demonstrate that using the X-ray OAM can increase the measured asymmetry ratio. Moreover, combining the OAM and SAM (spin angular momentum) provides stronger dichroic signals than linearly polarized light, and offers a powerful scheme for chiral discrimination.

Distinguishing different excitation pathways in two-dimensional terahertz-infrared-visible spectroscopy

In condensed molecular matter, low-frequency modes (LFMs) associated with specific molecular motions are excited at room temperature and determine essential physical and chemical properties of materials. LFMs, with typical mode energies of up to ∼500 cm^-1 (62 meV), contribute significantly to thermodynamic parameters and functions (e.g., heat capacity and entropy) and constitute the basis for room temperature molecular dynamics (e.g., conformational fluctuations and change). LFMs are often analyzed indirectly by the measurement of their effect on specific high-frequency modes (HFMs); the LFM-HFM coupling is reflected in the lineshape, as well as in the spectral and angular diffusion of the HFM. Two-dimensional terahertz-infrared-visible (2D TIRV) spectroscopy allows measuring the LFM-HFM coupling directly and can thereby provide new insights into the strength and nature of the coupling and the character of LFMs. However, the interference between the different signals generated by different excitation pathways can complicate 2D TIRV spectra, preventing a straightforward analysis. Here, we develop an experimental method to distinguish different excitation pathways in 2D TIRV spectroscopy and plot them separately in different quadrants of a 2D spectrum. We validate this method by measuring the spectra of CaF₂ and nitrogen gas. For CaF₂, only sum-frequency mixing between infrared and terahertz fields generates the signal. In contrast, for N₂, only difference-frequency mixing is observed. We then use this method to separate sum- and difference-frequency pathways in the 2D TIRV spectrum of liquid water, verifying the previous interpretation of the lineshape of the 2D TIRV spectrum of water.

Investigating ultrafast two-pulse experiments on single DNQDI fluorophores: a stochastic quantum approach

Ultrafast two-pulse experiments on single molecules are invaluable tools to investigate the microscopic dynamics of a fluorophore. The first pulse generates electronic or vibronic coherence and the second pulse probes the time-evolution of the coherence. A protocol that is able to simulate ultrafast experiments on single molecules is applied in this study. It is based on a coupled quantum-mechanical description of the fluorophore and real-time dynamics of the system vibronic wave packet interacting with an electric field, described by means of the stochastic Schrödinger equation within the Markovian limit. This approach is applied to the DNQDI fluorophore, previously investigated experimentally [D. Brinks et al., Nature, 2010, 465, 905-908]. We find this to be in good agreement with the experimental outcomes and provide microscopic and atomistic interpretation.

Ultrafast structural molecular dynamics investigated with 2D infrared spectroscopy methods

Ultrafast, multi-dimensional infrared (IR) spectroscopy has been advanced in recent years to a versatile analytical tool with a broad range of applications to elucidate molecular structure on ultrafast timescales, and it can be used for samples in a many different environments. Following a short and general introduction on the benefits of 2D IR spectroscopy, the first part of this chapter contains a brief discussion on basic descriptions and conceptual considerations of 2D IR spectroscopy. Outstanding classical applications of 2D IR are used afterwards to highlight the strengths and basic applicability of the method. This includes the identification of vibrational coupling in molecules, characterization of spectral diffusion dynamics, chemical exchange of chemical bond formation and breaking, as well as dynamics of intra- and intermolecular energy transfer for molecules in bulk solution and thin films. In the second part, several important, recently developed variants and new applications of 2D IR spectroscopy are introduced. These methods focus on (i) applications to molecules under two- and three-dimensional confinement, (ii) the combination of 2D IR with electrochemistry, (iii) ultrafast 2D IR in conjunction with diffraction-limited microscopy, (iv) several variants of non-equilibrium 2D IR spectroscopy such as transient 2D IR and 3D IR, and (v) extensions of the pump and probe spectral regions for multi-dimensional vibrational spectroscopy towards mixed vibrational-electronic spectroscopies. In light of these examples, the important open scientific and conceptual questions with regard to intra- and intermolecular dynamics are highlighted. Such questions can be tackled with the existing arsenal of experimental variants of 2D IR spectroscopy to promote the understanding of fundamentally new aspects in chemistry, biology and materials science. The final part of the chapter introduces several concepts of currently performed technical developments, which aim at exploiting 2D IR spectroscopy as an analytical tool. Such developments embrace the combination of 2D IR spectroscopy and plasmonic spectroscopy for ultrasensitive analytics, merging 2D IR spectroscopy with ultra-high-resolution microscopy (nanoscopy), future variants of transient 2D IR methods, or 2D IR in conjunction with microfluidics. It is expected that these techniques will allow for groundbreaking research in many new areas of natural sciences.

