Absorption cross-section classical theory

With the formalism of the time-dependent perturbation theory for interaction between an electron and the classical radiation field, within the electric dipole and Bom-Oppenheimer approximations, the isotropic absorption cross section is given by [17]... 

Figure 5.2 Absorption cross section of SissHse calculated using (1) tight-binding approach with local field effects (solid thick line), (2) the tight-binding energy levels with a classical model for the surface polarization contribution (dashed line) and (3) a time-dependent local density approximation (TDLDA) within density functional theory (solid thin line). TDLDA results from ref. 39.

The scattering cross section depends on the matrix element (8.9) of the polarizability tensor and contains furthermore the frequency dependence derived from the classical theory of light scattering. One obtains [8.15] analogously to the two-photon absorption cross section (Sect.7.4) ... 

We commence by deriving the absorption cross-section of a classical electric dipole oscillator.. The result should be similar to that obtained on the basis of the quantum theory and is of further interest since the frequency dependence of the cross-section is predicted in a simple way. Next we obtain the relations between the spontaneous emission transition probability, and the... 

Einstein coefficients for absorption and stimulated emission, denoted by and respectively. The expressions for B j, and Bj are then confirmed by means of quantum mechanics using time-dependent perturbation theory. This enables the probability of stimulated emission and absorption of radiation to be given in terms of the oscillator strengths of spectral lines. Finally we show that there is close agreement between the classical and quantum-mechanical expressions for the total absorption cross-section and explain how the atomic frequency response may be introduced into the quantum-mechanical results. 

Absorption and photodissociation cross sections are calculated within the classical approach by running swarms of individual trajectories on the excited-state PES. Each trajectory contributes to the cross section with a particular weight PM (to) which represents the distribution of all coordinates and all momenta before the vertical transition from the ground to the excited electronic state. P (to) should be a state-specific, quantum mechanical distribution function which reflects, as closely as possible, the initial quantum state (indicated by the superscript i) of the parent molecule before the electronic excitation. The theory pursued in this chapter is actually a hybrid of quantum and classical mechanics the parent molecule in the electronic ground state is treated quantum mechanically while the dynamics in the dissociative state is described by classical mechanics. 

Baev et al. review a theoretical framework which can be useful for simulations, design and characterization of multi-photon absorption-based materials which are useful for optical applications. This methodology involves quantum chemistry techniques, for the computation of electronic properties and cross-sections, as well as classical Maxwell s theory in order to study the interaction of electromagnetic fields with matter and the related properties. The authors note that their dynamical method, which is based on the density matrix formalism, can be useful for both fundamental and applied problems of non-linear optics (e.g. self-focusing, white light generation etc). 

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