  In this article, we discuss how a kind of hybrid computation, which employs
symbolic, numeric, classic, and quantum algorithms, allows us to conduct
Hartree-Fock electronic structure computation of molecules. In the proposed
algorithm, we replace the Hartree-Fock equations with a set of equations
composed of multivariate polynomials. We transform those polynomials to the
corresponding Gr\"obner bases, and then we investigate the corresponding
quotient ring, wherein the orbital energies, the LCAO coefficients, or the
atomic coordinates are represented by the variables in the ring. In this
quotient ring, the variables generate the transformation matrices that
represent the multiplication with the monomial bases, and the eigenvalues of
those matrices compose the roots of the equation. The quantum phase estimation
(QPE) algorithm enables us to record those roots in the quantum states, which
would be used in the input data for more advanced and more accurate quantum
computations.
