Spinor energies, atoms

Differences in spinor energies Ae (in eV) and radial maxima Ar (in bohr) of valence Pj 2 P3/2 spinors for p-block elements from atomic DHF calcula-... 

Table 8 Second-order many-body perturbation theory corrections to beryllium-like ions using non-relativistic (E ), Dirac-Coulomb (E ) and Dirac-Coulomb-Breit (E ) hamiltonians, obtained using the atomic precursor to BERTHA, known as SWIRLES. Basis sets are even-tempered S-spinors of dimension N= 17, with exponent sets, Xi generated by Xi = abi-i, with a = 0.413, and p = 1.376. Angular momenta in the range 0 < / < 6 have been included in the partial wave expansion of each second-order energy, and the total relativistic correction toE has been collected as Ef. All energies in hartree.

The energy spectrum of atoms and ions with j j coupling can be found using the relativistic Hamiltonian of iV-electron atoms (2.1)-(2.7). Its irreducible tensorial form is presented in Chapter 19. The relativistic one-electron wave functions are four-component spinors (2.15). They are the eigenfunctions of the total angular momentum operator for the electron and are used to determine one-electron and two-electron matrix elements of relativistic interaction operators. These matrix elements, in the representation of occupation numbers, are the parameters that enter into the expansions of the operators corresponding to physical quantities (see general expressions (13.22) and (13.23)). 

The DC-CASCI (12,4) calculation was performed to construct reference functions. The active space includes the molecular spinors, which have atomic nature of 6p1/2 and 6p3/2 of Pb. This was followed with the DC-CASPT2 (24, 12, 160) level of calculation and the potential energy curves are illustrated in Figure 6-5a. This figure includes all the states which go to the first, second, and third dissociation channels, except 1U(II) state, which had intruder state problem. For simplification... 

In this notation the presence of two upper and two lower components of the four-component Dirac spinor fa is emphasized. For solutions with positive energy and weak potentials, the latter is suppressed by a factor 1 /c2 with respect to the former, and therefore commonly dubbed the small component fa, as opposed to the large component fa. While a Hamiltonian for a many-electron system like an atom or a molecule requires an electron interaction term (in the simplest form we add the Coulomb interaction and obtain the Dirac-Coulomb-Breit Hamiltonian see Chapter 2), we focus here on the one-electron operator and discuss how it may be transformed to two components in order to integrate out the degrees of freedom of the charge-conjugated particle, which we do not want to consider explicitly. 

In the work of Larsson and Pyykko [51] the energy dependence of the atomic orbitals is defined by a polynomial fit of the charge dependence of the energy of any one AO on the population of the other AOs, as derived from Dirac-Fock calculations [49], Orbital populations are obtained by the Mulliken approximation. The orbital energies are corrected for the Coulomb interaction of the total charge on other atoms. The wavefunctions are of double-zeta variety, the radial parameters being optimised separately for the different spinors. 

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.

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