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The Journal of Chemical Physics

A quantum electrodynamical study of intermolecular line broadening and line shift

Nick P. Blake

Published Nov 1, 1990• 6 citations
Publisher

Abstract

In this paper quantum electrodynamics is used to investigate the effects of radiation field-induced energy transfer as a mechanism for the manifestation of frequency shift and both homogeneous and inhomogeneous line broadening. Using resolvent operator methods and the Power–Zienau–Woolley multipolar Hamiltonian to derive an effective Hamiltonian, it is illustrated how all coherent and incoherent field-induced line shift and line broadening including Förster energy transfer, dispersion, and cooperative Rayleigh scattering can be described solely in terms of a single multipolar interaction. The detailed analysis reveals that a description of intermolecular interactions in terms of virtual photon coupling leads a detailed range dependence that is not only different to nonretarded semiclassical theory, but also complex. This paper therefore illustrates for the first time that all intermolecular processes contribute to both line broadening and energy shift, and that the imaginary contributions to virtual photon coupling lead to significant retardation effects in condensed media. Finally by applying the results within a density matrix framework important aspects regarding the statistical averaging of these additional terms in the Hamiltonian are discussed, and it is illustrated how intermolecular coupling leads to non-Lorentzian, redshifted, asymmetric line profiles.

DOI
10.1063/1.458987

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•Sep 1, 2016•0 citations
In this paper quantum electrodynamics is used to investigate the effects of radiation field-induced energy transfer as a mechanism for the manifestation of frequency shift and both homogeneous and inhomogeneous line broadening. Using resolvent operator methods and the Power–Zienau–Woolley multipolar Hamiltonian to derive an effective Hamiltonian, it is illustrated how all coherent and incoherent field-induced line shift and line broadening including Förster energy transfer, dispersion, and cooperative Rayleigh scattering can be described solely in terms of a single multipolar interaction. The detailed analysis reveals that a description of intermolecular interactions in terms of virtual photon coupling leads a detailed range dependence that is not only different to nonretarded semiclassical theory, but also complex. This paper therefore illustrates for the first time that all intermolecular processes contribute to both line broadening and energy shift, and that the imaginary contributions to virtual photon coupling lead to significant retardation effects in condensed media. Finally by applying the results within a density matrix framework important aspects regarding the statistical averaging of these additional terms in the Hamiltonian are discussed, and it is illustrated how intermolecular coupling leads to non-Lorentzian, redshifted, asymmetric line profiles.
In this paper a quantum electrodynamical theory of cooperative Raman scattering in fluids is developed. The process is one in which pairs of molecules undergo concerted Raman transitions via an intermolecular energy transfer mechanism. The formalism is also extended to chromophore pairs in which intramolecular energy transfer is involved. Using the Power-Zienau-Woolley multipolar Hamiltonian, intensity equations of the Kramers-Heisenberg type are derived which are valid over all regions of molecular separation. In the near zone it is illustrated how the intensity of cooperative scattering is simply that obtained by consideration of the instantaneous response of one molecule to the scalar field generated by another. Within this regime it is shown that the rate of scattering has the familiar ${R}^{\mathrm{\ensuremath{-}}6}$ dependence on intermolecular separation R associated with interacting induced dipoles. In the far zone it is demonstrated that the fully retarded and causal interaction between molecules is mediated through virtual photon coupling, and that this leads to a result for the intensity that dies off with ${R}^{\mathrm{\ensuremath{-}}2}$. This long-range-interaction contribution to the overall scattering intensity is generally ignored in collisional treatments of intermolecular interactions, yet is likely to be of more significance than higher-order multipole terms in near-zone calculations. Finally it is shown how bands in a Raman spectrum due to cooperative scattering effects may be identified by their unique pressure characteristics, or by observation of light scattering that occurs outside the interaction volume of the laser beam. In the case where the scattering occurs between molecules or chromophores held in a fixed orientation with respect to one another, it is demonstrated how detection may be facilitated through the manifestation of a differential scattering effect.
Abstract A canonical transformation is performed on the conventional Hamiltonian for the electromagnetic radiation field and an assemblage of neutral molecules in interaction. The new Hamiltonian has interaction energies expressed in terms of the electromagnetic fields alone and these energies have direct physical significance. The terms linear in e are the multipole interactions, both electric and magnetic, and the term quadratic in e is a generalization of the elementary diamagnetic energy shift. The intermolecular Coulomb energies have cancelled with transverse polarization fields in the new Hamiltonian, although the intramolecular Coulomb potentials are left unaffected. The equations of motion that follow from the new Hamiltonian are deduced. They are the so-called atomic-field equations for the Maxwell fields and Schrödinger equation for an electron wave in a transverse electromagnetic field. The former are the microscopic analogues of Maxwell’s equations in a medium (not restricted to dipole polarization fields) and the latter are dependent on the field strengths alone (not explicit functions of the vector potential).
