Strong orthogonal functions

Miere the superscripts indicate the zero- and single-interdiange terms, evaluated in the usual way from the density functions in (14.5.9), while 2 ab stands for the second-order sum in (14.5.3). Since the zero-interchange terms give the same densities as for strong-orthogonal functions (minus the exchange part) yields a first approximation... 

To build up vin the cluster function (1) we use the functions (PvA vA2---fvBi 9vs2 -- all of which satisfy the strong orthogonality condition in the sense of to (2), but do not satisfy the strong orthogonality needed for (1) We therefore consider the linear combination... 

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) ... 

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 only approximation to be admitted at this stage will be that inherent in the separability anaatz (1) with the constraint of strong orthogonality. In this case there is a corresponding separability of the density functions, embodied in two theorems [1,2] for a separable system, comprising subsystems A, B,. . R,., , the one-body density matrix (spin included) takes the form... 

However, this is only possible if the core and valence functions are strongly orthogonal, i.e. 

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. 

Then the separability theorem [48,49] states that the strong orthogonality is equivalent to dividing the complete space of one-electron functions into mutually orthogonal subspaces ... 

In this case the spin and symmetry of the function eq. (2.104) coincide with the spin and symmetry of the wave functions of the ( /-system 4>)(. An assumption that the functions dn and1 satisfy the strong orthogonality condition of eq. (1.185) together with the variational principle yields a pair of the coupled equations for the functional multipliers ... 

The use of the Serber basis for spin functions combined with the strong orthogonality condition (100) ensures that separated pair functions (98)... 

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... 

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