Strong orthogonality constraint

We now substitute (9) and (10) into (1) and (2). It is at once obvious that, unless we impose the so-called strong orthogonality constraint... 

In the first section of this work, in order to obtain maximum simplicity of interpretation, we chose to impose the strong orthogonality constraint on our model wave functions any two separate-group functions will be constrained by Eq. (11) ... 

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. 

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. 

In general, the so-called strong orthogonality constraint on two separate-pair wavefunctions is... 

Typically, GVB-PP calculations are carried out within an orthogonal orbital basis, which is called the strong orthogonality constraint. This second simplification ensures that all orbitals are orthogonal to each other unless they are singlet paired. However, the strong orthogonality constraint was simply introduced for... 

Extensive introductions to the effective core potential method may be found in Ref. [8-19]. The theoretical foundation of ECP is the so-called Phillips-Kleinman transformation proposed in 1959 [20] and later generalized by Weeks and Rice [21]. In this method, for each valence orbital (pv there is a pseudo-valence orbital Xv that contains components from the core orbitals and the strong orthogonality constraint is realized by applying the projection operator on both the valence hamiltonian and pseudo-valence wave function (pseudo-valence orbitals). In the generalized Phillips-Kleinman formalism [21], the effect of the projection operator can be absorbed in the valence Pock operator and the core-valence interaction (Coulomb and exchange) plus the effect of the projection operator forms the core potential in ECP method. 

This point is at the origin of one of the difficulties one meets when the strong orthogonality constraint is relaxed. As mentioned above, the non-vanishing overlap of the one-electron functions belonging to different groups... 

The second simplification, which is introduced for computational convenience, is the strong orthogonality constraint, whereby all the orbitals in Eq. [89] are required to be orthogonal to each other unless they are singlet... 

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