Classical formalism

We will examine a model MC with two quantum wells (QWs) with polarized in-plane exciton resonances and located near z = 0 within a distance d A (Fig. 13.12). We can consider both QWs as one very thin layer with polarization 

Equation (13.100) gives the dispersion of polaritons for a cavity with two QWs. The function F(uj, k) represents the modification of the electric field by the microcavity and may easily be generalized for example if d is of the order of A. In an empty cavity F(uj, k) yields the cavity photon dispersion ui = Q = Q(k). If then l(k), uiw and ujf are close we expand F(u , k) for to ss 

2 Ti31 is the polariton splitting if only the semiconductor QW is present, 2 T23 if only the organic QW is present. Equation (13.104) gives the dispersion of the cavity polaritons. We assumed above that e e to neglect the radiative width of states. However, eqn (13.100) also contains information on the radiative widths. For example, taking for the factor r (eqn 13.98), the more exact expression r — e-i L(l + 2/ /q), we obtain for H(fc) = fV(k) — ifl (fc) for the lowest cavity modes 

To discuss phonon relaxation processes, it is better to develop a quantum-mechanical treatment still with e e. 

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The present appendix represents a detailed derivation of the kinetic equations of the fluctuating liquid cage model in the classical formalism. A natural generalization is done for the case of partially ordered media, e.g. nematic liquid crystals. One of the simplest ways to take into account the back reaction is demonstrated, namely to introduce friction. 

A combination of different approaches is at present the most convenient strategy. Most of the work done an complex material models adopts a classical formalism, disregarding for the moment quantum aspects, while there are significant progresses in quantum description of simple models [26]. 

Alternatively, fundamental parameter methods (FPM) may be used to simulate analytical calibrations for homogeneous materials. From a theoretical point of view, there is a wide choice of equivalent fundamental algorithms for converting intensities to concentrations in quantitative XRF analysis. The fundamental parameters approach was originally proposed by Criss and Birks [239]. A number of assumptions underlie the application of theoretical methods, namely that the specimens be thick, flat and homogeneous, and that, for calibration purposes, the concentrations of all the elements in the reference material be known (having been determined by alternative methods). The classical formalism proposed by Criss and Birks [239] is equivalent to the fundamental influence coefficient formalisms (see ref. [232]). In contrast to empirical influence coefficient methods, in which the experimental intensities from reference materials are used to compute the values of the coefficients, the fundamental influence coefficient approach calculates... 

Tapia, O. Solvent effect theories quantum and classical formalisms and their applications in chemistry and biochemistry, J.Math. Chem., 10 (1992), 139-181... 

In the classical formalism it is assumed that bimolecular electron transfer occurs in a precursor complex in which the inner-coordination shells of the reactants are in contact, that is, r - , where a2 and a3 are the hard-sphere radii of the reactants (16). Under these conditions the... 

We next consider the expression for k in the classical formalism. According to the Franck-Condon principle, internuclear distances and nuclear velocities do not change during the actual electron transfer. This requirement is incorporated into the classical electron-transfer theories by postulating that the electron transfer occurs at the intersection of two potential energy surfaces, one for the reactants... 

While such results can also be inferred from the classical formalism [F. H. Crawford. Phys. Rev. 72, 521A (1947)], they have a particularly transparent basis in the metric space Ms-... 

The classical formalism quantifies the above observations by assuming that both the ground-state wave functions and the excited state wave function can be written in terms of antisymmetrized product wave functions in which the basis functions are the presumed known wave functions of the isolated molecules. The requirements of translational symmetry lead to an excited state wave function in which product wave functions representing localized excitations are combined linearly, each being modulated by a phase factor exp (ik / ,) where k is the exciton wave vector and Rt describes the location of the ith lattice site. When there are several molecules in the unit cell, the crystal symmetry imposes further transformation properties on the wave function of the excited state. Using group theory, appropriate linear combinations of the localized excitations may be found and then these are combined with the phase factor representing translational symmetry to obtain the crystal wave function for the excited state. The application of perturbation theory then leads to the E/k dependence for the exciton. It is found that the crystal absorption spectrum differs from that of the free molecule as follows ... 

Classically, formal oxidation numbers ranging from —3 (e.g., in NH3) to +5 (e.g., in HN03) have been assigned to nitrogen though useful in balancing redox equations, they have no physical significance. 

The value of 4TrNr i Sr/lOMs typically 2 x 10 M s , whereas the value of Z used in the classical formalism is 10" M s. This value of Z seems low, because the rate constants for a diffusion-controlled reaction of two uncharged reactants are ca. 1 X 10 M s, and there are ca. 10 collisions in the solvent cage per encounter. On this basis, Z should be closer to 10 M s". ... 

The driving-force dependence of the quenching rate constant and the redox potential of the ES couple can be obtained from Eq. (d) or (0 if the free-energy dependences of AGj3 and AG are known. In terms of the classical formalism, AG23 is given by... 

Various efforts along these lines have attempted to derive the Boltzmann equation from first principles. However, a number of assumptions come into play in all these derivations, which renders even the more formal analyzes somewhat ad hoc. Therefore, many practitioners do not consider the classical formalism worthwhile. 

In a case where the number of reactants (proton emitters) per site is small (most likely one,) the number of identical observed sites is high, and the event is highly synchronized (the perturbation is short with respect to the relaxation time), the difference between the rate constants calculated according to classical formalism or stochastic approach is less than 15% (Vass, 1980). Thus, in most cases the classical formalism can be employed, but its applicability should always be examined. 

The phenomena described above have all one characteristic in common they can be properly described with a sufficiently accurate ab initio calculation with a standard package. These calculations, based in the Bom-Oppenheimer approximation, consider explicitly only the quantum mechanical nature of electrons, while dealing with the nuclei in a classical formalism. In contrast, there are a number of phenomena in transition metal polyhydrides that cannot be described with this scheme. We are going to present them briefly in the remaining of this section. The rest of the chapter will present a detailed explanation of their origin. 

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