Orthogonality, orbital constraints

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

So far we have considered the more usual orthogonal orbital type wave-functions in which the constraints are those of orbital orthogonality. However, for wavefunctions in which orbital orthogonality is not required (or for more general wavefunctions) the above discussion need not apply since in these cases it is possible to use an unconstrained minimization method directly on the functional... 

At this stage the HF method is unrestricted since no peculiar relation exists between the spin orbitals except for their orthogonality. However, constraints may be introduced (the HF scheme becoming therefore of a restricted type) these are of two basic types > ... 

Once an exchange-correlation functional has been selected, the computational problem is very similar to that encountered in wave mechanics HF theory determine a set of orthogonal orbitals that minimizes the energy. Since the 7[p] (and xc[p]) functional depends on the total density, a determination of the orbitals involves an iterative sequence. The orbital orthogonality constraint may be enforced by the Lagrange method (Section 12.5), again in complete analogy with wave mechanics HF methods (eq. (3.34)). 

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... 

Such constraints imply the non-orthogonality of the orbitals. The optimal virtual orbitals A and are determined accordingly to the approximation that they separately maximize the dispersion contribution of each of the NA NB two configuration wave functions... 

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 is not due, however, to the orthogonality constraint between 4 f and 5 f orbitals, contrary to a general belief (private communication from J. P. Desclaux)... 

Before deriving the Hartree-Fock equations for the excited state orbitals, we consider the orthogonality constraints imposed on these orbitals. 

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