Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Molecular Physics

Vibrational relaxation of a molecule in a dense medium

Abraham Nitzan, Joshua Jortner

Published Mar 1, 1973• 202 citations
Publisher

Abstract

In this paper we consider some features of vibrational relaxation of a guest molecule in a host matrix. The model system involves a harmonic molecule interacting with a harmonic medium. The molecule-medium coupling was handled by the rotating wave approximation considering linear terms in the intramolecular displacements and high terms in the medium displacements. Three specific models for the molecule-medium coupling were considered, which involve single phonon decay, vibron-phonon decay and multiphonon decay. Within the framework of the random phase approximation the Heisenberg equations of motion for the system could be expressed in terms of a unified scheme which is valid for both single phonon and multiphonon processes. Explicit solutions were derived utilizing the Wigner-Weisskopf approximation. This generalized formalism was applied for the study of the time evolution of the distribution, the cooling and the heating processes of the oscillator by a thermal field and for the coupling between vibrational relaxation and infra-red emission.

DOI
10.1080/00268977300100651

Related papers

Theory of Absorption and Dispersion Profiles of Molecular Transitions in Dense Media

H. Paeeschke, K.‐E. Süsse, D.‐G. Welsch +1 more

Abstract Starting from quantum theoretical principles the phase relaxation is described of a molecule localized in a dense medium by an integro‐differential equation. Using this equation, which leads to a non‐exponential phase decay, general formulas are derived for calculating the absorption and dispersion profiles of vibronic and vibrational transitions of the guest molecule. In detail the calculations are performed for the case that the acoustic phonons of the medium are responsible for the phase relaxation. Especially it is shown, that in comparison with the Lorentz profile the wings of a real absorption profile become steeper with increasing ratio between dephasing rate y and Debye frequency wD‐ Only for γ/wD ≪ 1 the absorption profiles are expected to be Lorentz profiles. The dispersion is found to increase with increasing ratio γ/wD. Both the real absorption and dispersion curves are asymmetric ones.

physica status solidi (b)•Aug 1, 1977•2 citations

Theory of Resonance Fluorescence from Vibronic Systems

K.‐E. Süsse, W. Vogel, D.‐G. Welsch

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.

Physica Scripta•Sep 1, 1983•5 citations

Time evolution of the average energy of a relaxing molecule

V. M. Kenkre, V. Seshadri

Evolution equations for the average energy of a harmonic oscillator in interaction with a bath are derived and solved for two cases: (a) when the probability evolution obeys a master equation with a true decay representing a sink-process like radiative emission occurring simultaneously with vibrational relaxation, and (b) when the Markoffian approximation in the Heisenberg-equation analysis of relaxation is not made, allowing for short-time effects.

Physical Review A•Jan 1, 1977•19 citations

Unified master equation for molecules in phonon and radiation baths

C. H. Raymond Ooi, K. J. Cedric Chia

We have developed a unified quantum optical master equation that includes the dissipative mechanisms of an impurity molecule in crystals. Our theory applies generally to polyatomic molecules where several vibrational modes give rise to intramolecular vibrational redistributions. The usual assumption on identical shapes of the nuclear potentials in ground and excited electronic states and the rotating wave approximation have been relaxed, i.e. the vibrational coordinates are different in the ground and excited states, with counter-rotating terms included for generality. Linear vibrational coupling to the lattice phonons accounts for dissipations via non-radiative transitions. The interaction of a molecule with photons includes Herzberg-Teller coupling as the first order non-Condon interaction where the transition dipole matrix elements depend linearly on vibrational coordinates. We obtain new cross terms as the result of mixing the terms from the zeroth-order (Condon) and first-order (non-Condon) approximations. The corresponding Lamb shifts for all Liouvilleans are derived explicitly including the contributions of counter-rotating terms. The computed absorption and emission spectra for carbon monoxide is in good agreement with experimental data. We use our unified model to obtain the spectra for nitrogen dioxide, demonstrating the capability of our theory to incorporate all typical dissipative relaxation and decoherence mechanisms for polyatomic molecules. The molecular quantum master equation is a promising theory for studying molecular quantum memory.

