Orbital average polarization

Note that core-polarization refers here to the entire correlation effect, polarization by outer electron instantaneously at rest. This should not be confused by the orbital-average polarization, - i.e. by the spherical potential of the entire 2s orbital which vanishes if complete H.F. is carried out on Li. 

It is customary to talk about van der Waals attractions between two atoms, say He He, in the Heitler-London (H.L.) description. The attraction near the minimum is due partly to the distortion of H.L. AO s (in the orbital average polarization effect), but especially at large intemuclear distances r, it arises from simultaneous excitations in both atoms (the usual London dispersion force). The two effects have been compared for Hg in the triplet state as a function of r. 

Figure 10. Orbit-averaged total flux and circular polarization spectra of the high-field magnetic variable V884 Her from 1998 September 14. The weak level of circular polarization for a magnetic cataclysmic variable and the highly structured appearance are seen, particularly for X< 5000 A. Terrestrial absorption features affecting the spectral flux are indicated [32],...

To illustrate how well DFT or ab initio methods predict the dipole moments. Table 1 illustrates the comparison between theory and experiment for eight small molecules. The error statistics are summarized in Table 2. In general, the quality of the basis set plays an important role in the prediction of dipole moments. We see that the 6-3IG basis set provides poor predictions, even when applied with a QCISD level of theory. The performances of the double-zeta basis set plus polarization functions (6-3IG, DZVPD (double-zeta valence orbitals plus polarization and diffuse functions on heavy atoms), and cc-pVDZ (correlation-consistent polarized valence double-zeta)) are poorer than those from the polarized triple-zeta basis sets. The only exception is B-P/DZVPD (B-P = Becke-Perdew), from which we obtained an average absolute deviation of 0.040 debye, lower than that (0.053 debye) from B-P/TZVPD (triple-zeta valence orbitals plus polarization and diffuse functions on heavy atoms). It can be seen that the inclusion of correlation effects through either ab initio or DFT approaches significantly improves the agreement. 

The 59Co hyperfine matrix components must have identical signs in order that the average values match the observed isotropic couplings we assume the signs are negative since the isotropic couplings almost certainly arise from polarization of inner shell s orbitals (see below). 

The relative magnitudes of the resulting average mixing coefficients Ay and Ay per and Bi average AO, respectively, give a measure of the polarity of the Bi — Bi MOs. In simple cases (say, HC1) the concepts of orbital polarity and electron polarity (as a measure of ionicity) coincide. The extension of the latter inference to more complicated molecular systems is far from trivial, although such a conclusion is commonly accepted for qualitative purposes. 

The adequacy of the spin-averaged approach has been confirmed in self-consistent spin-density-functional calculations for H in Si by Van de Walle et al. (1989). The deviation from the spin-averaged results is expected to be largest for H at the tetrahedral interstitial (T) site, where the crystal charge density reaches its lowest value. For neutral H at the T site, it was found that inclusion of spin polarization lowered the total energy of the defect only by 0.1 eV. The defect level was split into a spin-up and a spin-down level, which were separated by 0.4 eV. These results are consistent with spin-polarized linearized-muffin-tin-orbital (LMTO) Green s-function calculations (Beeler, 1986). 

Recent work improved earlier results and considered the effects of electron correlation and vibrational averaging [278], Especially the effects of intra-atomic correlation, which were seen to be significant for rare-gas pairs, have been studied for H2-He pairs and compared with interatomic electron correlation the contributions due to intra- and interatomic correlation are of opposite sign. Localized SCF orbitals were used again to reduce the basis set superposition error. Special care was taken to assure that the supermolecular wavefunctions separate correctly for R —> oo into a product of correlated H2 wavefunctions, and a correlated as well as polarized He wavefunction. At the Cl level, all atomic and molecular properties (polarizability, quadrupole moment) were found to be in agreement with the accurate values to within 1%. Various extensions of the basis set have resulted in variations of the induced dipole moment of less than 1% [279], Table 4.5 shows the computed dipole components, px, pz, as functions of separation, R, orientation (0°, 90°, 45° relative to the internuclear axis), and three vibrational spacings r, in 10-6 a.u. of dipole strength [279]. 

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