Balance equations quantum mechanical treatment

We have so far limited ourselves to a classical description, the natural requirement for which is the condition /, / — oo. In order that the description is valid for any angular momentum value, it is necessary to employ the quantum mechanical approach. We presume that the reader is acquainted with the density matrix (or the statistic operator) introduced into quantum mechanics for finding the mean values of the observables averaged over the particle ensemble. Under the conditions and symmetry of excitation considered here one must simply pass from the prob- 

We will now introduce the following notation. Let the ground (initial) state of the molecule with rotational quantum number J be characterized by the ground state density matrix (reserving notation /mm for the excited state), where the indices // cover the set M of magnetic quantum numbers. The quantum mechanical equivalent of (3.4) for the situations presented in Figs. 2.5 and 2.6 is 

It follows that on switching to a vector model of the quantum angular momentum we obtain G(9, p) = (Jz/ J )2 = cos2 0. 

The population of the magnetic sublevel M = p emerges as a stationary solution of Eq. (3.9) and equals 

The expressions for pm in the case of linear or circular polarized P- or 72-absorption may also be easily obtained by substitution of the respective actbp (presented in the Table C.2 of Appendix C) into (3.11) (the explicit form of (C b/3)2 can be found in Table 3.6 as coefficients multiplied by x). The cases of linear and circular polarized P-absorption are demonstrated in Fig. 3.6(6) and 3.6(c). 

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The detailed-balance corrected quantum-classical method described in section 8.2 is capable of giving this surface temperature dependence [62]. More rigorous methods involve the solution of the Liouville-von Neumann equation [159]. However, this appears to be close to impossible for systems involving the number of atoms and processes of interest for molecule-surface scattering [160]. At least this is so unless approximations are introduced, for instance, combining classical and quantum mechanical treatments or self-consistent field approximations [161]. 

III.D) where the solute is treated quantum mechanically and the solvent molecules classically [186-197]. The second approach [185] may be implemented in an entirely classical framework (e.g., through the solution of the Poisson equation or the introduction of the generalized Born model in molecular mechanics) or in a quantum mechanical framework where the wavefunction of the solute is optimized self-con-sistently in the presence of the reaction field which represents the mutual polarization of the solute and the bulk solvent. Due to the complexity of solvation phenomena, both approaches contain a number of severe approximations, and if a quantum chemical description is employed at all, it is usually restricted to the solute molecule. When choosing such a quantum chemical description from the usual alternatives ab initio, DFT, or semiempirical methods) it should be kept in mind that ab initio or DFT calculations may provide an accuracy that is far beyond the overall accuracy of the underlying solvation model. For a balanced treatment it may be attractive to employ efficient semiempirical methods provided that they capture the essential physics of solvation. The performance and predictive power of such semiempirical solvation models may then be improved further by a specific parametrization. 

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