Multimode Brownian Oscillator Research Articles

Ohmic Brownian Oscillator Approach to Hole-Burning and Photon-Echo Spectroscopies

The multimode Brownian oscillator (MBO) model has been at the forefront in interpreting the subsystem−bath interaction manifestations in optical spectroscopy for probing homogeneous structure of chromophores in crystals and amorphous solids. The spectroscopic consequences of employing the underdamped MBO model with Ohmic dissipation in linear absorption, photon-echo, and hole-burning data of chromophores in solid hosts at low temperatures are investigated. The zero-phonon line (ZPL) in homogeneous linear absorption spectrum, the slow-decay component (due to ZPL) in photon-echo signal, and the zero-phonon hole (ZPH) in hole-burned spectra in host molecular solids at low temperatures are usually resolved from the multiphonon-transitions structure. In the MBO model, the harmonic vibrations (Brownian oscillators) are linearly coupled to bath modes. This coupling, with Ohmic dissipation, results in a maximum contribution of the bath modes to the ZPL region. This contribution affects the width of the ZPL profil...

The underdamped Brownian oscillator model with Ohmic dissipation: Applicability to low-temperature optical spectra

The multimode Brownian oscillator (MBO) model with Ohmic dissipation has frequently been used to interpret photon echo and spectral data on solvation dynamics of chromophores in liquids at room temperature. We report on the applicability of this model to high-resolution linear electronic absorption spectra of chromophores in solid hosts at low temperatures, where the zero-phonon line (ZPL) is resolved from phonon sideband structure. The results are also relevant to frequency and time domain nonlinear spectra. In the MBO model, active BOs (phonons) are linearly coupled to bath modes. This coupling endows the bath modes with absorption intensity which, with Ohmic dissipation (white light spectrum for the bath modes), results in the bath modes contributing to absorption in the region of the ZPL. Experimental results for a multitude of molecular systems indicate that the ZPL profile is determined by electronic dephasing, which is not accounted for in the MBO model. Thus, it is important to assess the contribution of the MBO bath modes to the ZPL profile. To this end, closed-form, finite temperature expressions for the underdamped MBO (UMBO) model are derived for the linear response function, linear absorption spectrum, and width and Franck–Condon factor of the ZPL. It is proven formally that the UMBO ZPL width is zero at T=0 K. Results of calculations for model systems whose parameter values (BO damping constant, frequency and Huang–Rhys factor) are typical of real systems are presented. It is concluded that Ohmic dissipation leads to unphysically large ZPL widths as well as asymmetric ZPL profiles that appear not to have been observed. Moreover, the ZPL width adds to those of the multi-BO (phonon) transitions. Thus, use of the UMBO model with Ohmic dissipation to interpret data on relaxation dynamics of nuclear modes may result in erroneous conclusions. It is shown that Franck–Condon factors of the ZPL obtained with the UMBO model can differ significantly from those calculated with the conventional formula.

Ultrafast Protein Dynamics of Bacteriorhodopsin Probed by Photon Echo and Transient Absorption Spectroscopy

Bacteriorhodopsin (bR) is an efficient light-driven proton pump which shows a trans-cis isomerization reaction of its retinal chromophore after light absorption. BR exhibits a large reorganization energy λ of 2520 cm-1 on optical excitation. In this paper, we have studied the nature, origin, and dynamical aspects of this extensive reorganization. We report the results of a femtosecond three-pulse echo peak shift (3PEPS), transient grating (TG) and transient absorption (TA) study, complemented with those of steady-state absorption and fluorescence spectroscopy in wild-type bR and the D85S mutant in its blue and purple, halide-pumping forms. We have simulated the results in the context of the multimode Brownian oscillator (MBO) formalism. A simple model that incorporates retinal's known intramolecular vibrations, which represent 1094 cm-1 or reorganization energy, and a single Gaussian protein relaxation with a decay of 50 fs representing 1430 cm-1 of reorganization energy, yielded satisfactory results for ...