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.

Two-dimensional stimulated resonance Raman spectroscopy study of the Trp-cage peptide folding

We report a combined molecular dynamics (MD) and ab initio simulation study of the ultrafast broadband ultraviolet (UV) stimulated resonance Raman (SRR) spectra of the Trp-cage mini protein. Characteristic two dimensional (2D) SRR features of various folding states are identified. Structural fluctuations erode the cross peaks and the correlation between diagonal peaks is a good indicator of the α-helix formation.

Two-Dimensional Stimulated Ultraviolet Resonance Raman Spectra of Tyrosine and Tryptophan; A Simulation Study

We report an ab-initio simulation study of the ultrafast broad bandwidth ultraviolet (UV) stimulated resonance Raman spectra (SRRS) of L-tyrosine, L-tryptophan and trans-L-tryptophan-L-tyrosine (WY) dipeptide. Two-pulse one-dimensional (1D) SRRS and three-pulse 2D SRRS that reveal inter- and intra-residue vibrational coorelations are simulated using electronically resonant or preresonant pulse configurations that select the Raman signal and discriminate against excited state pathways. Multimode effects are incorporated via the cumulant expansion. The 2D SRRS technique is more sensitive to residue couplings than spontaneous Raman.

Quantum theory of time-resolved femtosecond stimulated Raman spectroscopy: direct versus cascade processes and application to CDCl3

We present a quantum mechanical wave packet treatment of time-resolved femtosecond stimulated Raman spectroscopy (FSRS), or two-dimensional (2D) FSRS, where a vibrational coherence is initiated with an impulsive Raman pump which is subsequently probed by FSRS. It complements the recent classical treatment by Mehlenbacher et al. [J. Chem. Phys. 131, 244512 (2009)]. In this 2D-FSRS, two processes can occur concurrently but with different intensities: a direct fifth-order process taking place on one molecule, and a cascade process comprising two third-order processes on two different molecules. The cascade process comprises a parallel and a sequential cascade. The theory is applied to the 2D-FSRS of CDCl(3) where calculations showed that: (a) the cascade process is stronger than the direct fifth-order process by one order of magnitude, (b) the sidebands assigned to C-Cl E and A(1) bends, observed on both sides of the Stokes C-D stretch frequency, are not due to anharmonic coupling between the C-D stretch and the C-Cl bends, but are instead due to the coherent anti-Stokes Raman spectroscopy (CARS) and coherent Stokes Raman spectroscopy (CSRS) fields produced in the first step of the cascade process, (c) for each delay time between the femtosecond impulsive pump and FSRS probe pulses, the line shape of the sidebands shows an inversion symmetry about the C-D stretch frequency, and this is due to the 180(∘) phase difference between the CARS and CSRS fields that produced the left and right sidebands, and (d) for each sideband, the line shape changes from positive Lorentzian to dispersive to negative Lorentzian, then to negative dispersive and back to positive Lorentzian with the period of the bending vibration, and it is correlated with the momentum of the wave packet prepared on the ground-state surface by the impulsive pump along the sideband normal coordinate.

