Angular functions nonrelativistic

In the next section we will examine the properties of the angular functions. The radial functions will be the subject of the subsequent sections, where we also make comparisons with nonrelativistic radial functions and expectation values to gain some insight into the effects of relativity on electronic structure. 

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 no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

Consider first corrections to the energy levels with nonvanishing angular momenta. The respective wave functions vanish at the origin in coordinate space, hence only small photon momenta contribute to the integral, and one can use the first two terms in the nonrelativistic expansion of the polarization operator... 

The second order perturbation theory term with two one-loop self-energy operators does not generate any logarithm squared contribution for the state with nonzero angular momentum since the respective nonrelativistic wave function vanishes at the origin. Only the two-loop vertex in Fig. 3.24 produces a logarithm squared term in this case. The respective perturbation potential determined by the second term in the low-momentum expansion of the two-loop Dirac form factor [111] has the form... 

In the case of states with nonvanishing angular momenta the small distance contributions are effectively suppressed by the vanishing of the wave function at the origin, and the perturbation theory becomes convergent in the nonrelativistic region. Then this nonrelativistic approach leads to an exact result for the recoil correction of order (Zo ) (m/M)m for the P states [30]... 

Here ln[fco(o)/Ry] is the nonrelativistic Bethe logarithm [10] and is the only quantity in Eq. (1) that depends on the whole wave function and requires significant computational efforts. Recently, following Schwartz (see [11]) a new method has been elaborated that is applicable to both one- and two-electron systems of an arbitrary angular momentum [12]. The final accuracy of expansion (1) is a5 In a 1.0 10-10. 

It is clear from Eq. (22) that a different REP arises for each pseudospinor. The complete REP is conveniently expressed in terms of products of radial functions and angular momentum projection operators, as has been done for the nonrelativistic Hartrce Fock case (23). Atomic orbitals having different total angular momentum j but the same orbital angular momentum / are not degenerate in j-j coupling. Therefore the REP is expressed as... 

Table VI - Mulliken atomic orbital populations and net charges for [VV XgXg]2- complexes from DV calculations. NR is nonrelativistic model R is relativistic. Angular momentum subshells j= 1/2, 3/2, 5/2 and an extended ligand basis set of valence ns, np, nd and (n+l)s optimized functions are indicated.

The correct nonrelativistic limit as far as the basis set is concerned is obtained for uncontracted basis sets, which obey the strict kinetic balance condition and where the same exponents are used for spinors to the same nonrelativistic angular momentum quantum number for examples, see Parpia and Mohanty (1995) and also Parpia et al. (1992a) and Laaksonen et al. (1988). The situation becomes more complicated for correlated methods, since usually many relativistic configuration state functions (CSFs) have to be used to represent the nonrelativistic CSF analogue. This has been discussed for LS and j j coupled atomic CSFs (Kim et al. 1998). 

The Pauli approximation may be used in conjunction with this method by neglecting the small component spinors Q) of the Dirac equation, leading to RECPs expressed in terms of two-component spinors. The use of a nonrelativistic kinetic energy operator for the valence region, and two-component spinors leads to Hartree-Fock-like expressions for the pseudoorbitals. Note that the V s (effective potentials) in this expression are not the same for pseudo-orbitals of different symmetry. Thus the RECPs are expressed as products of angular projectors and radial functions. In the Dirac-Fock approximation, the orbitals with different total j quantum numbers, but which have the same / values are not degenerate, and thus the potentials derived from the Dirac-Fock calculations would be y-dependent. Consequently, the RECPs can be expressed in terms of the /y-dependent radial potentials by equa-... 

In atom/ions with N = 5 due to the presence of angular orbitals in the wave function, L — S there are many kinds of configurations to be constructed, whose contribution to the nonrelativistic energy is important. In particular, the ground states of B and C atoms are of P-symmetry. The Hy-CI program is general for any type of orbital, although in Hy-CI we use unnormalized s-, p- and d-orbitals. 

Therefore we mention only two examples. The nonrelativistic 3 parameter for s-electrons is equal to 2, while the inclusion of both relativistic and correlational corrections leads in the case of 5s-electrons in Xe to a very complicated curve >, given in Fig. 2 together with recent experimental data. The action of both 5p and 4d excitations upon 5s-electrons is very essential. The inclusion of many-electron correlations leads to a very complex behaviour of D(a)) as a function of O), D((jo) even changing its sign several times. The detailed information on D(a)) may be obtained in a so called complete experiment For photoionization it includes measurement of partial subshell cross section and angular distribution, and of electron spin direction, i.e. its polarization. The polarization follows all variations of D(o)), Recently, extensive calculations of polarization parameters for noble gases were performedand the RPAE predictions are now confirmed expe-... 

A part of the relativistic corrections due to a heavy neighboring atom can be approximated by third-order perturbation theory with nonrelativistic functions. The contribution to shielding comes from the cross term involving the spin-orbit coupling, I S interaction, and the external field-orbit interaction. In nonrelativistic terms, the shielding mechanism is that the external field induces an orbital angular momentum on the electrons of the heavy atom (Br or I), this produces a polarization in the electron spin by spin-orbit coupling, and the spin polarization is transferred to the resonant nucleus by a Fermi contact and by a nuclear spin-electron spin dipolar... 

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