Evaluation of Molecular Integrals by Solid Spherical Harmonic Expansions

The use of expansions of solid spherical harmonics, which are the functions $${\mathrm{(r}}^n\hspace{0.17em}\mathrm{or}{\mathrm{\hspace{0.17em}r}}^{\text{−n−1}}{\mathrm{)\hspace{0.17em}P}}_n^m\mathrm{\hspace{0.17em}(}\cos \mathrm{\theta )(}\cos \mathrm{m\phi \hspace{0.17em}}\mathrm{or}\hspace{0.17em}\sin \mathrm{m\phi ),}$$ is discussed with respect to molecular integrals, and it is found that terms of the delta function type must be included in the expansion for the r⁻ⁿ⁻¹ case if n—m≥2. These expansions are especially useful for operators involving inverse powers of r such as electromagnetic interaction operators. General formulas are given for integrals in which two atomic orbitals, e.g., Slater orbitals, are on one center, and one of these operators is on the other center. Formulas are also given for the corresponding one‐center integrals, and methods of evaluating three‐center integrals and other two‐center integrals are discussed briefly.