Long-lived coherences for homogeneous line narrowing in spectroscopy

Line broadening, which can arise from inhomogeneities or homogeneous relaxation effects that lead to finite lifetimes of quantum states, is the Achilles' heel of many forms of spectroscopy. We show that line broadening may be considerably reduced by exploiting long lifetimes associated with superpositions of quantum states with different symmetry, termed long-lived coherences. In proton NMR of arbitrary molecules (including proteins) in isotropic solution, the slow oscillatory decays of long-lived coherences can yield spectra with very high resolution. This improvement opens the way to high-field magnetic resonance of molecular assemblies that are almost an order of magnitude larger than could be hitherto studied. Coherences between states of different symmetry may be useful in other forms of spectroscopy to cancel unwanted line broadening effects.

Semiclassical nonlinear response functions for coupled anharmonic vibrations

Observables in linear and nonlinear infrared spectroscopy may be computed from vibrational response functions describing nuclear dynamics on a single electronic surface. We demonstrate that the Herman-Kluk (HK) semiclassical approximation to the quantum propagator yields an accurate representation of quantum coherence effects in linear and nonlinear response functions for coupled anharmonic oscillators. A considerable numerical price is paid for this accuracy; the calculation requires a multidimensional integral over a highly oscillatory integrand that also grows without bound as a function of evolution times. The interference among classical trajectories in the HK approximation produces quantization of good action variables. By treating this interference analytically, we develop a mean-trajectory (MT) approximation that requires only the propagation of classical trajectories linked by transitions in action. The MT approximation accurately reproduces coherence effects in response functions of coupled anharmonic oscillators in a regime in which the observables are strongly influenced by these interactions among vibrations.

Controlling multidimensional off-resonant-Raman and infrared vibrational spectroscopy by finite pulse band shapes

Closed expressions are derived which incorporate pulse shaping effects in femtosecond nonlinear optical signals involving various combinations of temporally well-separated vibrationally resonant infrared and electronically off-resonant Raman pulses. Combinations of broadband and narrow band pulses that yield multidimensional extensions of coherent anti-Stokes Raman and sum frequency generation spectroscopy are presented.

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.

Simultaneous time and frequency detection in femtosecond coherent Raman spectroscopy. I. Theory and model calculations

For coherent Raman spectroscopies, common femtosecond pulses often lie in an intermediate regime: their bandwidth is too wide for measurements in the frequency domain, but their temporal width is too broad for homodyne measurements in the time domain. A recent paper [S. Nath et al., Phys. Rev. Lett. 97, 267401 (2006)] showed that complete Raman spectra can be recovered from intermediate length pulses by using simultaneous time and frequency detection (TFD). Heterodyne detection and a phase-stable local oscillator at the anti-Stokes frequency are not needed with TFD. This paper examines the theory of TFD Raman in more detail; a companion paper tests the results on experimental data. Model calculations illustrate how information on the Raman spectrum is transferred from the frequency domain to the time domain as the pulse width shortens. When data are collected in both dimensions, the Raman spectrum is completely determined to high resolution, regardless of the probe pulse width. The loss of resolution in many femtosecond coherent Raman experiments is due to the restriction to one-dimensional data collection, rather than due to a fundamental restriction based on the pulse width.

Simultaneous time and frequency detection in femtosecond coherent Raman spectroscopy. II. Application to acetonitrile

The preceding paper showed that, in principle, a high-resolution coherent Raman spectrum can be recovered using femtosecond probe pulses by combined detection in both time and frequency. This measurement is possible even when the pulses are too broad in frequency for conventional frequency-domain spectroscopy and too broad in time for conventional time-domain spectroscopy. In this paper, the method is tested on experimental coherent anti-stokes Raman spectroscopy data from acetonitrile. Compared to theoretical models, experimental data are complicated by noise and incomplete knowledge of the pulse structure. Despite these complications, most of the information in the Raman spectrum is recovered from the data: weak transitions are detected and natural-linewidth resolution is achieved across an 800 cm(-1) spectral range. However, circumstances in which experimental limitations result in missed features or ambiguities in the recovered spectrum are also identified. These results suggest where improvements in measurement and data analysis can be made.

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.

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.

