Strong orthogonality

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... 

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... 

The second term of (13) can be evaluated with the help of (8) and the knowledge that the RS can only involve, at most, two-electron operators. If any VRS contains only two-electron operators and the R are constrained by the strong orthogonality condition (11) then it is obvious that only the first two terms in the expansion (8) of Ax give rise to non-zero contributions to... 

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. 

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... 

Consider the RDMs obtained from the separable wavefunction in Eq. (12). Since a and b are strongly orthogonal, it follows from Eq. (8) that ( a b a flj I a b) = 0 unless 0, and (f)j are associated with the same subsystem. Thus the 1-RDM separates into subsystem 1-RDMs,... 

The interesting scenario is when two of the four indices in this equation refer to subsystem A and the other two refer to subsystem B. Suppose, for definiteness, that (j)j e Ha and 4>k 4>i Then the strong orthogonality of a and b implies that D,y y = 0. More interesting is the case when 0,, (f)f. G Ha and G Hb. In this case Dy / is generally nonzero hence the 2-RDM mixes indices from different non-interacting subsystems, and thus fails to be additively separable. What about Ay / According to Eq. (14), D, / = 0 since i and / refer to different subsystems, and therefore Ay H = Dij-u P /- The 2-RDM part... 

The earliest such attempts go back to 1953, when strongly orthogonal antisymmetrized geminal products (SOAGP) were employed [1, 2]. A strongly orthogonal geminal is such that Jgi(l,2)g2(2,3)d2 = 0, while the weaker... 

Recently, an alternative scheme based on singlet-type strongly orthogonal geminals (SSG) was proposed [5]. In this scheme, the wavefunction is split into gem-inal subspaces depending on the number of spin-up or spin-down electrons, n and n, respectively, while the wavefunction is filled up with one Slater determinant. 

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... 

Formally, the above results depend only on strong orthogonality but separability is a valid and useful concept only when the ansatz (1) is variationally optimized. The optimization problem is considered in the next Section. 

At the end of the cycle update the full matrix T making all corrections simultaneously, noting that this will result in a slight loss of strong-orthogonality. 

Start a new cycle, restoring strong-orthogonality, as above and continue to convergence... 

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