Non-relativistic numerical radial orbitals

In the case of N = 41 + 2 these expressions lead to exact values of the interactions considered for closed shells. 

The system of equations (28.10), each describing a shell of equivalent electrons n,/,Ni with boundary conditions P(n,/, 0) = P(n,/, oo) = 0 is usually solved numerically by computer. The methods of solving the 

The Hartree-Fock approach is also called the self-consistent field method. Indeed, the potential of the field in which the electron nl is orbiting is also expressed in terms of the wave functions we are looking for. Therefore the procedure for determination of the radial orbitals must be coordinated with the process of finding the expression for the potential starting with the initial form of the wave function, we find the expression for the potential needed to determine the more accurate wave function. We must continue this process until we reach the desired consistency between these quantities. 

The main difficulty in solving the Hartree-Fock equation is caused by the non-local character of the potential in which an electron is orbiting. This causes, in turn, a complicated dependence of the potential, particularly of its exchange part, on the wave functions of electronic shells. There have been a number of attempts to replace it by a local potential, often having an analytical expression (e.g. universal Gaspar potential, Slater approximation for its exchange part, etc.). These forms of potential are usually employed to find wave functions when the requirements for their accuracy are not high or when they serve as the initial functions. 

Let us consider a few examples. If we replace the non-local potential by a local one, then we arrive at the homogeneous equation for each shell 

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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 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. 

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