Coulombic potential energy integrals and approximations

Theorems are derived which establish a method of approximating two‐particle Coulombic potential energy integrals, ${\mathrm{[\phi }}_a{\text{(1)\hspace{0.17em}r}}_{12}^{\text{−1}}{\mathrm{\hspace{0.17em}|\phi }}_b\text{(2)]}$ , in terms of approximate charge densities ${\phi }_a\prime$ and ${\phi }_b\prime$ . Rigorous error bounds, ${\mathrm{|[\phi }}_a{\text{(1)|r}}_{12}^{\text{−1}}{\mathrm{|\phi }}_b{\text{(2)]−[φ}}_a{\text{′(1)|r}}_{12}^{\text{−1}}{\mathrm{|\phi }}_b\text{′(2)]| ≤slant δ}$ , are simply expressed in terms of information calculated separately for the pair of densities ${\phi }_a$ and ${\phi }_b\prime$ and the pair ${\phi }_b$ and ${\phi }_b\prime$ . From the structure of the bound, a simple method of optimizing charge density approximations such that δ is minimized is derived. The framework of the theory appears to be well suited for application to the approximation of electron repulsion integrals which occur in molecular structure theory, and applications to the approximation of integrals over Slater orbitals or grouped Gaussian functions are discussed.