Lowdin’s symmetric orthogonalization

To restore orthogonality, while preserving as far as possible the desirable features of hi and h2, we carry out Lowdin s symmetric orthogonalization procedure12... 

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-... 

U. As U is neither normalized nor orthogonal, we introduce an overlap matrix S defined in Equation 15 and use Lowdin s symmetric orthogonalization scheme to... 

The formal properties of operator L eq 2.18 (known as the symplectic structure ) allow the introduction of a variational principle eq D3, " a scalar product (eq Bl), and ultimately to reduce the original non-Hermitian eigenvalue problem (eq 2.18) to the equivalent Hermitian problem which may be solved using standard numerical algorithms (Appendices B—E). For example, F is a Hermitian operator. Lowdin s symmetric orthogonalization procedure " " leads to the Hermitian eigenvalue problem as well (eq E5), which may be subsequently solved by Davidson s algorithm (Appendix E). The spectral transform Lanczos method developed by Ruhe and Ericsson is another example of such transformation. 

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 ... 

In addition Kondo and Yamashita (22), and others (15,17,18) have used Lowdin s work by applying the symmetrical orthogonalization technique to the chemical shifts in the alkali halides. All of these results agree remarkably well with the experimental data. 

Orthogonalize (using Lowdin (symmetric) orthogonalization) the following Is (core), 2s (valence), and 3s (Rydberg) STO s for the Li atom given ... 

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 ... 

Fig. 3.2. The symmetric (or Lowdin s) orthogonalization of the normalized vectors u and v. The vectors are pushed off by the same angle in such a way as to assure u and v to become orthogonal.

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... 

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