Rotating atom approximation

Ref. 144.) A perturbed rotating atom approximation is employed in the evaluation of the magnitudes of the dipole-dipole interaction (6 = 0 see Eq. (20)). Theoretical values by Katsuura at 500 meV employing rectilinear trajectories are also indicated. (From Ref. 143.)... 

To obtain a rough estimate of the excitation transfer cross section, consider first the adiabatic approximation, neglecting the Coriolis coupling (the so-called rotating-atom approximation ). The transfer is then associated with mere interference between gerade and ungerade states. As an example, the 2 time-dependent wave function can be written as... 

This assumption is acceptable for the near-resonant case (18, 19, 46, 47). Together with this assumption we assume that the direction of P-state points to the other atom throughout collision (the rotating atom approximation). The rotating atom approximation has been employed and discussed by Bates et al. (6,7,8) in some collision problems. In our excitation transfer problem this approximation would also be available. 

Judging from the calculation of the coupled equation (without the rotating atom approximation) in the resonant case and the discrete-continuum case, the error from the rotating atom approximation would not change the order of the magnitude (67, 68, 70). With these approximations, the process can be described in terms of the resonance defect [o> = (Et — Ef)/h] and the transition matrix dipole moments of A and B fiA and /xB. By Nakamura s calculation (50), the transfer probability is written as... 

Muller et al. focused on polybead molecules in the united atom approximation as a test system these are chains formed by spherical methylene beads connected by rigid bonds of length 1.53 A. The angle between successive bonds of a chain is also fixed at 112°. The torsion angles around the chain backbone are restricted to three rotational isomeric states, the trans (t) and gauche states (g+ and g ). The three-fold torsional potential energy function introduced [142] in a study of butane was used to calculate the RIS correlation matrix. Second order interactions , reflected in the so-called pentane effect, which almost excludes the consecutive combination of g+g- states (and vice-versa) are taken into account. In analogy to the polyethylene molecule, a standard RIS-model [143] was used to account for the pentane effect. 

It should be noted that result given by equation (91) coincides with that calculated under the assumption that the orientation of atomic orbitals remains fixed during collision ( fixed-atom approximation ) [84, 87]. This is to be expected, because the strong Coriolis mixing of molecular functions in the rotating frame prevents rotation of orbitals in a space fixed frame. 

Four-atom reactions came into focus with the development by Clary of the Rotating Bond Approximation (RBA)[10. 11] and Bowman s reduced-dimensionality adiabatic bend (RD-AB) calculations of four-atom reactions. In the latter three stretching vibrational motions are treated explicitly quantum dynamically while the bending degrees of freedom are treated adiabatically and one diatom is assumed to be a spectator[12, 13, 14, 15, 16], The RBA may be seen as an extension... 

We consider a two-level atom with excited and ground states e) and g) when in a photonic crystal coupled to the field of a discrete (or defect) mode and to the photonic band structure in the vacuum state. The hamiltonian of the system in the rotating-wave approximation assumes the form [Kofman 1994]... 

Manipulation of Trapped Atomic Ions Hamiltonian in the rotating-wave approximation is given by... 

The Ewing model is based on a three-atom approximation of the dimer. As such it is best applied to dimers in which the initial excitation is in the bond that is part of the van der Waals bond. Hence, in the previously cited example of HCN—HCCH, the model is much more likely to succeed in the case of the linear isomer in which the acetylenic C—H stretch has been excited rather than the C—H stretch on the HCN molecule and to the T-shaped isomer in which the C—H stretch in the HCN molecule has been excited. Suppose that the C—H stretch in HCN at ca. 3300 cm" is excited in the T-shaped isomer. If the 2045 mode in the HCN product is excited during the vibrational predissociation, then just 500 cm" remains in translation. Thus we obtain a A x = 5.5 and predict a lifetime of about 1 nsec, which is precisely the measured lifetime of 1.3 nsec. However, it is evident that as the complexity of the molecules increases, there will be numerous combinations of product vibrations and rotations which could be excited. Which one is dominant is difficult to determine a priori. 

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