  This article is an introduction to a new approach to first principles
electronic structure calculation. The starting point is the
Hartree-Fock-Roothaan equation, in which molecular integrals are approximated
by polynomials by way of Taylor expansion with respect to atomic coordinates
and other variables. It leads to a set of polynomial equations whose solutions
are eigenstate, which is designated as algebraic molecular orbital equation.
Symbolic computation, especially, Gr\"obner bases theory, enables us to rewrite
the polynomial equations into more trimmed and tractable forms with identical
roots, from which we can unravel the relationship between physical parameters
(wave function, atomic coordinates, and others) and numerically evaluate them
one by one in order. Furthermore, this method is a unified way to solve the
electronic structure calculation, the optimization of physical parameters, and
the inverse problem as a forward problem.

Computational Physics;Symbolic Computation