Symmetrically orthogonalized AOs

Let us consider the simple case of the H atom and its variational approximation at the standard HF/3-21G level, for which we can follow a few of the steps in terms of corresponding density-matrix manipulations. After symmetrically orthogonalizing the two basis orbitals of the 3-21G set to obtain orthonormal basis functions A s and dA, we obtain the corresponding AO form of the density operator (i.e., the 2 x 2 matrix representation of y in the... 

To alleviate a number of these problems, Lowdin proposed that population analysis not be carried out until the AO basis functions tp were transformed into an orthonormal set of basis functions / using a symmetric orthogonalization scheme (Lowdin 1970 Cusachs and Politzer 1968)... 

Now, if the differential overlaps of the original AO basis are approximated by the formula eq. (2.28), it turns out that applying the Lowdin transformation S 2 to the set of the AOs makes the products i.e. the differential overlaps of the symmetrically orthogonal OAOs vanishing ... 

Lowdin s symmetric orthogonalization method is an often employed technique for the generation of orthonormal molecular basis sets. Since within most LCAO MO methods, density matrices are determined by AO basis set coefficients, and idempotency of density matrices is a property easily controlled on an orthonormal basis, Lowdin s transforma-... 

The density matrix can be transformed from a non-orthogonal AO to an orthogonal AO (OAO) basis by adopting the Lowdin s symmetric ortho-gonalization procedure in order to minimize the distortion of the new basis relative to the original AO. The transformation matrix ... 

All the various options influence the result of a population analysis. However, before a final assessment can be made we must address the more basic problem of a category for the population analysis. The most popular analysis is the Mulliken analysis for nonorthogonal AOs and equal partitioning of overlap between the two atoms involved. If the ZDO assumption is maintained for the Fock matrix eigenvalue problem, an orthogonalized AO basis must be considered. In such semiempirical methods the symmetric orthogonalization is most frequently implied. 

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... 

The most important feature of the SINDOl method is that an orthogonalization transformation of the basis functions is taken into account exphcitly in solving the HF LCAO equations. The one-electron integral matrix H is transformed to Lowdin symmetrically orthogonahzed [225] AOs = S / x... 

Population analysis in semiempirical methods fall into two categories. Methods including overlap in the Fock equations use the Mulliken population analysis. The majority of semiempirical methods uses the ZDO approximation, and the net charges are interpreted on the basis of symmetrically orthog-onalized AOs. It is pointed out that this interpretation is not exactly valid, because of truncation and empirical adjustment. But the corresponding nonsymmetrical orthogonalization is not uniquely defined. Charge models based on semiempirical wave functions play an important role in the calculation of molecular electrostatic potentials for reactivity. 

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