Fourier transform general formula for systematic potentials

For calculating molecular integrals of systematic potentials, a three-dimensional (3D) Fourier transform general formula can be derived, by the use of the Abel summation method. The present general formula contains all 3D Fourier transform formulas which are well known as Bethe-Salpeter formulas (Bethe and Salpeter, Handbuch der Physik, Bd. XXXV, 1957) as special cases. It is shown that, in several of the Bethe-Salpeter formulas, the integral does not converge in the meaning of the Riemann integral but converges in the meaning of a hyper function as the Schwartz distribution. For showing an effectiveness of the present general formula, the convergence condition of molecular integrals is derived generally for all of the present potentials. It is found that molecular integrals can be converged in the meaning of the Riemann integral for the present potentials, except for those for extra super singular potentials. It is also found that the convergence condition of molecular integrals over the Slater-type orbitals is exactly the same as that of the corresponding integrals over the Gaussian-type orbitals for the present systematic potentials. For showing more effectiveness, the molecular integral over the gauge-including atomic orbitals is derived for the magnetic dipole-same-dipole interaction.