Symbolic Computation of Molecular Integrals

The purpose of the present article is to give an account of the symbolic computation of molecular integrals. This sort of computation is a requisite for a scheme of quantum chemistry, which has been developed by the authors of the present article. This scheme enables us to simulate the quantum state of molecules through computational algebraic geometry. The requisite is that all of the molecular integrals should be prepared as analytical functions. Then those functions are approximated by multivariate polynomials, and the Hartree-Fock-Roothaan equation is given by a set of polynomial equations. We apply the technique of computational algebraic geometry, such as Gr\"obner basis and primary ideal decomposition, to solve the equation. The library is intended for the generation of molecular integrals by Gaussian-type orbitals (GTO). In this article, we shall explain how the library shall generate the analytic molecular integrals and what are the points treated carefully.