Electron radial integrals

Some frequently occurring two-electron radial integrals are given special names, as listed in Table 15.2. 

Here, k is an integer of values 2,4 and 6,/ are the coefficients representing the angular part of the wave function [29] and are the electrostatic Slater two-electron radial integrals given by Equation 1.17. 

Here / is a nuelear spin, F is the full momentum of the system, and J is the full electron momentum. The hyperfine splitting constants are expressed through the standard radial integrals ... 

A final point about basis functions concerns the way in which their radial parts are represented mathematically. The AOs, obtained from solutions of the Schrbdin-ger equation for one-electron atoms, fall-off exponentially with distance. Unfoitu-nately, if exponentials are used as basis functions, computing the integrals that are required for obtaining electron repulsion energies between electrons is mathematically very cumbersome. Perhaps the most important software development in wave function based calculations came from the realization by Frank Boys that it would be much easier and faster to compute electron repulsion integrals if Gaussian-type functions, rather than exponential functions, were used to represent AOs. 

We shall not consider here the more complex cases if necessary, they may be found in [9, 11] or deduced utilizing methods described there. The expressions for matrix elements of the majority of energy operators (1.16) in terms of radial integrals and transformation matrices or 3n/-coefficients for complex electronic configurations may be found in [14]. 

As was mentioned in Chapter 2, there exists another method of constructing the theory of many-electron systems in jj coupling, alternative to the one discussed above. It is based on the exploitation of non-relativistic or relativistic wave functions, expressed in terms of generalized spherical functions [28] (see Eqs. (2.15) and (2.18)). Spin-angular parts of all operators may also be expressed in terms of these functions (2.19). The dependence of the spin-angular part of the wave function (2.18) on orbital quantum number is contained only in the form of a phase multiplier, therefore this method allows us to obtain directly optimal expressions for the matrix elements of any operator. The coefficients of their radial integrals will not depend, except phase multipliers, on these quantum numbers. This is the case for both relativistic and non-relativistic approaches in jj coupling. 

In the central field approximation, when radial wave functions not depending on term are usually employed, the line strengths of any transition may be represented as a product of one radial integral and of a number of 3n./-coefficients, one-electron submatrix elements of standard operators (C(fc) and/or L(1 S(1)), CFP (if the number of electrons in open shells changes) and appropriate algebraic multipliers. It is usually assumed that the radial integral does not depend on the quantum numbers of the vec-... 

The one-electron submatrix element of operator rk 1 stands for the radial integral of this quantity and the appropriate radial wave functions. Indeed, we can easily show that the right side of formula (27.7) equals zero for k = 0. Therefore, Ml-transitions occur only between levels of one and the same configuration. 

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. 

Usually the dependence of the radial integrals of electronic transitions on spectral terms is neglected, and their value averaged with respect to these terms is employed. However, there are cases when the role of this dependence may increase substantially and may even become decisive. This is the case, when the radial integral itself becomes small and then it is rather sensitive even to insignificant improvements of wave function. 

The results of Table 28.1 show that for xenon, the 4/-electron is localized in the outward potential well far from the nucleus. Therefore, its radial orbital overlaps with those of inward electrons only a little. For this reason all integrals are very small and are of the same order for all LS values of the configuration considered. However, for barium we have a completely different picture, illustrating the collapse phenomenon. Only for the term 1P does the electron remain not collapsed for the rest of the terms the values of radial integrals have increased by 2-5 orders, evidencing... 

Crystal-field parameters would be expected to change across the series roughly in proportion to the radial integrals r2 and r4 (and r6 for 4f electrons). These integrals decrease dramatically across the lanthanide series for the 4f electrons (because their orbitals contract dramatically) but only by a few percent for the 5d electron. Thus, crystal-field parameters determined for Ce3+ may be used across the lanthanide series, with only a small scaling factor for the heavy ions. 

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