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The Journal of Chemical Physics

A theoretical study of resonance Raman scattering from molecules. II. Temperature effect

Yuichi Fujimura, S. H. Lin

Published Nov 1, 1979• 38 citations
Publisher

Abstract

The temperature effect on the resonance Raman scattering from molecules is investigated. Considering a total system consisting of a molecular system, a heat bath, and the weak radiation field, a general expression for the relaxation processes is derived utilizing the Liouville operator technique for the time evolution of the density matrix of the total system. By using the perturbation method with respect to the radiation and matter up to second order, rate constants for zero photon (radiationless processes), one-photon and two-photon processes are derived after taking the long time limit in the Markoff approximation. Assuming a weak interaction between the molecular system and heat bath, a temperature-dependent resonance Raman scattering cross section is formulated in the adiabatic basis set for the molecular systems. In the displaced harmonic oscillator model, we obtain an analytical expression for the temperature-dependent Raman scattering cross section for the multimode case, which is suitable for both weak coupling and strong coupling cases. Analytical expressions for the Stokes and anti-Stokes bands of the jth order vibrational transition in the one oscillator model can easily be obtained from the general expression for the resonance Raman scattering cross section derived. An approximate Stokes to anti-Stokes ratio is given up to the second order of the dimensionless displacement Δ. The resonance Raman scattering cross section and the excitation energy profiles are numerically calculated as a function of physical constant such as temperature, molecular vibrational frequency and Δ. It is shown that the excitation profiles of the anti-Stokes band of the jth order vibrational transition have the similar envelope of the corresponding Stokes band shifted by jh/ω. The inverse Raman effect which is closely related to the ordinary Raman effect is also discussed.

DOI
10.1063/1.438830

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In this paper we apply the T matrix formalism of scattering theory to derive general expressions for the absorption cross sections, the cross sections for resonance fluorescence and the emission quantum yields from large molecules in the statistical limit. In the simple case of an isolated molecular resonance both the absorption line shape and the photon scattering cross section exhibit a Lorentzian distribution on the photon energy, the emission quantum yields are distributed among the ground state vibronic levels according to their radiative widths and, most important, the emission quantum yields are independent of the photon energy and of the spectral width of the exciting light. We were able to derive general expressions for the resonance scattering from a pair of overlapping resonances, including radiative corrections to infinite order. The absorption cross section does not vanish in the region of destructive interference but assumes a finite value which depends on the radiative widths. A sharp maximum in the partial and in the total emission quantum yields is exhibited in the destructive interference regions. This general scheme was applied to a pair of zero order discrete states, one of which is optically active, which interact with an optically inactive quasicontinuum. The energy dependent quantum yield depends on the total width of the radiatively impotent state and may exhibit a minimum. We have demonstrated that when interference effects are involved the decay characteristics of this system will differ for coherent and for narrow band excitation. The general formalism was utilized to derive approximate relations for the resonance fluorescence cross sections and for the quantum yield in the case of a Fano absorption line shape which are valid away from the interference region. Finally, we have applied the general theoretical scheme to the case of the direct photodissociation spectrum of molecules. We have demonstrated that a finite energy dependent emission quantum yield will be observed when a molecule is optically pumped into a dissociative continuum.

