Rotating wave approximation rotation factor

We have used the Born Oppenheimer approximation to factor 4 0/3, I,ma into electronic and nuclear parts and have further assumed that the former are orthogonal to enable us to reduce V. Both wave functions may be approximated by products of electronic, nuclear rotation and vibrational wave functions. The last of these may be factored out at once, and... 

Finally, it is also interesting to compare the result (9.65) to the result (8.106) ofthe very different semiclassical formalism presented in Section (8.3.3). If we identify y of the present treatment with the factor Zgoi/< Eq. (8.96) the two results are identical for e = hco ksT. The rotating wave approximation used in the model (9.44) cannot reproduce the correct result in the opposite, classical, limit. Most studies of vibrational relaxation in molecular systems are done at temperatures considerably lower than s/ks, where both approaches predict temperature-independent relaxation. We will see in Chapter 13 that temperature-dependent rates that are often observed experimentally are associated with anhannonic interactions that often dominate molecular vibrational relaxation. 

The excitation of a molecule may result in a change of its electron and rotational-vibrational quantum numbers. In the adiabatic approximation," the total wavefunction of a molecule can be presented as a product of the electron wave and the rovibrational wavefunction. In those cases where the former is weakly affected by the changes in the relative position of the nuclei (this is usually the case with lower vibrational levels), we can use the Condon approximation considering the electron wavefunction only at equilibrium configuration of the nuclei. In this case the oscillator strength factorizes into an electron oscillator strength and the so-called Frank-Condon factor, which is the overlap integral of the vibrational wavefunctions of the initial and the final states of the molecule.115,116... 

Within the harmonic approximation for the bending motion the rotational FC factors are proportional to the square of the bending wave-function with argument (7 — je) = jft/mcj, modulated by a sinusoidal factor with wavelength Aj 7r/7e-... 

The approximation involved in factorization of the total wave function of a molecule into electronic, vibrational and rotational parts is known as the Bom-Oppenheimer approximation. Furthermore, the Schrodinger equation for the vibrational wave function (which is the only part considered here), transformed to the normal coordinates Qi (which are linear functions of the "infinitesimal displacements q yields equations of the harmonic oscillator t5q>e. For these reasons Lifson and Warshel have stressed that the force-field calculations should not be considered as classical-me-... 

Optical activity of natural products may depend on chemical factors such as asymmetric carbon atoms, restricted rotation, etc. These may be termed primary structural features. There are also secondary structures, e.g., helices or random coils, that may confer chirality to a natural product. Optical rotatory dispersion (ORD, i.e., rotation of plane-polarized radiation over a range of wave-lengths usually from approximately 200 to approximately 500/im) has been used in studies of the conformations of many different molecules, including polymers, proteins, and polypeptides [90]. 

An alternative method is based on the approximation discussed in 12-4, where was expressed as a product of rotational, vibrational and electronic factors. Analytical expressions for f - and fYib. lyQ obtamcd by the methods of wave mechanics these expressions are rather complicated, being expressed as series expansions, and will not be quoted here. 

Within the framework of the Born-Oppenheimer, binary encounter, plane wave impulse and target Kohn-Sham [KS] (or target Hartree-Fock [HF]) approximations, and disregarding rotational wave functions, differential (e, 2e) ionization cross-sections are proportional to the square of a structure factor F (p), which is given by [1-3, 64] ... 

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