A landmark in the development of quantum electrodynamics was the discovery that emission–reabsorption of virtual photons modifies the value of energy levels in an atom from those computed by using Dirac’s equation (Lamb shift). An early result of statistical quantum electrodynamics was that the exchange of virtual photons in an ensemble of identical atoms leads as well to a change in the frequency of the radiation emitted from this system (cooperative Lamb shift). Dicke’s discovery that coherence effects lead to the shortening of the emission lifetime from a small sample by a factor equal to the number of atoms in the ensemble (superradiance or cooperative decay rate) was an early landmark in quantum optics. Both cooperative decay rate and cooperative Lamb shift were shown to have the same physical origin—the exchange of virtual photons, a process described by the Lienard–Wiechert dipole–dipole interaction. This effective potential is the kernel of the integral equation describing the dynamics of the system. This complex long-range kernel gives, for both cooperative quantities, strong dependence on the geometry of the atomic cloud. I summarize the known expressions for the initial cooperative decay rate and the cooperative Lamb shift in different geometries. The results for both the scalar photon and the vector photon (electrodynamics) theories for experimentally realizable systems of either uniform or phased polarization are given.
In this paper it is demonstrated how the phenomena of cooperative Raman scattering and cooperative two-photon absorption may be described in a unified manner, through canonically transforming the system Hamiltonian. The unique features associated with the virtual photon coupling between centres are discussed. In particular it is illustrated how in the near-zone, close proximity leads to two-body two-photon events involving interaction-induced dipoles. In the far-zone, it is shown how the coupling between molecules may be understood in terms of a radiative dipole effect. Attention is focused upon the behaviour of this coupling for distributive and cooperative mechanisms in fluids, and the relative importance of each process discussed, particularly with regard to their very different range dependences. A detailed analysis of the coupling reveals that in two-photon absorption the distributive mechanism dominates, whilst in Raman scattering cooperative energy transfer is favoured. Finally recent progress in the field of synergism is reviewed along with the characteristics associated with synergism in Raman scattering and two-photon absorption.
The close-coupled theory of atomic collisions in the presence of a radiation field may be used to calculate the distribution of final atomic states which results from absorption of polarized light during a collision. The theory applies equally well to optical collisions (line broadening) and to radiative collisions (laser-induced collisional energy transfer). For an optical collision the detuning $\ensuremath{\omega}\ensuremath{-}{\ensuremath{\omega}}_{\ensuremath{\infty}}$ is restricted to be larger than either the Rabi frequency or the widths due to natural, Doppler, or pressure broadening. The radiation field is assumed to be weak enough that the transition probabilities are linear in field intensity. The molecular picture is emphasized in which the wave function is expanded in a basis of field-free molecular states and the Hamiltonian is blocked in accordance with molecular quantum numbers. The quantities needed to predict experimental observables are reduced radiative scattering SSS-matrix elements ${s}_{\ensuremath{\omega}}$ in the asymptotic Hund's case-(e) representation. The theory of product orientation and alignment is equivalent to that which has been developed for molecular photodissociation. It requires coherent sums of ${s}_{\ensuremath{\omega}}$ matrix elements from different transition branches, that is, for PPP-, QQQ-, and RRR-type transitions corresponding to changes of -1, 0, and +1 in total molecular angular momentum. On the other hand, the total absorption coefficient and the branching ratio to different final fine-structure states irrespective of orientation or alignment depend on incoherent sums of the same matrix elements.
This thesis employs molecular quantum electrodynamical theory to analyse the interactions between light and matter for four main processes. The first to be considered is Raman scattering, where the effect of the electrodynamic environment for the centre of spectroscopic interest is considered. This is achieved by engaging a retarded dipole-dipole interaction between the centre and a neighbouring molecule. Physically, this is explained by a virtual photon between the pair of centres. The results predict characteristic new lines on the Raman spectrum, for the species, arising from the engagement of selection rules not limited to those of a two-photon process. \nThe second process, is hyper-Rayleigh scattering, in which a single multipolar coupling is considered in place of the more familiar electric dipole. This modification to the theory subverts the standard selection rules for a three-photon process, which can allow for second-harmonic emission to be generated by a centre of high symmetry, such as a centrosymmetric molecule. \nThe third process offers another means of subverting the standard selection rules for second-harmonic generation, namely by incorporating six-wave mixing. First, the general mechanism is developed and pertinent results are expressed for the widely-deployed depolarisation ratio. Following this, structured light is considered and by utilising orbital angular momentum (OAM) conservation arguments, the pair of harmonic photons are found to display quantum entanglement. Moreover, the relative magnitudes of the possible emissions are found to correspond directly to that of the binomial coefficients. \nThis thesis concludes with a family of novel structures capable of directly generating OAM light. This work exploits symmetry characteristics for a delocalised excitonic structure that can allow for a more complex multipolar emission than that of any isolated centres. The phase of the exciton is shown to display an azimuthal phase progression, a vortex feature most commonly associated with Laguerre-Gaussian light.