This work completes the construction of a purely algebraic version of the theory of finite dimensional electronic Fock spaces endowed with a new law of composition (star product). For a fixed number of electrons the corresponding sector of the Fock space becomes a commutative algebra and its ideals are determined by the excitation level used. New approach to non-linear methods of quantum chemistry based on the systematic use of the star product is developed. Efficient computer implementation of multiplication in the aforementioned algebras is described. Quality of different polynomial approximations of electronic wave functions is illustrated with concrete examples