Radial spinors

Given a trial orthonormal set of one-particle radial spinors r) (6... 

Table I also contains an analysis of the orbital character of these five energy levels. These were determined from the four-component spinors by neglecting the two lower, "small," components, and by assuming that the radial functions depend only upon , i.e. that the radial functions for pi/2 and p3/2> or for da/2 and ds/2> are the same. The orbitals may then be written in "Pauli" form as products of (complex) spherical harmonics and spin functions. Populations are equal to the squares of the absolute magnitudes of the coefficients listed in Table I. [For all but 17e3g, an additional orbital (not shown) is occupied which has the same energy but the opposite spin pattern (i.e. a and 3 are interchanged).]...

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... 

The appearance of the general expression (11) is at first sight far from obvious, although its construction can be described quite transparently [7, 22.6] see also [8, 3]. Here we just highlight key elements. For the upper 2-spinor, f3 = +1, the term with the factor (1 — /3) vanishes what remains is equivalent to a radial function exp(—a r ) multiplied by a single... 

At the restoration stage, a one-center expansion in the spherical harmonics with numerical radial parts is most appropriate both for orbitals (spinors) and for the description of external interactions with respect to the core regions of a considered molecule. In the scope of the discussed two-step methods for the electronic structure calculation of a molecule, finite nucleus models and quantum electrodynamic terms including, in particular, two-electron Breit interaction may be taken into account without problems [67]. 

The coupled radial equations (4.185) are the relativistic analogue of (4.19) for bound states and (4.57) for scattering states. In order to set up partial-wave integral equations corresponding to (4.121) we need the partial-wave form of the free-electron state (3.170). This is set up by generalising (4.56) to include the spin and using it in the partial-wave expansion of (3.170), which becomes a four-component spinor. 

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. 

L-spinors, [87], are related to Dirac hydrogenic functions in much the same way as Sturmian functions [93,94] are related to Schrodinger hydrogenic functions. The radial parts are given by [87, III]... 

G-spinors are appropriate for distributed charge nuclear models, and are much the most convenient for relativistic molecular calculations. Whereas neither L-spinors nor S-spinors satisfy the matching criterion (140) for finite c (although they do in the nonrelativistic limit), G-spinors are matched according to (140) for all values of c. The radial functions can be written... 

The matrix form of the atomic Dirac-Hartree-Fock (DHF) equations was presented by Kim [37,95], who used a basis set of modified radial Slater-type functions, without the benefit of a balancing presciption for the small component set. A further presentation of the atomic equations was made by Kagawa [96], who generalized Kim s work to open shells and discussed matrix element evaluation. An extension to include the low-ffequency form of the Breit interaction self-consistently in an S-spinor basis was presented by Quiney [97], who demonstrated that this did not produce variational collapse. Our presentation of the DHFB method is based on [97-99]. 

In the case of atoms Eq. (Ill) represents the complete spinor, whereas linear combinations of these single-centre spinors can be used to build up molecular spinors (or crystal spinors) in multi-centre problems. We briefly give an overview on the equations which finally have to be solved for these radial functions in non-relativistic and relativistic calculations. The form... 

Here P and Q are the radial large and small components of the wavefunction, the angular functions are 2-component spinors, the quantum number k = 2 - j) j + 1/2), -j < rrij < j, and the phase factor i is introduced for convenience in some atomic applications because it makes the radial Dirac equation real. 

While the angular parts (including spin) of the spinors emerge naturally from the spherical symmetry of the problem, it is slightly more complicated to find the radial parts P(r) and Q r). For the case of a point nucleus, these can be shown to take the form of the product of an exponential function, a power of r and a polynomial,... 

Figure 8. Radial densities of the 4f, 5d and 6s valence spinors of jgCe in the 4f 5d 6s ...

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 ... 

If spin-orbit effects are considered in ECP calculations, additional complications for the choice of the valence basis sets arise, especially when the radial shape of the / -f-1/2- and / — 1/2-spinors differs significantly. A noticeable influence of spin-orbit interaction on the radial shape may even be present in medium-heavy elements as 53I, as it is seen from Fig. 21. In many computational schemes the orbitals used in correlated calculations are generated in scalar-relativistic calculations, spin-orbit terms being included at the Cl step [244] or even after the Cl step [245,246]. It therefore appears reasonable to determine also the basis set contraction coefficients in scalar-relativistic calculations. Table 9 probes the performance of such basis sets for the fine structure splitting of the 531 P ground state in Kramers-restricted Hartree-Fock [247] and subsequent MRCI calculations [248-250], which allow the largest flexibility of... 

Similarly as in Section 1.2, one starts from atomic AE reference calculations at the independent-particle level (some kind of quasi-relativistic HF or fully relativistic DHF). The first step now in setting up pseudopotentials consists in a smoothing procedure for valence orbitals/spinors ( pseudo-orbital transformation ). In the DHF case, to be specific, the radial part ( )/ of the large component of the energetically lowest valence spinors for each //-combination is transformed according to... 

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