Electronic similarity matrices

The aja, operator tests whether orbital i exists in the wave function, if that is the case, a one-electron orbital matrix element is generated, and similarly for the two-electron terms. Using the Hamiltonian in eq. (C.6) with the wave function in eq. (C.4) generates the first quantized operator in eq. (C.3). 

The two-electron density matrix elements are given in similar fashion ... 

It turned out, however, that for the d-shells the recipe [92] of constructing the two-electron density matrix does not work for a major part of the atomic electronic terms of the transition metal ions [93]. Further studies revealed that constructs similar to [92] are, nevertheless, possible also for some other terms for which the name of the non-Roothaan terms [94,95] was coined not very conveniently (the point is that the Roothaan and non-Roothaan terms together do not exhaust the entire set of terms). The Roothaan and non-Roothaan terms together are those where it is possible... 

Finally, it should be noted that treating one of the otherwise equivalent electrons in equ. (7.46) individually is frequently used for calculating matrix elements with a one-electron operator (e.g., the photon operator) acting on equivalent electrons. Similarly, if two-electron operators play a role, like in the Coulomb interaction between electrons, then it is convenient to separate two electrons from the equivalent electrons. This is done using the coefficients of fractional grandparentage (for more details see [Cow81]). 

From Eq. [239], it is apparent that the size of a particular is not only determined by the magnitude of the electronic coupling matrix element but also by the overlap of the vibrational wave functions v,- and i/. Squared overlap integrals of the type (Xi/, (Q) IXt/ (Q))q 2 are frequently called Franck-Con-don (FC) factors. In contrast to radiative processes, FC factors for nonradiative transitions become particularly unfavorable if two states differing considerably in their electronic energies exhibit similar shapes and equilibrium coordinates of their potential curves. Due to the near-degeneracy requirement, an upper state vibrational wave function, with just a few nodes... 

The encounter complexes exhibit high degrees of charge-transfer [20, 91], and on the basis of absorption and emission data electronic coupling matrix elements for similar complexes (exciplexes) have been determined [205] which are comparable to those of mixed-valence metal complexes commonly used as prototypical models for the bridged-activated complex in inner-sphere electron transfers [2, 26, 197]. Accordingly, we ascribe the unusually high rate constants, their temperature-independence, and their on-Marcus behavior to an inner-sphere electron transfer process [31]. 

This equation represents the general solution of the non-orthogonality problem , since the construction of the matrix G " (A) does not involve N operations. All the other density matrices needed in the evaluation of the matrix of the Hamiltonian (36) may be derived similarly. Thus, for example, the required one-electron density matrix elements are given by... 

Even if other methodologies may provide better results, it must be stated that the methodology presented in this work, and that includes descriptor generation, similarity matrix transformation, and statistical procedure, has not been altered in any way to take into account the nature of the studied system. In this way, the exposed QSAR protocol is potentially capable of handling and characterizing different molecular biological activities from diverse molecular sets without introducing further information than those provided by quantum similarity, which is based on electronic... 

The bipyratnidal carbon cluster 7 is electronically similar to 6 here however the two apical carbons are approximately sp rather than sp, with the two apical p atomic orbitals probably forming a long C-C bond. This molecule is likely to be particularly reactive and may be hmited to matrix isolation studies. Also like 6, molecule 7 may be preparable from a substituted pyramidane (via an intermediate disubstituted derivative of 6). 

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