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The Journal of Physical Chemistry A

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

Mohamad M. Toutounji

Published Sep 6, 2002• 10 citations
Publisher

Abstract

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 profile, which should only be determined by pure electronic dephasing as dictated by experiments. It is therefore important to study how the MBO model bath modes contribute to the ZPL, ZPH, and slow-decay component profiles. Analytical expressions for the linear absorption spectrum and width and Franck−Condon factor of the ZPL are derived. Homogeneous linear absorption spectra, two-pulse photon-echo, and hole-burning calculations are carried out with model systems of which the parameter values are typical for real systems. The MBO model ZPL, ZPH, and slow-decay component were not seen in linear absorption, hole-burning, and two-pulse photon-echo profiles, respectively. The hole-burned spectrum is produced by blending the line broadening function, g ( t; T ), of the MBO model and Small hole-burning formula. This full (inclusion of Matsubara series) form of g ( t; T ) has not been exploited before in any spectroscopic calculation. It is concluded that the MBO model ZPL and ZPH widths and the electronic exponential decay are better exhibited in the corresponding profiles when using non-Ohmic spectral density.

DOI
10.1021/jp025686r

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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 profile, which should only be determined by pure electronic dephasing as dictated by experiments. It is therefore important to study how the MBO model bath modes contribute to the ZPL, ZPH, and slow-decay component profiles. Analytical expressions for the linear absorption spectrum and width and Franck−Condon factor of the ZPL are derived. Homogeneous linear absorption spectra, two-pulse photon-echo, and hole-burning calculations are carried out with model systems of which the parameter values are typical for real systems. The MBO model ZPL, ZPH, and slow-decay component were not seen in linear absorption, hole-burning, and two-pulse photon-echo profiles, respectively. The hole-burned spectrum is produced by blending the line broadening function, g ( t; T ), of the MBO model and Small hole-burning formula. This full (inclusion of Matsubara series) form of g ( t; T ) has not been exploited before in any spectroscopic calculation. It is concluded that the MBO model ZPL and ZPH widths and the electronic exponential decay are better exhibited in the corresponding profiles when using non-Ohmic spectral density.
While different expressions of homogeneous linear electronic dipole moment time correlation functions and their respective absorption lineshapes of anharmonic systems were derived and discussed at length in part I [Toutounji, M. J. Phys. Chem. B 2010, in press] of this study, the electronic dephasing (zero-phonon line width), caused by pseudolocal phonons, and vibrational relaxation time scales were treated equally, hence, unphysical. In this study electronic dephasing and vibrational structure are correctly accounted for in harmonic and anharmonic multimode systems. Model calculations are presented. Anharmonic hole-burned spectra are calculated using linear anharmonic absorption lineshapes. An analytical expression of hole-burned absorption line shape in terms of the linear dipole moment time correlation function, leading to a hole-burned absorption line shape expressed in time-domain, is reported. An elementary method is utilized to treat nonlinear spectra in anharmonic molecules. The link between a hole-burned spectrum and a 2-pulse photon echo profile is established by combining the theoretical framework of Mukamel and that of Hays-Small, thereby calculating photon echo signals of anharmonic systems. Model photon echo calculations of a solvable harmonic model and an anharmonic system are reported. Applications to Al-phthaolcyanine tetrasulphonate (APT) in hyperquenched glassy ethanol and special pair bacterial reaction center are presented and discussed in detail.
An approximate excited state vibrational Hamiltonian, He, which accounts for both linear and diagonal quadratic electron-phonon coupling is derived and shown to be adequate for mode frequency changes up to ~30%. The associated linear response function which is valid for finite temperature in the absence of damping is obtained. 2-point correlation function or, equivalently, the line broadening function, g(t;T), that provides a good physical description of pure electronic dephasing and vibrational damping for linearly coupled modes is postulated. A linear response function that accounts for pure electronic dephasing and damping for systems with either linearly or quadratically coupled modes is proposed. Applications of the new linear response function include the single site absorption and hole-burned spectra of the special pair band of the bacterial reaction center and the temperature dependence of the single site absorption spectrum of Al-phthalocyanine tetrasulphonate in glassy ethanol. A 4-point correlation function which is valid for finite temperature in the absence of damping using the approximate excited state vibrational Hamiltonian, He, is derived. An impulsive stimulated photon echo signal for a single-mode system with very large inhomogeneous broadening is calculated using the 4-point correlation function. An expression for ordinary photon echo with two finite pulses is derived. 4-point correlation function for linearly coupled modes is expressed in terms of g(t;T) and can be used to calculate all non-linear spectroscopic signals. Applications of the 4-point correlation function are limited to impulsive 2-pulse photon echo (IPE) signals. IPE signals are generated for different values of the Huang-Rhys factor (S) at a fixed temperature and at different temperatures for a fixed value of S in order to see how the echo profile changes as one varies S or temperature. An initial fast non-exponential decay, which reflects the contribution of the phonon sideband, followed by vibrational oscillations (quantum beats) due to linear electron-phonon coupling and a slow exponential decay, on picosecond time scales, due to the zero-phonon line are observed in the IPE signals. Application of IPE to the special pair absorption band of the bacterial reaction center is also included.
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.
Fluorescence—line—narrowing, spectral hole—burning and single molecule spectroscopy are methods of selective spectroscopy using zero-phonon lines (ZPL) to make selection in molecular ensemble. However if electron-phonon coupling is strong or sample temperature is near to room temperature, ZPL are absent and the optical band consists of electron-phonon transitions which are called phonon-side-band (PSB). In these cases the above mentioned methods of selective spectroscopy do not work. Can we get an information concerning a homogeneous optical band from inhomogeneously broadened bands in these cases? The answer is yes if we use femtosecond 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.
We continue the discussion of the nonperturbative theory of zero-phonon impurity lines in crystals formulated by Osad’ko and more recently by us. In that paper, a diagrammatic analysis was undertaken which led to analytic results for the thermal width and shift of the zero-phonon line. Here we evaluate these results for two model phonon densities of states: the Debye model for acoustic phonons, and a sharply peaked density of states appropriate for optical phonons. For Debye phonons we find that at low temperatures the standard (perturbative) theory of optical dephasing is qualitatively correct, but at high temperatures it becomes very inaccurate. Moreover, if one of the excited state phonon frequencies tends to zero (a soft mode), then perturbation theory breaks down completely. For optical phonons our results give an Arrhenius temperature dependence for the linewidth and place an upper bound on the value of the preexponential. In paper II of this series we will discuss dephasing by pseudo-local phonons, and in paper III we will compare our results to absorption, photon echo, and hole burning experiments.