The Journal of Chemical Physics•Oct 1, 1972•58 citations
The temperature effect on the resonance Raman scattering from molecules is investigated. Considering a total system consisting of a molecular system, a heat bath, and the weak radiation field, a general expression for the relaxation processes is derived utilizing the Liouville operator technique for the time evolution of the density matrix of the total system. By using the perturbation method with respect to the radiation and matter up to second order, rate constants for zero photon (radiationless processes), one-photon and two-photon processes are derived after taking the long time limit in the Markoff approximation. Assuming a weak interaction between the molecular system and heat bath, a temperature-dependent resonance Raman scattering cross section is formulated in the adiabatic basis set for the molecular systems. In the displaced harmonic oscillator model, we obtain an analytical expression for the temperature-dependent Raman scattering cross section for the multimode case, which is suitable for both weak coupling and strong coupling cases. Analytical expressions for the Stokes and anti-Stokes bands of the jth order vibrational transition in the one oscillator model can easily be obtained from the general expression for the resonance Raman scattering cross section derived. An approximate Stokes to anti-Stokes ratio is given up to the second order of the dimensionless displacement Δ. The resonance Raman scattering cross section and the excitation energy profiles are numerically calculated as a function of physical constant such as temperature, molecular vibrational frequency and Δ. It is shown that the excitation profiles of the anti-Stokes band of the jth order vibrational transition have the similar envelope of the corresponding Stokes band shifted by jh/ω. The inverse Raman effect which is closely related to the ordinary Raman effect is also discussed.
A theory for the resonance light scattering in a molecular system interacting with the heat bath is developed, taking into account the multilevel structure of the molecular system. An expression for the cross section of the resonance light scattering process is derived by using the Liouville operator method in the Markoff approximation; the cross section consists of three terms: resonance Raman scattering, resonance fluorescence, and their mixing. The mixing term vanishes in the limit of the zero vibrational dephasing constant. Analytical expressions for the resonance Raman scattering and resonance fluorescence cross sections are derived in the displaced harmonic oscillator model. Model calculations of the vibrational band structure in the resonance light scattering cross section are performed by using the expressions derived. Differences of the band intensity distribution in the resonance fluorescence from that in the resonance Raman scattering are illustrated.
We consider light scattering from a multilevel system (e.g., a molecule) interacting with several heat baths. The Liouville space approach is employed and weak incident radiation intensity is assumed, whereas the light scattering process is described in the lowest appropriate order in the interaction between the molecule and the incident and emitted photons. The interaction with other radiation field modes as well as with the thermal baths is treated more fully, such as to account for the correct damping terms and their temperature dependence. The formalism is suited for light scattering from an impurity molecule in both gas and dense phases. The thermal baths induce relaxation within the manifold of excited states as well as transitions to nonradiative channels. Damping due to such transitions (as well as radiative damping) corresponds to interactions with zero temperature baths. The physical interpretation of such a bath depends on the nature of the particular damping process. Thermal relaxation within the excited manifold is induced by interaction with the environment represented by a bath of temperature T. Part of this interaction which is diagonal in the molecular space corresponds to the so-called ’’quasielastic collisions’’ and induces modulations of the molecular energy levels and broadening of the relaxed emission. The remainder, nondiagonal part induces actual transitions between the excited molecular energy levels. The formalism yields in a single calculation expressions for the cross section for both the coherent (Raman) scattering process and the relaxed luminescence and enables us to discuss the effects of different factors such as temperature and quenching processes on the relative yields for these processes. The formalism also allows for interference between different levels of the excited manifold due to coupling of several levels to the same bath, and makes it possible to study the effect of thermal relaxation on these interference phenomena.
The problem of resonant interaction between an electromagnetic radiation field and a vibronic system, such as a molecule or a localized electronphonon system, is considered. The vibronic coupling is assumed to result from displaced and distorted potential energy surfaces for the vibrational motion in the electronic quantum states involved in the resonant interaction with the radiation field. Furthermore, the vibrational energy relaxation is taken into account without assumptions about Markovian-type relaxation. In order to study correlation properties of higher than second order in the electric field strength of the light emitted a method of calculation of the expectation value of an arbitrary product combination of effective time evolution operators of the vibronic system is developed. The calculations are performed by utilizing Feynman's disentangling and operator ordering techniques and yield closed solutions in the low-temperature limit. The theory is used for studying the intensity and the normally ordered intensity correlation function of the resonance fluorescence light from a two-atom molecule excited by a relatively weak laser field, which undergoes phase fluctuations. It is shown that the vibronic coupling can give rise to an anticorrelation effect which is substantially stronger than in the case of resonance fluorescence from an atom. For the case of small vibrational damping a general formula which directly connects the asymptotic long-time behaviour of the normalized intensity correlation function with the vibrational relaxation in the molecule is deduced. It should be convenient for the determination of very small vibrational decay rates.