Scientific Reports•Nov 21, 2022•1 citation

Theory of the interplay of luminescence and vibrational relaxation: A master-equation approach

V. Seshadri, V. M. Kenkre

A theory is developed to describe the interplay of vibrational relaxation and luminescence occurring simultaneously in a molecule, in terms of a master equation involving true sinks of probability. Specifically, the basic equation is the Montroll-Shuler equation augmented by the addition of sink terms which can be nonlinear as well as linear in the vibrational energy. These terms describe radiative and nonradiative decay and expressions for the former are derived explicitly in terms of Franck-Condon factors. Exact solutions in terms of known functions are obtained in the linear case for several physically motivated initial vibrational distributions, viz., $\ensuremath{\delta}$-function, Boltzmann, Poisson, and Laguerre. Two perturbation schemes are developed to analyze the nonlinear case, one of which is useful for small nonlinearities and the other for arbitrarily large nonlinearities but for low temperatures. Illustrative applications of the theory include the exact calculation of time- and frequency-resolved emission spectra (linear decay calculation) and the perturbation analysis of quantum yields in the presence of strongly energy-dependent nonradiative transitions (nonlinear decay calculation).

Physical Review A•Jan 1, 1978•21 citations

Random coupling models for intramolecular dynamics. II. Kinetic equations for collisionless multiphoton excitation of large molecules

Benny Carmeli, Abraham Nitzan

Multiphoton excitation and dissociation of large molecules under collisionless conditions is discussed in terms of an intercontinuum random coupling model. The mathematical approach described in a previous paper is used to obtain the general solution for a system of consecutively coupled discrete states, quasicontinuous manifolds, and continuous (dissociative) manifolds of molecular levels (eigenstates of the total molecular Hamiltonian), where the radiative coupling matrix elements are assumed to be given as a linear combination of smoothly varying and randomly varying (over level indices in the molecular manifolds) components. In the range of discrete molecular levels the time evolution is coherent and described in terms of the optical Bloch equation. In the quasicontinuous and continuous ranges the time evolution may be described in terms of Markoffian kinetic equations for the number of photons absorbed by the molecule, provided that the intramolecular vibrational relaxation widths associated with the optically active molecular modes is much larger than the Rabi frequency associated with the excitation of these modes. The kinetic evolution itself consists of direct multiphoton excitation processes (simultaneous transitions from the upper discrete levels to all higher energy molecular manifolds) resulting from the smooth component in the radiative coupling, and a consecutive excitation process described by the Pauli master equation with rates given by the golden rule expression. The interaction which enters into the golden rule expression is the variance in the radiative coupling. The direct excitation component contributes a negligible part of the overall excitation even if the random and smooth radiative coupling components are comparable. The resulting incoherent time evolution of the multiphoton excitation process is consistent with available experimental results. Coherent effects in the time evolution are expected for higher radiation field intensities, where the Rabi frequency becomes comparable to the intramolecular vibrational relaxation rate.

