Summation system elements

In general, the complete system frequency response is obtained by summation of the log modulus of the system elements, and also summation of the phase of the system elements. 

TheMP2 wave functions we have introduced in a preceding subsection are just the application of this method to another problem, that of the electronic correlation. In this case, the simpler unperturbed system is the HF approximation, the corrections are limited to the second order, and the corresponding contributions to the energy are expressed as a simple summation of elements. The MP2 method is currently used in the PT description of the intermolecular potential in such cases, two different applications of PT are used at the same time. 

The interpretation of the density (and efficiency) matrices is rather obvious the diagonal elements represent the probabilities of finding a certain magnetic quantum number of the ensemble, and the non-diagonal elements contain the phase information of the system for the corresponding different magnetic quantum numbers. However, in calculations with these matrices one still has to work out the cumbersome summations over these quantum numbers. It is more convenient to replace the density (and efficiency) matrices by statistical and efficiency tensors, also called state multipoles. 

The unitary transformation from the basis of the CMOs to the basis of the LMOs of the /-system does not change the covalent contribution to the effective crystal field. According to numerical estimates the resonance integrals (,/. between d-AO and LPs of the donor atoms by I0-H00 times overcomes the resonance integrals between d- AO and any other LMOs and thus dominates the resonance interaction of the d- and /-systems. So, as it has been shown in [71], restricting the summation in eq. (4.82) by the sum of diagonal elements (L = L) over only the LPs results in error in the estimated splitting of the (/-levels of 0.1 eV. This precision is comparable to that of the EHCF method itself. This estimate is described by the formula ... 

Also in response theory the summation over excited states is effectively replaced by solving a system of linear equations. Spin-orbit matrix elements are obtained from linear response functions, whereas quadratic response functions can most elegantly be utilized to compute spin-forbidden radiative transition probabilities. We refrain from going into details here, because an excellent review on this subject has been published by Agren et al.118 While these authors focus on response theory and its application in the framework of Cl and multiconfiguration self-consistent field (MCSCF) procedures, an analogous scheme using coupled-cluster electronic structure methods was presented lately by Christiansen et al.124... 

The second term in Eq. (12) has the form characteristic for atomic polarization, the summation in this term being performed over the vibronic states of the ground electronic term. The occurrence of this term is due to the Jahn-Teller effect, since in the absence of the effect the matrix elements < o o- ne) are identically equal to zero. Its contribution to the atomic polarization is determined by the magnitude of vibronic interaction and, generally speaking, it is not small. Visually, the vibronic contribution to the atomic polarization can be explained by the increase of the mobility and hence the polarizability of the vibrational system due to the Jahn-Teller effect. 

In considering systems where there are very many Maxwell elements employed in the model, that is, z in equation (3-34) is large, it is often convenient to replace the summation in the equation by an integration. Thus ... 

Although the analysis in terms of the propagators for independent motion gL is convenient for displaying the content of the kinetic theory expression for the rate kernel, calculations based on (10.4), which contains the propagator for the correlated motion of the AB pair, are probably more convenient to carry out. In kinetic theory, such rate kernel expressions are usually evaluated by projections onto basis functions in velocity space. (We carry out such a calculation in Section X.B). Hence the problem reduces to calculation of matrix elements of (coupled AB motion in a nonreactive system) and subsequent summation of the series. This emphasizes the point that a knowledge of the correlated motion of a pair of molecules for short distance and time scales is crucial for an understanding of the dynamic processes that contribute to the rate kernel. 

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