Nodal structure, spinor

Figure 13. Valence spinors of the Db atom in the 6d 7s ground state configuration from average-level all-electron (AE, dashed lines) multiconfiguration Dirac-Hartree-Fock calculations and corresponding valence-only calculations using a relativistic energy-consistent 13-valence-electron pseudopotential (PP, solid lines). A logarithmic scale for the distance r from the (point) nucleus is us in order to resolve the nodal structure of the all-electron spinors. The innermost parts have been truncated.

It may be asked how accurate energy-consistent pseudopotentials will reproduce the shape of the valence orbitals/spinors and their energies. Often radial expectation values < r > are used as a convenient measure for the radial shape of orbitals/spinors. Due to the pseudo-valence orbital transformation and the simplified nodal structure it is clear that values n < 0 are not suitable, since the resulting operator samples the orbitals mainly in the core region. Table 2 lists orbital energies, < r > and < > expectation values for the Db [Rn] 5f 6d ... 

In plane-wave calculations of solids and in molecular dynamics, the separable pseudopotentials [93,492,515] are more popular now because they provide linear scaling of computational effort with the basis-set size in contrast to the radially local RECPs. Moreover, the nonlocal Huzinaga-type ab-initio model potentials [521-523] conserving the nodal structure for the valence spinors are often applied. Contrary to the... 

Another feature that emerges from these plots is the loss of nodal structure. Because the spin-up and spin-down components of each spinor have nodes in different places, the directional properties of the angular functions are smeared out compared with the properties of the nonrelativistic angular functions. Only for the highest m value does the spinor retain the nodal structure of the nonrelativistic angular function, and that is because it is a simple product of a spin function and a spherical harmonic. The admixture of me and me + I character approaches equality as I increases and as me approaches zero, resulting in a loss of spatial directionality. The implications of this loss of directionality for molecular structure could be significant, particularly where the structure is not determined simply from the molecular symmetry or from electrostatics. 

The molecular spinors are expanded in terms of the four quaternion units (1, i, j, k). Two-dimensional contour maps of the large components are created in this work for the molecular spinors so to illustrate the nodal structure, and we review the relation between the quaternion representation and the normal four-component complex representation. [32] Each quaternion unit belongs to one of the boson irreducible representations (boson irreps) of C2v provided that the small components are neglected. 

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