A simple description of the stimulated Raman emission of Stokes radiation, and of the induced absorption found by Stoicheff at the anti-Stokes frequency, is based on the semiclassical theory of radiation. This theory gives the well-known formulas for absorption and stimulated emission when carried to the first order in the time-dependent perturbation due to the interaction of the light beam with the molecules. If there are two light beams of frequencies ωL and ω, the second-order perturbation may be associated with stimulated emission at the Stokes frequency ω=ω−1=ωL—ωR and with induced absorption at the anti-Stokes frequency ω=ω1=ωL+ωR, where ωL is the laser frequency and ωR a Raman-active vibrational frequency; in both cases, there is a molecular transition to the excited vibrational state. The highly directional and very sharp anti-Stokes emission at ω1 arises from the conversion by molecules in intense beams at ωL and ω−1 of two ωL photons into an ω−1, ω1 pair. Like spontaneous emission from excited states, the production of this anti-Stokes radiation requires the full quantum theory of radiation for an adequate description. The emission of ω1 and the observed directional absorption of ω−1 due to the reverse conversion ω−1+ω1→2ωL are explained. The anti-Stokes emission is proportional to the Stokes intensity and to the square of that of the laser, and to the square of the molecular number density; its sharpness is due to the absence of a molecular transition. The generation of photon pairs ω−n+ωn, where ωn=ωL+nωR, can occur for all values of n, but it is most efficient for n=1, due to resonance in the appropriate molecular hyperpolarizability and to the intensity of the Stokes beam at ω−1.
A theoretical study is presented of the effects of collisional perturbations on the time-resolved resonant and near-resonant photon scattering from a molecular system. The photon counting rate is formulated in terms of tetradic Green's functions in Liouville space. The collisional and radiative damping effects are handled by an effective-Liouvillian formalism. The formal expressions are disentangled by invoking a series of successive approximations, assuming short correlation times of the thermal bath, neglecting relaxation processes in the ground electronic state, considering rotational relaxation as providing the dominant damping mechanism in the excited electronic manifold, specializing to weak electromagnetic fields, neglecting some off-resonance processes and disregarding collision-induced radiative processes. The photon counting rate is expressed as a triple convolution of the photon counting rate for the free molecule with the Doppler Gaussian profile and with a Lorentz profile. The latter incorporates purely collisional effects due to intrastate cross relaxations and proper T2{T}_{2}T2​ processes. The theory accounts well for the recent experimental data of Williams, Rousseau, and Dworetsky on collisionally perturbed I2{\mathrm{I}}_{2}I2​ molecules.
In this paper we apply the T matrix formalism of scattering theory to derive general expressions for the absorption cross sections, the cross sections for resonance fluorescence and the emission quantum yields from large molecules in the statistical limit. In the simple case of an isolated molecular resonance both the absorption line shape and the photon scattering cross section exhibit a Lorentzian distribution on the photon energy, the emission quantum yields are distributed among the ground state vibronic levels according to their radiative widths and, most important, the emission quantum yields are independent of the photon energy and of the spectral width of the exciting light. We were able to derive general expressions for the resonance scattering from a pair of overlapping resonances, including radiative corrections to infinite order. The absorption cross section does not vanish in the region of destructive interference but assumes a finite value which depends on the radiative widths. A sharp maximum in the partial and in the total emission quantum yields is exhibited in the destructive interference regions. This general scheme was applied to a pair of zero order discrete states, one of which is optically active, which interact with an optically inactive quasicontinuum. The energy dependent quantum yield depends on the total width of the radiatively impotent state and may exhibit a minimum. We have demonstrated that when interference effects are involved the decay characteristics of this system will differ for coherent and for narrow band excitation. The general formalism was utilized to derive approximate relations for the resonance fluorescence cross sections and for the quantum yield in the case of a Fano absorption line shape which are valid away from the interference region. Finally, we have applied the general theoretical scheme to the case of the direct photodissociation spectrum of molecules. We have demonstrated that a finite energy dependent emission quantum yield will be observed when a molecule is optically pumped into a dissociative continuum.