Strongly orthogonal orbitals

In this section we examine this orthogonality constraint in order to evaluate its consequences for a theory of valence. Is it a substantive formal constraint on the type of model we may use does it restrict the type of physical phenomenon we can describe or is it simply a technical constraint on the method of calculation or what In fact we shall find that the strong orthogonality constraint is central to any orbital basis theory of molecular electronic structure. It has a bearing on the applicability of the model approximations we use, on the validity of most numerical approximations used within these models and (apart from the simplest MO model) has a dominant effect on the technical feasibility of the methods of solution of the equations generated by our models. Thus, it is of some importance to try to separate these various effects and attempt to evaluate them individually. 

In practice it is most common and convenient to side-step one of the problems associated with strong orthogonality. We can work with an orbital basis which satisfies (11) independently of the choice of the physical structure of the groups of electrons a basis for which (11) is guaranteed for all R, S. Any orbital basis which forms an orthogonal set will fulfill this condition all overlap integrals are then zero.6)... 

Because subsystems A and B do not interact, it must be that T a consists of a determinantal expansion in functions taken solely from the set Ha, and similarly uses only those spin orbitals in Br. It follows that T a and are strongly orthogonal [53]. Two antisymmetric functions f x, ..., Xp) and g yi,..., yg) are said to be strongly orthogonal if... 

The second simplification, which is introduced for computational convenience, is the strong orthogonality (SO) constraint, by which all the orbitals in... 

This strong orthogonality constraint, while seemingly a restriction, is usually not a serious one, since it applies to orbitals that are not expected to overlap significantly. On the other hand, the orbitals (

perfectly paired GVB wave function generated under the constraint of zero-overlap between the orbitals of different pairs. 

Switching to a basis of orbitals allows us to proceed toward constructing the description of a molecular electronic structure based on the GF approximation under well acceptable restrictions. Indeed, let us assume that the multipliers in eq. (1.181) satisfy the additional condition of strong orthogonality ... 

The most common approach is to note that since the pair functions vv/ may not contain any interparticle co-ordinates (as then the strong orthogonality requirement cannot be sustained), they may be expressed as bilinear expansions in one-electron orbitals ... 

The further assumption that the orbitals are orthogonal except for those which are singlet paired (the strong orthogonality approximation), leads to the strongly orthogonal, perfect pairing (SOPP) wave function... 

Triplet-triplet, or T-T energy transfer is a transition of the type ( D A) — A ). It is overall spin allowed, however a coupling such as ( D MD I D> is zero because of the orthogonality of the spin states. The coupling that promotes such a transition amounts to simultaneous exchange of two electrons in opposite directions between the donor and the acceptor molecules, and depends strongly on orbital overlap between the two molecules. 

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