Delocalized excitation wave function

To make excited state functions with correct symmetry it is necessary to take linear combinations of the localized excitation functions, and these must transform as do the representations of the space group of the crystal. We arrive in this way at delocalized excitation wave functions, Eq. (3),... 

Semiconductor materials are rather unique and exceptional substances (see Semiconductors). The entire semiconductor crystal is one giant covalent molecule. In benzene molecules, the electron wave functions that describe probabiUty density ate spread over the six ting-carbon atoms in a large dye molecule, an electron might be delocalized over a series of rings, but in semiconductors, the electron wave-functions are delocalized, in principle, over an entire macroscopic crystal. Because of the size of these wave functions, no single atom can have much effect on the electron energies, ie, the electronic excitations in semiconductors are delocalized. 

Only for a special class of compound with appropriate planar symmetry is it possible to distinguish between (a) electrons, associated with atomic cores and (7r) electrons delocalized over the molecular surface. The Hiickel approximation is allowed for this limited class only. Since a — 7r separation is nowhere perfect and always somewhat artificial, there is the temptation to extend the Hiickel method also to situations where more pronounced a — ix interaction is expected. It is immediately obvious that a different partitioning would be required for such an extension. The standard HMO partitioning that operates on symmetry grounds, treats only the 7r-electrons quantum mechanically and all a-electrons as part of the classical molecular frame. The alternative is an arbitrary distinction between valence electrons and atomic cores. Schemes have been devised [98, 99] to handle situations where the molecular valence shell consists of either a + n or only a electrons. In either case, the partitioning introduces extra complications. The mathematics of the situation [100] dictates that any abstraction produce disjoint sectors, of which no more than one may be non-classical. In view if the BO approximation already invoked, only the valence sector could be quantum mechanical9. In this case the classical remainder is a set of atomic cores in some unspecified excited state, called the valence state. One complication that arises is that wave functions of the valence electrons depend parametrically on the valence state. 

Ionic The function, in each case, with the next highest weight, 3 %, is ionic and involves a single excitation into the 2s AO. This contributes to adjusting the electron correlation and also contributes to adjusting the size of the wave function along the lines of the scale adjustment of the Weinbaum treatment. As we have shown, it also contributes to delocalization. 

The admixture of singly-excited configurations O, to Og to represent the HF wave function indicates electron transfer from bonding to antibonding orbitals and, accordingly, can be interpreted in terms of electron delocalization, which can be measured by the ratio CjlCG. Alternatively, one can define interbond populations by equation 1647 80 ... 

Several methods exist for calculating g values. The use of crystal field wave functions and the standard second order perturbation expressions (22) gives g = 3.665, g = 2.220 and g = 2.116 in contrast to the experimentaf values (at C-band resolution) of g = 2.226 and g 2.053. One possible reason for the d screpancy if the use of jperfXirbation theory where the lowest excited state is only 5000 cm aboye the ground state and the spin-orbit coupling constant is -828 cm. A complete calculation which simultaneously diagonalizes spin orbit and crystal field matrix elements corrects for this source of error, but still gives g 3.473, g = 2.195 and g = 2.125. Clearly, covalent delocalization must also be taken into account. 

The rates km cover the km o accounting for the excited state decay of chromophore m (by radiative as well as non-radiative transitions) and the SC ) originated by inter-system crossing to triplet states (ISC rate). The simple km do not include the effect of excited state wave function delocalization and a possible decay out of exciton states [45], Therefore, we shortly demonstrate the computation of the photon emission part of the km including such a delocalization effect (determination of excitonic augment rates). It will be important for the mixed quantum classical simulations discussed in the following (for more details see also [11]). 

In the MO theory, the most reliable approach for the study of reaction pathways perhaps is CASSCF [12, 13], but multi-VBSCF is essentially at the same level with CASSCF [14]. Since a VB wave function can be expanded into the combination of numerous Slater determinants that are used to define configurations in the MO theory, the VB theory provides a very compact, accurate description for chemical reactions. While both MO and VB theories have their own advantages as well as disadvantages, in our opinions, there are some areas where the VB theory is particularly superior to the MO theory 1) the refinement of molecular mechanics force field 2) the development of empirical or semi-empirical VB approaches 3) the impact of intermolecular charge transfer or intramolecular electron delocalization on the structure and properties 4) the validation of classical chemical theories and concepts at the quantitative level 5) the elucidation of chemical reactions and excited states intuitively. 

There are at least three types of cluster expansions, perhaps the most conventional simply being based on an ordinary MO-based SCF solution, on a full space entailing both covalent and ionic structures. Though the wave-function has delocalized orbitals, the expansion is profitably made in a localized framework, at least if treating one of the VB models or one of the Hubbard/PPP models near the VB limit -and really such is the point of the so-called Gutzwiller Ansatz [52], The problem of matrix element evaluation for extended systems turns out to be somewhat challenging with many different ideas for their treatment [53], and a neat systematic approach is via Cizek s [54] coupled-cluster technique, which now has been quite successfully used making use [55] of the localized representation for the excitations. 

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