Many-electron atoms, radial wave functions

Fig. 3 shows a qualitative graphical representation of hydrogen-like wave functions for one-electron atoms which have to be replaced for many-electron atoms at least by Slater-type 107) analytical wave functions ifnlm (1) which are approximate as they contain no nodes in the radial part R ,. 

RADIAL WAVE FUNCTIONS FOR MANY-ELECTRON ATOMS... 

Other prescriptions for the exponents, have been advanced over the years. Clementi and Raimondi (8) proposed in 1963 that the best exponents should be based on the criterion that the atomic energy should be minimized. Clementi, too, (9,10) and others (11) have investigated the use of more than one Slater function to obtain a better representation of the radial wave functions for many-electron atoms. 

For a many-electron atom, the self-consistent-field (SCF) method is used to construct an approximate wave function as a Slater determinant of (one-electron) spin-orbitals. The one-electron spatial part of a spin-orbital is an atomic orbital (AO). We took each AO as a product of a spherical harmonic and a radial factor. As an initial approximation to the radial factors, we can use hydrogenlike radial functions with effective nuclear charges. 

The radial wave functions /t i(r) do not have closed-form expressions in many-electron atoms (Section 2.3), and so A , is not given by simple formulas in such atoms. Note the sensitivity of A i to the atomic number Z this gives rise to large spin-orbit coupling in heavy atoms. 

A natural starting point for the discussion of radial basis functions is the one-electron hydrogenic system, for which the wave function can be written in a closed analytical form. Indeed, we would naively expect the simple hydrogenic eigenstates - the exact solutions for a one-electron atom - to represent an ideal set of functions in terms of which we may expand the more complicated orbitals of many-electron atoms. Somewhat surprisingly, perhaps, the hydrogenic orbitals have certain deficiencies when used as basis functions for many-electron atoms. Nevertheless, from a consideration of the hydrogenic wave functions, we should learn much about the desirable analytical properties of radial atomic basis functions. 

In this last section, we shall present some examples of results of numerical atomic structure calculations to demonstrate properties of radial functions and the magnitude of specific effects like the choice of the finite nucleus model or the inclusion of the Breit operator. It should be emphasized that the reliability of numerical calculations is solely governed by the affordable length of the Cl expansion of the many-electron wave function since the numerical solution techniques allow us to determine spinors with almost arbitrary accuracy. Expansions with many tens of thousands of CSFs can be routinely handled (with the basis-set techniques of chapter 10, expansions of billions of CSFs are feasible via subspace iteration techniques [372]). 

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