The Journal of Chemical Physics•Feb 1, 1980•40 citations
In this paper we consider some features of vibrational relaxation of a guest molecule in a host matrix. The model system involves a harmonic molecule interacting with a harmonic medium. The molecule-medium coupling was handled by the rotating wave approximation considering linear terms in the intramolecular displacements and high terms in the medium displacements. Three specific models for the molecule-medium coupling were considered, which involve single phonon decay, vibron-phonon decay and multiphonon decay. Within the framework of the random phase approximation the Heisenberg equations of motion for the system could be expressed in terms of a unified scheme which is valid for both single phonon and multiphonon processes. Explicit solutions were derived utilizing the Wigner-Weisskopf approximation. This generalized formalism was applied for the study of the time evolution of the distribution, the cooling and the heating processes of the oscillator by a thermal field and for the coupling between vibrational relaxation and infra-red emission.
Abstract Starting from quantum theoretical principles the phase relaxation is described of a molecule localized in a dense medium by an integro‐differential equation. Using this equation, which leads to a non‐exponential phase decay, general formulas are derived for calculating the absorption and dispersion profiles of vibronic and vibrational transitions of the guest molecule. In detail the calculations are performed for the case that the acoustic phonons of the medium are responsible for the phase relaxation. Especially it is shown, that in comparison with the Lorentz profile the wings of a real absorption profile become steeper with increasing ratio between dephasing rate y and Debye frequency wD‐ Only for γ/wD ≪ 1 the absorption profiles are expected to be Lorentz profiles. The dispersion is found to increase with increasing ratio γ/wD. Both the real absorption and dispersion curves are asymmetric ones.
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.
We have developed a unified quantum optical master equation that includes the dissipative mechanisms of an impurity molecule in crystals. Our theory applies generally to polyatomic molecules where several vibrational modes give rise to intramolecular vibrational redistributions. The usual assumption on identical shapes of the nuclear potentials in ground and excited electronic states and the rotating wave approximation have been relaxed, i.e. the vibrational coordinates are different in the ground and excited states, with counter-rotating terms included for generality. Linear vibrational coupling to the lattice phonons accounts for dissipations via non-radiative transitions. The interaction of a molecule with photons includes Herzberg-Teller coupling as the first order non-Condon interaction where the transition dipole matrix elements depend linearly on vibrational coordinates. We obtain new cross terms as the result of mixing the terms from the zeroth-order (Condon) and first-order (non-Condon) approximations. The corresponding Lamb shifts for all Liouvilleans are derived explicitly including the contributions of counter-rotating terms. The computed absorption and emission spectra for carbon monoxide is in good agreement with experimental data. We use our unified model to obtain the spectra for nitrogen dioxide, demonstrating the capability of our theory to incorporate all typical dissipative relaxation and decoherence mechanisms for polyatomic molecules. The molecular quantum master equation is a promising theory for studying molecular quantum memory.
A theory is developed to describe the interplay of vibrational relaxation and luminescence occurring simultaneously in a molecule, in terms of a master equation involving true sinks of probability. Specifically, the basic equation is the Montroll-Shuler equation augmented by the addition of sink terms which can be nonlinear as well as linear in the vibrational energy. These terms describe radiative and nonradiative decay and expressions for the former are derived explicitly in terms of Franck-Condon factors. Exact solutions in terms of known functions are obtained in the linear case for several physically motivated initial vibrational distributions, viz., $\ensuremath{\delta}$-function, Boltzmann, Poisson, and Laguerre. Two perturbation schemes are developed to analyze the nonlinear case, one of which is useful for small nonlinearities and the other for arbitrarily large nonlinearities but for low temperatures. Illustrative applications of the theory include the exact calculation of time- and frequency-resolved emission spectra (linear decay calculation) and the perturbation analysis of quantum yields in the presence of strongly energy-dependent nonradiative transitions (nonlinear decay calculation).
Multiphoton excitation and dissociation of large molecules under collisionless conditions is discussed in terms of an intercontinuum random coupling model. The mathematical approach described in a previous paper is used to obtain the general solution for a system of consecutively coupled discrete states, quasicontinuous manifolds, and continuous (dissociative) manifolds of molecular levels (eigenstates of the total molecular Hamiltonian), where the radiative coupling matrix elements are assumed to be given as a linear combination of smoothly varying and randomly varying (over level indices in the molecular manifolds) components. In the range of discrete molecular levels the time evolution is coherent and described in terms of the optical Bloch equation. In the quasicontinuous and continuous ranges the time evolution may be described in terms of Markoffian kinetic equations for the number of photons absorbed by the molecule, provided that the intramolecular vibrational relaxation widths associated with the optically active molecular modes is much larger than the Rabi frequency associated with the excitation of these modes. The kinetic evolution itself consists of direct multiphoton excitation processes (simultaneous transitions from the upper discrete levels to all higher energy molecular manifolds) resulting from the smooth component in the radiative coupling, and a consecutive excitation process described by the Pauli master equation with rates given by the golden rule expression. The interaction which enters into the golden rule expression is the variance in the radiative coupling. The direct excitation component contributes a negligible part of the overall excitation even if the random and smooth radiative coupling components are comparable. The resulting incoherent time evolution of the multiphoton excitation process is consistent with available experimental results. Coherent effects in the time evolution are expected for higher radiation field intensities, where the Rabi frequency becomes comparable to the intramolecular vibrational relaxation rate.