Atomic wavefunctions, effect

Peterkop, R. and Rabik, L. (1971). Effect of inaccuracy of atomic wavefunction in variational calculations of scattering length. J. Phys. B At. Mol. Phys. 4 1440-1449. 

Lo et a/,102 have calculated spin-orbit coupling constants for first- and second-row atoms and for the first transition series, results agreeing with the work of Blume and Watson. Karayanis103 has extended the calculation to triply ionized rare earths. However, with very heavy atoms relativistic effects on the part of the wavefunction near the nucleus become severe, leading to a breakdown of the conditions under which simple perturbation theory ought to be applied. Lewis and co-workers104 have used relativistic self-consistent Dirac-Slater and Dirac-Fock wavefunctions to evaluate spin orbit coupling... 

The main effect is the so-called direct relativistic effect, a strong decrease of the radial distribution of the atomic wavefunctions with an angular momentum 1/2,... 

The second insight that can be obtained from the electronic band structure is the mobility of charge carriers, which is related to the width of the conduction and valence bands. For Si the bands are rather broad, spanning more than 10 eV. This is a direct consequence of the extensive overlap of the sp orbitals on neighboring atoms. More overlap between atomic wavefunctions results in broader bands and easier transport of free charge carriers through the material. This can be quantified via the curvature of the individual bands, which is directly related to the effective mass and mobility of the charge carriers ... 

In the continuum limit a K) —> 0) eqn (E.3) is identical to the effective one-dimensional equation for the radial part of the three-dimensional hydrogen atom wavefunction, u r) = r (r), for the case of zero angular momentum, where 4> r) is the radial wavefunction (see Cohen-Tannoudji et al. (1977, p. 792)). This equation was studied in detail by Loudon (1959). It is useful to treat the even and odd parity solutions separately. 

Thus, we have a linear variation procedure that, in effect, allows for AO expansion and contraction. It is akin to optimizing an orbital exponent, but it does not require nonlinear variation. Of course, one still has to choose the values of the larger and smaller of Fig. 11-1. This is normally done by optimizing the fit to very accurate atomic wavefunctions or by a nonlinear variation on atoms. A basis set in which every minimal basis AO is represented by an inner-outer pair of STOs is often referred to as a double-zeta basis set. 

For closed-shell singlet species of overall symmetry (such as F ), the NAOs are perfectly equivalent to final natural orbitals of the atomic wavefunction. However, for open-shell atoms, the NAOs (I/0-2.2) are very slightly different from the final atomic natural orbitals (I/0-2.3 and FO-2.4). For qualitative conceptual purposes, however, the differences are immaterial. We shall henceforth consider NAOs to be effectively equivalent to the physical natural orbitals of the atomic wavefunction. 

---