Collective Oscillations and the Linear and Two-Dimensional Infrared Spectra of Inhomogeneous β-Sheets

We numerically calculate the collective amide I oscillations and the associated linear and two-dimensional infrared (2DIR) spectra for model antiparallel beta-sheets and study the effect of inhomogeneity. To visualize the collective vibrational exciton states, a new method is introduced, which proves very useful in classifying the optically dominant states with respect to their symmetry properties and phase relations, even in the absence of exact symmetries. We find that energy (diagonal) and interaction (off-diagonal) disorder may have profoundly different effects on the main peaks in the linear spectrum. We also show that in the 2DIR spectra energy disorder leads to diagonal stretching of the diagonal peaks, while the cross-peaks are typically stretched more horizontally. This offers an explanation for the recently observed overall Z-shape in experimental spectra. Finally, we find that the anharmonic splitting between associated positive and negative features in the 2DIR spectra scales inversely proportionally with the exciton delocalization size imposed by the disorder, thus offering a spectroscopic ruler for this size.

Spectrally resolved femtosecond 2-colour 3-pulse photon echoes: a new spectroscopic tool to study molecular dynamics

We present a new multidimensional femtosecond spectroscopy technique based on spectrally resolved 2-colour 3-pulse photon echoes for investigating molecular dynamics in a variety of systems including proteins. In this technique the sample is illuminated by two femtosecond 'pump' pulses with wave vectors k1, k2 and wavelength lambda(pump) and a femtosecond 'probe' pulse with wave vector k3 and wavelength lambda(probe). Nonlinear signals are generated in the phase-matching directions k4 = - k1 + k2 + k3 and k6 = - k3 + k1 + k2. These signals are analysed in spectrometers equipped with CCD detectors and the spectra of the signals are recorded for various values of (i) the delay t12 between pulses 1 and 2, (ii) the delay t23 between pulses 2 and 3, and (iii) the wavelengths lambda(pump), lambda(probe). The technique has been used for studying vibrational and electronic dynamics of dye molecules, such as cresyl violet in methanol, and ultra-fast transient processes that occur during the photo-dissociation of carbonmonoxy myoglobin (MbCO) into myoglobin (Mb) and CO.

Spectrally resolved femtosecond two-color three-pulse photon echoes: Study of ground and excited state dynamics in molecules

We report the use of spectrally resolved femtosecond two-color three-pulse photon echoes as a potentially powerful multidimensional technique for studying vibrational and electronic dynamics in complex molecules. The wavelengths of the pump and probe laser pulses are found to have a dramatic effect on the spectrum of the photon echo signal and can be chosen to select different sets of energy levels in the vibrational manifold, allowing a study of the dynamics and vibrational splitting in either the ground or the excited state. The technique is applied to studies of the dynamics of vibrational electronic states in the dye molecule Rhodamine 101 in methanol.

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.

Heterodyne-detected fifth-order nonresonant Raman scattering from room temperature CS2

Actively phase-locked heterodyne-detected fifth-order nonresonant Raman scattering from room temperature CS2 has been measured. The experimental signals have similar magnitudes, shapes, and sign changes as calculated responses obtained via molecular dynamics simulations [S. Saito and I. Ohmine, Phys. Rev. Lett. 88, 207401 (2002)]. The measured signals contain sign changes that appear to be associated with the coupling of rotational motions both to each other and to translational motions.

Off-resonant fifth-order response function for two-dimensional Raman spectroscopy of liquids CS2 and H2O

The off-resonant fifth-order response functions for two-dimensional (2D) Raman spectroscopy of molecular liquids CS2 and H2O are investigated by using molecular dynamics calculation. This spectroscopy, able to deal with a phase space dynamics, shows the existence of nodal lines in several polarization tensor elements [see L. Kaufman et al., Phys. Rev. Lett. 88, 207402 (2002) for experimental results]. The nodal property is found to arise from the dynamical couplings among rotational modes, not accounted for in a normal mode picture. The effects of anharmonicities and "mode coupling through polarizability" are also investigated by comparing the 2D Raman signal with a constant temperature velocity reassignment echo method.