Using the finite-difference time-domain pulse propagation method to simulate time-resolved THz experiments

The finite-difference time-domain (FDTD) method has been applied to time-resolved THz spectroscopy (TRTS) experiments. Time-resolved THz spectroscopy utilizes an optical pump pulse to excite the sample, followed by a far-infrared (FIR) probe pulse with frequency components that span from 10 to 100 cm−1. The subpicosecond evolution of the FIR spectrum is obtained as a function of time after the visible photoexcitation event. Significant challenges arise in interpreting these experimental results due to the very different frequencies of the pump and probe pulses. Therefore, it is essential to simulate the experiment. The method described entails numerically propagating both the THz probe pulse and the visible pump pulse simultaneously, keeping track of the transiently induced polarization from absorption of the visible pulse. Group velocity mismatch between the visible and THz pulse and a transiently changing response function are completely accounted for in the calculation. Furthermore, a spatially varying polarization can be included to account for a nonuniform excited region of the sample under investigation. The response function of the material is described as a multimode Brownian oscillator that can describe dispersive media in a very general sense. In particular, the overdamped, underdamped, and critically damped cases are all included, as well as special cases such as a Debye or Drude response. As a specific example, we present results of modeling a TRTS experiment of photoexcitation of a dye in solution, namely, 2,11,20,29-tetra-tert-butyl-2,3-napthalocyanine, dissolved in toluene. We carry out a nonlinear least squares fit of a parameterized model to the measured data to show that the FDTD–TRTS method is able to accurately reproduce the features observed in the measured data set.

Wavelength-resolved stimulated photon echoes: Direct observation of ultrafast intramolecular vibrational contributions to electronic dephasing

Novel wavelength-resolved stimulated photon echo measurements on a dye molecule in solution are presented. Data are simulated within the multimode Brownian oscillator model using the spectral density of de Boeij et al. [J. Phys. Chem. 100, 11806 (1996)] for the same solute–solvent system. For photon echo population times <50 fs there are considerable differences between the measured and calculated data. Aided by further simulations, we conclude that these discrepancies result from dephasing dynamics of high frequency intramolecular vibrational modes not included in the previously derived spectral density.

Ultrafast solvation dynamics explored by femtosecond photon echo spectroscopies.

Chemical reaction and optical dynamics in the liquid phase are strongly affected by specific solute-solvent interactions. The dynamical part of this coupling leads to energy fluctuations. The associated energy gap dynamics can be probed by using various nonlinear optical spectroscopies. We discuss various forms of photon echo--time-integrated, time-gated, and heterodyne-detected photon echo--as well as Fourier transform spectral interferometry. It is shown that for solutions of the dye molecule DTTCI, a system-bath correlation function can be acquired that provides a quantitative description of all (non)linear spectroscopic experiments. The deduced correlation function is projected onto the multimode Brownian oscillator model, which allows for a physical interpretation of the multiple-time correlation function and a determination of the spectral density relevant to the solvation process. The following applications of photon echo to condensed phase dynamics are discussed: enhanced vibrational mode suppression, Liouville pathways interference, and dynamical Stokes shift. Recent results of echo-peak shift experiments on the hydrated electron are also presented. The review concludes that photon echo should be useful as a novel tool to explore transition state dynamics.

Heterodyne-detected stimulated photon echo: applications to optical dynamics in solution

Heterodyne detection of the stimulated photon echo (HSPE) is discussed and applied to explore molecular solvation dynamics. With this technique the in-phase and in-quadrature parts of the induced transient nonlinear polarization can be time-gated. A third-order perturbative description of the HSPE is presented, which shows that the transient polarization provides direct information on the real and imaginary components of the nonlinear optical response functions of the system. In the spectral diffusion limit and for impulsive excitation conditions, analytical expressions for the HSPE signal are derived and model calculations are presented. This novel technique is applied to a nonlinear optical study of the dye molecule DTTCI dissolved in ethylene glycol. Several effects are analyzed in detail as, for instance, the intriguing interference effect between the conventional and virtual echo contributions to the total HSPE signal. Furthermore, it is shown that from HSPE measurements the instantaneous frequency of the emitted (nonlinear polarization induced) signal can be derived and that its time-dependent shift projects directly onto the dynamical Stokes shift and the energy reorganization process of the coupled solute–solvent complex. It will also be shown that the experimental data can be simulated using the multimode Brownian oscillator model.

Viscoelastic Continuum Model of Nonpolar Solvation. 1. Implications for Multiple Time Scales in Liquid Dynamics

A continuum model of nonpolar solvation is presented. Coupling of the solute to the solvent is assumed to occur through a change in the solute's size or shape upon electronic excitation. Both spherical and nonspherical changes in the solute are treated. The time-dependent shear and longitudinal moduli of the solvent determine the solvation response function. Unlike prior continuum models of solvation, both a rapid, viscosity-independent inertial component and a slower, viscosity-dependent diffusive component emerge from the model, even when only one time scale is assumed in the moduli. The origin of multiple time scales in this model, which has a single solvent coordinate with complex dynamics, is contrasted with treatments such as the multimode Brownian oscillator models, which postulate multiple solvent coordinates, each with simple dynamics.

