Abstract: In this article, two groups {E, C₂} and {E, σ_v} are proposed to describe the orbital symmetry of the whole reaction paths with respect to the conrotatory and disrotatory electrocyclic reaction of butadiene. Furthermore, the symmetries of the two groups are extended to the systems as a many-body treatment based on spin-free theory. The effect of full configuration interaction is investigated using the Huckel-Hubbard approximation. The many-body treatment is generalized to the case in which the Gel'fand bases take the molecular orbitals as the starting point.We obtain singlet and triplet reaction paths-energy reaction surfaces, from which the activiation energies of thermal and photochemical reactions can be calculated. The results are compared with the values of other authors'and can explain the selection rules established by Woodward and Hoffmann.