Orthogonalisation

H- analyt-gauss two-electron atomic system, importance of the orthogonalisation 22,23... 

The theory described in the previous section is now applied to beryllium metal. Accurate low temperature data was taken from the paper of Larsen and Hansen [20]. (But note that in (20) I used the structure factors multiplied by 1000, as given in then-paper.) For the orthogonalisation, all nearest neighbours we included within the first shell. There were 12 atoms. A triple zeta basis set from Ref. [21] was used. There are 182 basis functions and 361 independent parameters in the wave function, whereas there are 58 experimental measurements. Figure 1 shows a plot of the x2 agreement statistic as a function of the parameter X for k = 0.2. Larger values of k caused... 

In the last section we have constantly referred to an orthogonalised basis and yet made no specific proposals for the generation of such a basis. We now take up this matter. If a matrix V transforms the overlap matrix S to unity in the equation... 

V = S-1/2 is also the one which is optimum in the sense of generating the closest orthogonal basis possible to the original set. We therefore concentrate on finding optimum non-orthogonal bases and then apply the Lowdin orthogonalisation technique. 

The use of the Lbwdin orthogonalisation technique (or any other method of or-thogonalisation) means inevitably that the final basis of orthogonalised hybrid atomic orbitals (OHAOs) does contain many-centre orbitals in the sense that each OHAO is mainly its HAO parent but necessarily contains (minimal) contributions from overlapping HAOs. 

Thus the differential overlap factor 02 for the orthogonalised GHOs is... 

The pairwise overlap, symmetrical orthogonalisation and Mulliken approximation together validate the NDO approximation — the orbital product in the orthogonalised GHO basis vanishes to the extent that the Mulliken approximation is realistic. This conclusion obviously has enormous consequences for any NDO approximation schemes. 

This analysis therefore provides some theoretical justification for the rather empirical scaling rules which have been used to simulate the effect of orthogonalisation in earlier work (5,24). 

Fig. 2. The energies of the states relative to the JT energy with L = odd (due to high spin term) and L = even (due to low spin terms) with one phonon excitation of nr-symmetry, neglecting orthogonalisation to the states with no phonon excitations. See text for details.

Fig. 3. As Fig. 2 but including orthogonalisation to the states with no phonon excitations. 

ODENDOR ODLTS ODMR OICTS OLCAO OMVPE OSC optically detected electron nuclear double resonance optical deep level transient spectroscopy optically detected magnetic resonance optical isothermal capacitance transient spectroscopy orthogonalised linear combination of atomic orbitals organo-metallic vapour phase epitaxy on-surface-cracking... 

R. Sureshkumar and A.N. Beris, Linear stability of viscoelastic Poiseuille flow using an Arnoldi based orthogonalisation algorithm, J. Non-Newtonian Fluid Mech., 56 (1995) 151-182. 

The n=l and 2 shells were frozen at the ground-state Hartree—Fock values. The orbital set included the 4s,3p,3d,4f and 5g natural orbitals and 3p,3d,4s,4p,4d,4f,5s,5p,5d,5f orbitals from frozen-core Hartree—Fock calculations to provide representations for states whose dominant configuration is 13s n/). This set was again augmented by extra ad hoc orbitals to increase flexibility. The full set contained 24 orbitals (6 s-type, 7 p-type, 6 d-type, 3 /-type, 2 g-type) which were all orthogonalised using the prescription for two orbitals a) and b)... 

For continuum channels the radial orbitals in (7.140) are not bounded and the integral is divergent. The choice of radial functions in (7.136) must be based on intuition obtained from a study of ionisation, which is treated in chapter 10. A necessary condition for a reasonable distorted wave in the distorted-wave Bom approximation for ionisation is that it should be orthogonal to the initial state in the ionisation amplitude. For computational simplicity we set Fop,7l equal to zero and orthogonalise the resulting Ricatti—Bessel functions to all the states of P space using (5.83). 

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