Femtosecond two-dimensional spectroscopy from anharmonic vibrational modes of molecules in the condensed phase

We have developed a theory of the fifth-order off-resonant spectroscopy to study the effect of anharmonicity of molecular vibrational modes. The anharmonicity, as well as nonlinear dependence of polarizability on nuclear coordinates, can be the origin of the fifth-order Raman signal. A profile of the signal varies depending on the relative importance of the two effects—the anharmonicity and the nonlinearity. The anharmonicity of a potential can be distinguished from the other effects such as the nonlinearity or the inhomogeneity of vibrational modes. In order to carry out calculations analytically, we employ the multimode Brownian oscillator model and treat anharmonicity as perturbation to the harmonic vibrational modes. A simple analytical expression for the fifth-order polarization is obtained through a diagrammatic technique, called Feynman rule on the unified time path. Physical pictures for the analytical expression are given for a single mode system through numerical calculations and through double-sided Feynman diagrams. Applications to CHCl3 and CS2 are made where the third-order experiments are used to extract parameters. In the CS2 case, the theoretical fifth-order signals are compared with recent experiment, which suggests some sign of anharmonicity.

Interplay of multiple vibrational spectral densities in femtosecond nonlinear spectroscopy of liquids

The multimode Brownian oscillator model of nonlinear response functions is generalized to include a multielectronic level system interacting with several spectral densities representing solvent and vibrational coupling to electronic energies, transition dipoles, and permanent dipoles. Applications to resonant and off-resonant transient grating as well as to infrared and fifth-order Raman photon echoes illustrate how the various spectral densities may be probed separately.

System−Bath Correlation Function Probed by Conventional and Time-Gated Stimulated Photon Echo

We show how in the framework of the multimode Brownian oscillator model the system−bath correlation function can be derived from conventional and time-gated stimulated photon echo experiments and consideration of the linear optical spectra. Experiments are performed on the infrared dye DTTCI in room temperature solutions of ethylene glycol, methanol, and acetonitrile. The obtained correlation function is the sum of several Brownian oscillators, of which four are attributed to intrachromophore vibrational dynamics and the other three to solute−solvent dynamics. The ultrafast part of the correlation function on the time scale of the excitation pulses is interpreted as a free induction decay-like effect due to impulsive excitation of spectrally broad underlying vibrational structure in the dye's electronic transition. The slower parts are assigned to multiple time scale solute−solvent dynamics. The effect of vibrational coherences on the echo measurements is also analyzed; this analysis permits the dissection of the correlation function into a part due to intrachromophore dynamics and a part due to solvation dynamics. The spectral densities associated with these latter oscillators are located in the far infrared, in the same spectral region as probed by the optical Kerr effect. The measurements, however, provide no definite answer to the question of whether these spectral densities are the same.

Interrogation of Vibrational Structure and Line Broadening of Liquid Water by Raman-Induced Kerr Effect Measurements within the Multimode Brownian Oscillator Model

The method used to deduce the spectral density distribution of intermolecular and intramolecular (vibrational) degrees of freedom in the liquid state from optical heterodyne detected optical (Raman-induced) Kerr effect (OHD-RIKE) measurements is reexamined within a multimode Brownian oscillator model. The ramifications of nonlinear coupling of the nuclear degrees of freedom to the medium polarizability are explored for discrimination between “homogeneous” and “inhomogeneous” contributions to the vibrational spectral density. Under physically reasonable assumptions, an estimation of the homogeneous contribution to the vibrational line shape can be made from the OHD-RIKE observable (if nonlinear coupling is nonnegligible). The model is developed generally, and calculations are applied specifically to temperature-dependent OHD-RIKE measurements of liquid water. The results indicate that the line broadening in the low-frequency vibrational distribution due to the hydrogen-bonded network structure of liquid wa...

Ultrafast electronic dephasing dynamics of large molecules in room temperature liquids studied by variable pulse-width tunable femtosecond transient absorption spectroscopy

Electronic dephasing dynamics of HITCI in ethylene glycol are investigated by degenerate femtosecond transient absorption spectroscopy. Pulse width and wavelength are varied to change the relative contributions of the coherent and population bleaching terms to the overall transient signal. Results from fitting the experimental data to a multi-mode Brownian oscillator model are presented. The data are also compared with photon-echo measurements.

