X-ray atomic orbital analysis. I. Quantum-mechanical and crystallographic framework of the method.

The scattering unit of X-ray crystal structure analysis is changed from atoms to the subshell electrons by X-ray atomic orbital analysis (XAO). All the atoms in the unit cell are divided into groups of subshell electrons in the XAO analysis. Each subshell is treated as an independent pseudo-atom, which enables the atomic orbitals (AO's) and the electron population of each AO expressed as a linear combination of s/p/d/f orbitals in each subshell to be determined. When the environmental condition of the sample is varied, the electron transfer among the AO's in the crystal can be traced with XAO. It is applicable mainly to analyses of the electron-density distribution in ionic solids including those with a nonstoichiometric structure. The expansion coefficients of each AO are calculated with the perturbation theory putting a point charge on each atom in the unit cell. This automatically makes the perturbation potential have the point-group symmetry of the atom in the crystal field. Then the coefficients of each AO are refined to fit to the observed structure factors keeping the orthonormal relationships among the AO's. Complex basis functions with alpha or beta spin as well as real ones are employed for heavy atoms and the relationships among the coefficients for the AO's of an electron in the crystal fields of the 32 point-group symmetries are derived for p, d and f orbitals. The AO's thus derived can be applicable to an anti-symmetrized multi-electron system, although X-ray diffraction cannot specify the atomic terms occupied when the crystal symmetry permits the atom to have many terms.