Radial integrals, electronic structure

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

In the angle integrated mode, electrons emitted in many different directions are accumulated simultaneously. To an appreciable extent this averages out the structure-dependent effects of step 4, and therefore emphasizes the information arising from the primary and secondary events of steps 2 and 3, such as the radial distances determined in fine-structure techniques like SEXAFS. 

Although the radial distribution function of an atom shows the shell structure, the electron probability density integrated over the angles and plotted versus r does not oscillate. Rather, for ground-state atoms this probability density is a maximum at the nucleus (because of the electrons) and continually decreases as r increases. Similarly, in molecules the maxima in electron probability density usually occur at the nuclei see, for example. Fig. 13.7. [For further discussion, see H. Weinstein, R Politzer, and S. Srebnik, Theor. Chim. Acta, 38,159 (1975).]... 

Pressure affects the energetic structure of the Ln ion and the energies of the 4f -4f transitions in two ways (1) the nephelauxetic [115] reduction of the Slater integrals and (2) the spin-orbit coupling constant. This effect was broadly discussed by Shen and Holzapfel [116] in the framework of the covalence model. In this framework, the radial-wave functions of the Ln ion, when it is embedded in the lattice, expand compared with those of a free ion due to the penetration of the ligand electrons into the Ln ion space. Penetration increases when pressure decreases the Ln ion-ligand distances. The second effect is related to the pressure-induced increase of the crystal-field strength. 

The radial distribution function D(r) represents the probability of finding an electron between the distances r and r + dr from the nucleus, regardless of directionThis radial density function reveals the atomic shell structure when plotted as function of r. Its integration over r gives the total number of electrons of the system... 

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