X-ray analysis of wavefunctions by the least-squares method incorporating orthonormality. I. General formalism

The least-squares method incorporating orthonormal relation between wavefunctions is formulated. The method fulfils the idempotency condition and allows the representation of wavefunctions to be obtained from charge-density distributions measured by the X-ray diffraction method. The present method can be applied to both the atomic-orbital (AO) and the molecular-orbital (MO) models. The scattering factor formalisms are also described for both models. One of the most important applications of the present method to the AO models is the determination of the d wavefunctions in a general crystal field. The restrictions among the coefficients of d wavefunctions in the 32 point-group-symmetry fields are derived. The effect of expansion and contraction of electron clouds in crystal fields is highly important, since it allows the possibility of determining all the unknown d wavefunctions. Even the wavefunctions of atoms with spherical electron configuration like Zn²⁺ and high spin Mn²⁺ ions can be determined when this effect is significant. From the scattering factors of two-center terms, the temperature factors for these terms are derived.