Absorption cross section classical

Derivation of the Beer-Lambert Law from considerations at a molecular scale is more interesting than the classical derivation (stating that the fraction of light absorbed by a thin layer of the solution is proportional to the number of absorbing molecules). Each molecule has an associated photon-capture area, called the molecular absorption cross-section a, that depends on the wavelength. A thin layer of thickness dl contains dN molecules. dN is given by... 

In Ref. [9] we demonstrated how one approaches the DCL for the CC absorption cross section, Eq. (31). In a first step, the overall time evolution operator exp(iHcct/k) has to be replaced by the 5-operator 5i(t, 0) which includes the difference Hamiltonian of the excited CC state and of the ground-state. Then, the vibrational Hamiltonian matrix appearing in the exponent of 5i(t, 0) is replace by an ordinary matrix the time-dependence of which follows from classical nuclear dynamics in the CC ground-state. The time-dependence of the dipole moment d follows from intra chromophore nuclear rearrangement and changes of the overall spatial orientation. At last, this translation procedure replaces the CC state matrix elements of the 5-operator by complex time-dependent functions... 

We define the total classical absorption cross section, save for an unimportant normalization constant, as... 

According to (5.15) and (5.21) the classical approximation of the absorption cross section, as function of the energy in the excited state, is given... 

Let us consider a one-dimensional model system with coordinate R as illustrated in Figure 13.1. Vq(R) and Vi(R) are the potentials in the ground and in the excited electronic states and >i(R) represents the initial wavefunction. In analogy to Equation (6.3), the classical absorption cross section may be approximately written as... 

According to the reflection principle, discussed in detail in Section 6.1, the absorption cross section is simply a reflection of the coordinate distribution, fj(/ ) 2, onto the energy axis mediated directly by the upper-state potential Vi(i2). Equation (13.2) establishes the unique relation between R and E. Within this simple classical picture each maximum and each minimum of E i(/ ) is uniquely mapped onto the energy axis as illustrated in Figure 13.1. 

Within the separable harmonic approximation, the < f i(t) > and < i i(t) > overlaps are dependent on the semi-classical force the molecule experiences along this vibrational normal mode coordinate in the excited electronic state, i.e. the slope of the excited electronic state potential energy surface along this vibrational normal mode coordinate. Thus, the resonance Raman and absorption cross-sections depend directly on the excited-state structural dynamics, but in different ways mathematically. It is this complementarity that allows us to extract the structural dynamics from a quantitative measure of the absorption spectrum and resonance Raman cross-sections. 

The theoretical approaches range from the simplest phenomenological models to complex quantum calculations. They can be split into two main strategies. The first one consists in keeping the classical Mie expression for the absorption cross section... 

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 reports the absorption cross section of a small silicon nanocrystal. It is clear that the tight-binding approach with inclusion of local field effects (calculated by inversion of the dielectric matrix) compares very well to the formulation with a classical model of the surface polarization, based on the Clausius Mossotti equa-... 

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.

In order to apply MD simulations to the calculation of absorption spectrum one needs a classical version of the absorption cross section. This has been the subject of many papers (see, e.g., ref. 99 and references therein), introducing equivalent formulations. We briefly sketch here the most direct one, focusing on a case in which the adiabatic approximation holds (i.e., only a single electronic excited state e)... 

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. 

As might be expected, the absorption cross-section has the Lorentzian frequency dependence which is characteristic of the classical oscillator. The intensity of a collimated beam of radiation, I (x), propagating through a gas of stationary classical atoms, density N per unit volume, would vary as... 

This result should be compared with the classical expression, equation (9.12). We see that the f-value of an absorption line can be interpreted as that fraction of the integral of the classical total absorption cross-section which is to be associated with the given transition. 

Despite the fact that relaxation of rotational energy in nitrogen has already been experimentally studied for nearly 30 years, a reliable value of the cross-section is still not well established. Experiments on absorption of ultrasonic sound give different values in the interval 7.7-12.2 A2 [242], As we have seen already, data obtained in supersonic jets are smaller by a factor two but should be rather carefully compared with bulk data as the velocity distribution in a jet differs from the Maxwellian one. In the contrast, the NMR estimation of a3 = 30 A2 in [81] brought the authors to the conclusion that o E = 40 A in the frame of classical /-diffusion. As the latter is purely nonadiabatic it is natural that the authors of [237] obtained a somewhat lower value by taking into account adiabaticity of collisions by non-zero parameter b in the fitting law. 

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. 

Classical absorption and photo dissociation cross sections... 

First, we provide the formal definitions of classical absorption and photodissociation cross sections and subsequently we describe a practical way in order to calculate them. 

Fig. 5.4. Comparison of the quantum mechanical and the classical absorption spectra for H2O in the second continuum. The quanta result is calculated by means of the time-independent close-coupling method and the classical curve is obtained in a Monte Carlo simulation. Both cross sections are normalized to the same area. The arrow indicates the threshold for H + OH(2E). Reproduced from Weide and Schinke (1989).

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