Radial orbitals numerical

The reader should note that no transformation operator 0A can alter the radial function J r) of an orbital and consequently the symmetry properties of the AOs are completely defined by the angular functions, Y O, ). Since these angular functions are the same in all one-electron product function approximations, the orbitals in all these approximations (Slater orbitals, numerical Hartree-Fock orbitals,... 

Usually the solutions of any version of the Hartree-Fock equations are presented in numerical form, producing the most accurate wave function of the approximation considered. Many details of their solution may be found in [45], However, in many cases, especially for light atoms or ions, it is very common to have analytical radial orbitals, leading then to analytical expressions for matrix elements of physical operators. Unfortunately, as a rule they are slightly less accurate than numerical ones. 

Here af and cf for the cases n = l + 1 are found from the variational principle requiring the minimum of the non-relativistic energy, whereas cf (n > l + 1) - form the orthogonality conditions for wave functions. More complex, but more accurate, are the analytical approximations of numerical Hartree-Fock wave functions, presented as the sums of Slater type radial orbitals (28.31), namely... 

The main advantage of analytical radial orbitals consists in the possibility to have analytical expressions for radial integrals and compact tables of numerical values of their parameters. There exist computer programs to find analytical radial orbitals in various approximations. Unfortunately, the difficulties of finding optimal values of their parameters grow very rapidly as the number of electrons increases. Therefore they are used only for light, or, to some extent, for middle atoms. Hence, numerical radial orbitals are much more universal and powerful. 

The relativistic theory and computation of atomic structures and processes has therefore attained some sort of maturity and the various codes now available are widely used. Those mentioned so far were based on ideas originating from Hartree and his students [28], and have been developed in much the same way as the non-relativistic self-consistent field theory recorded in [28-30]. All these methods rely on the numerical solution, using finite differences, of the coupled differential equations for radial orbital wave-functions of the self-consistent field. This makes them unsuitable for the study of molecules, for which it is preferable to expand the radial amplitudes in a suitably chosen set of analytic functions. This nonrelativistic matrix Hartree-Fock method, as it is often termed, was pioneered by Hall and Lennard-Jones [31], Hall [32,33] and Roothaan [34,35], and it was Roothaan s students, Synek [36] and Kim [37] who were the first to attempt to solve the corresponding matrix Dirac-Hartree-Fock equations. Kim was able to obtain solutions for the ground state of neon in 1967, but at the expense of some numerical instability, and it seemed at the time that the matrix Dirac-Hartree-Fock scheme would not be a serious competitor to the finite difference codes. 

The starting point to obtain a PP and basis set for sulphur was an accurate double-zeta STO atomic calculation4. A 24 GTO and 16 GTO expansion for core s and p orbitals, respectively, was used. For the valence functions, the STO combination resulting from the atomic calculation was contracted and re-expanded to 3G. The radial PP representation was then calculated and fitted to six gaussians, serving both for s and p valence electrons, although in principle the two expansions should be different. Table 3 gives the numerical details of all these functions. 

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