Description of nonlinear optical response using phase space wave packets

Closed-form expressions are derived for the nonlinear response function of a polyatomic molecule with two electronic states and linearly displaced harmonic potentials to arbitrary order in the radiation field. The underlying dynamics can be best visualized using wave packets in phase space. Using exact expressions for these wave packets, we discuss the possible factorization of the process into two parts representing a forward and a backward propagating wave packet. This factorization facilitates the description of sequential measurements, such as pumpprobe spectroscopy, in terms of a doorway wave packet prepared by the pump and a window prepared by the probe. We show that once a reduced description is adopted, where we trace over bath degrees of freedom, this factorization is possible only in certain limiting cases. I. Introduction The density matrix offers a natural and convenient means for calculating and interpreting ultrafast optical measurements.14 This is well recognized for studies in the condensed phase, since it allows performing thermal averaging and developing a reduced description in which bath coordinates have been eliminated and replaced by dephasing processes. However, even for isolated molecules in the gas phase, the density matrix formulation has a lot to offer compared with wave function based calculations.~ The density matrix follows naturally the time ordering of the various interactions with the fields. In the Wigner representation, it can be used to construct a phase-space wave packet representation of the nonlinear response which provides a powerful framework for developing classical and semiclassical approximations.2 The possibility of visualizing time domain spectroscopy in terms of wave packets in phase space is particularly valuable for sequential experiments such as pump-probe, transient grating, and impulsive Raman scattering conducted using well-separated laser pulsesbll Sequential experiments can be described in terms of overlap of two wave packets in phase space (thedoorway window picture).2J A full quantum treatment of these wave packets in the complete phase space (when all degrees of freedom are included) has been de~eloped.~ This level of description is restricted to small systems (e.g., pumpprobe spectroscopy of ICN in the gas pha~e).~ For larger systems and for spectroscopy in the condensed phase, we need to develop a reduced description in which only part of the phase space is treated explicitly and the remaining degrees of freedom are eliminated. Using a Langevin equation, Yan and Mukamel have calculated the wave packets for a multimode Brownian oscillator model.2 These calculations are limited by the Langevin equation which provides a semiclassical high-temperature approximation. We recently developed a theory for the nonlinear optical response of a multimode Brownian oscillator model which extends that theory in the following respects: First, by adopting a harmonic model for the bath degrees of freedom, we derived exact quantum expressions for the phase space wave packets which are valid at all temperatures and for the entire range of parameters. The solution was obtained using path integral techniques. Second, we calculated the response function to arbitrary order in the field so that we can describe processes higher than four-wave mixing. Third, we incorporated an arbitrary dependence of the transition dipole on nuclear coordinates (non-Condon effects). In this article, we use the exact solution of the multimode Brownian oscillator model to explore the applicability of the

Photon echoes of polyatomic molecules in condensed phases

A theory of optical echo spectroscopies of large polyatomic molecules in condensed phases is developed. Using phase space correlation functions, we examine the interrelationships among the following optical measurements: ordinary photon echo, stimulated photon echo, accumulated photon echo, incoherent accumulated photon echo, and pump–probe absorption. Conditions for the elimination of inhomogeneous broadening in these experiments are specified. A multimode Brownian oscillator model is used to account for high frequency molecular vibrations, as well as solvent modes, and electronic dephasing processes. The effects of quantum beats, spectral diffusion, and homogeneous dephasing on the echo signals are studied and compared in detail with pump–probe and hole burning spectroscopy.

Femtosecond pump-probe spectroscopy of polyatomic molecules in condensed phases.

A theory for ultrafast pump-probe spectroscopy of large polyatomic molecules in condensed phases is developed. A multimode Brownian oscillator model is used to account for high-frequency molecular vibrations and local intermolecular modes as well as collective solvent motions. A semiclassical picture is provided using the density matrix in Liouville space. The pump field creates a doorway state that propagates for a specified time interval, and the spectrum is calculated by finding its overlap with a window state, prepared by the probe pulse. The doorway and the window states are wave packets in phase space. For high-frequency modes and with long pulses they are expanded in the vibronic eigenstates, whereas for low-frequency modes and with impulsive pulses the Wigner (phase-space) representation is more adequate. Conditions for the observation of quantum beats, spectral diffusion, and solvation dynamics (dynamical Stokes shift) are specified.