Symmetric orthonormalization

There are numerous ways in which to orthonormalize a basis. Here we choose to employ the symmetric orthonormalization procedure described by Lowdin (51), which has the benefit over other orthogonality procedures that the new basis is as close as possible, in a least squares sense, to the original basis (52)... 

The spin-orbitals 17,) iGA + f form a basis set for the supersystem A —B. One possible procedure of realizing the transformation (218) as well as (219) is the Lowdin symmetric orthonormalization method137. ... 

In a fairly early discussion of solids Wannier[38] showed how linear combinations of the AOs could be made that rendered the functions orthogonal while retaining a relatively large concentration on one center. In more modern language we would now say that he used a symmetric orthonormalization of the AO basis. If we symbolize the overlap matrix for the AO basis by S, then any matrix N that satisfies... 

Set N = S 1A. This gives the symmetric orthonormalization, so-called because this AT is a symmetric matrix for real basis functions. 

We may demonstrate this difficulty by giving a result due to Slater. [39] Applying a symmetric orthonormalization to the basis normally used in the Heitler-London calculation we have a His function on each of two centers,... 

We add another comment about this example and note that using symmetric orthonormalization on the simple two AO basis for the triplet state of H2 gives the same answer as that obtained with unmodified orbitals. 

Figure 3.17 Symmetric orthonormalization of the sto-3g) basis sets for hydrogen Is and 2s radial functions and the comparisons with the exact radial functions from solution of the Schrodinger equation.

One further step is required to reach the PPP-model, and that is the approximation of linear combinations of atomic orbitals. The field operators are expanded in a set of symmetrically orthonormalized atomic spin orbitals of a" symmetry. Furthermore, each atomic orbital is associated with a separate atomic center r in the molecule. We write... 

In this section, we examine in some detail the case when the off-diagonal ma.-trix elements of the core operator 0rs are small in comparison with the electron repulsion integrals jrr- This corresponds to the situation of separated atoms, but it should be realized that other matrix elements also depend on the relative positions of the nuclei, both through the core operator and the symmetric orthonormalization procedure of the orbitals used in the expansion Eq. (11.10). In the following, the constant term of H PPP) is omitted, the third term is considered as a perturbation, while the second and fourth parts constitute the unperturbed hamiltonian for this case. 

A straightforward extension of the Hg model of Jankowski et al. [60] (see Sect. 3.3) is displacement of all four H2 molecules [64]. In particular to study size extensivity, all H2 molecules are symmetrically displaced to infinity or as a practical matter in this work to the D4 , dissociation geometry of a = 994ao. A C localized orbital (LO) set computed from the symmetrically orthonormalized atomic orbitals [18] is employed. This set corresponds to the orthonormal MOs that are closest in the least squares sense to the atomic orbitals. Thus, these molecular calculations are performed without a preliminary SCF or MCSCF orbital optimization step. The order of levels in the Shavitt... 

Due to this projection, the two eigenvectors contained in (5.71) are not strictly orthonormal, but can be made so by means of Lowdin symmetric orthonormalization... 

These transformations are in fact nothing more than the application of a symmetric orthonormalization followed by a transformation to the new metric. 

The creation operators aj, for nonorthogonal spin orbitals are defined in the same way as for orthonormal spin orbitals (1.2.5). As for orthonormal spin orbitals, the anticommutation relations of the creation operators and the properties of their Hermitian adjoints (the annihilation operators) may be deduced from the definition of the creation operators and from the inner product (1.9.2). However, it is easier to proceed in the following manner. We introduce an auxiliary set of symmetrically orthonormalized spin orbitals... 

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