Analytical radial 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. 

Analytical radial orbitals may be found from the variational principle starting with a given analytical form of the initial function. The accuracy of the final radial orbital obtained depends essentially on the type of initial function and on the number of parameters varied. With increase of the number of shells in an atom the number of varied parameters grows rapidly. This, in turn, complicates substantially the optimization procedure. For these reasons analytical radial orbitals are impractical for obtaining the energy spectra of middle and, particularly, heavy atoms. 

The simplest analytical radial orbitals may be found by solving the radial Schrodinger equation for a one-electron hydrogen-like atom with arbitrary Z. They are usually called Coulomb functions and are expressed... 

There exist a number of attempts to generalize hydrogenic radial orbitals to cover the case of a many-electron atom. In [180] the so-called generalized analytical radial orbitals (Kupliauskis orbitals) were proposed ... 

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 extension of the matrix solution of section 4.3 for one-electron bound states to the Hartree—Fock problem has many advantages. It results in radial orbitals specified as linear combinations of analytic functions, usually normalised Slater-type orbitals (4.38). This is a very convenient form for the computation of potential matrix elements in reaction theory. The method has been described by Roothaan (1960) for a closed-shell or single-open-shell structure. 

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 relativistic version (RQDO) of the quantum defect orbital formalism has been employed to obtain the wavefunctions required to calculate the radial transition integral. The relativistic quantum defect orbitals corresponding to a state characterized by its experimental energy are the analytical solutions of the quasirelativistic second-order Dirac-like equation [8]... 

Boys (1950) proposed an alternative to the use of STOs. All that is required for there to be an analytical solution of the general four-index integral formed from such functions is that the radial decay of the STOs be changed from e to e. That is, the AO-like functions are chosen to have the form of a Gaussian function. The general functional form of a normalized Gaussian-type orbital (GTO) in atom-centered Cartesian coordinates is... 

The radial functions f (r) will be different for different atoms. Only for the hydrogen atom is the exact analytical form of the i2((r) s known. For other atoms the f (r) s will be approximate and their form will depend on the method used to find them. They might be analytical functions (e.g. Slater orbitals) or tabulated sets of numbers (e.g. numerical Hartree-Fock orbitals). 

A. Radial Functions and Atomic Orbital Energies.—Self-consistent field (SCF) radial functions for vanadium 3d and 4s orbitals were taken from Watson s report.16 Watson gives no 4p function, so it is estimated as having approximately the same radial dependence as the 4s function. Analytic 2s and 2p oxygen SCF radial functions were obtained by fitting the numerical functions given by Hartree17 with a linear combination of Slater functions. These radial functions are summarized... 

The RQDO radial, scalar, equation derives from a non-unitary decoupling of Dirac s second order radial equation. The analytical solutions, RQDO orbitals, are linear combinations of the large and small components of Dirac radial function [6,7] ... 

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