On the relative merits of non-orthogonal and orthogonal Valence Bond methods illustrated on the hydrogen molecule

Valence bond (VB) is one of the cornerstone theories of quantum chemistry. Even if in practical applications the molecular orbital (MO) approach has obtained more attention, some basic chemical concepts (such as the nature of the chemical bond and the failure of the single determinant-based MO methods in describing the bond cleavage) are normally explained making use of the VB approach. From the didactic point of view, the VB method is introduced and compared to the MO theory in the study of the hydrogen molecule through the use of the 1s atomic orbitals as basis functions for the construction of the VB structures. In this article we shall show that this approach leads to an interpretative problem owing to the non-orthogonality of the VB structures. An alternative approach is presented, in which the VB structures are based on orthogonal atomic orbitals. This approach has the great advantage of attributing a unique nature to the VB structures and therefore to make the didactic presentation clearer. The reasonableness of the orthogonal VB strategy is confirmed by the comparison of the energy of the neutral and ionic VB structures with the 1Sigma_u and 3Sigma_u states of H2, which have a well defined nature (ionic and neutral, respectively). All the energy expressions are fully developed as functions of the internuclear distance R, and, whenever possible, the analytical expressions are given. Furthermore, the minimal valence complete active space self-consistent field (CASSCF) approach with a general atomic basis set is described and its relation with different variants of the orthogonal VB approach is discussed. We report also as Supplemental Material the analytic expression of the relevant integrals, a didactic example of the Löwdin orthogonalization, the energy expression for the MO approach (single determinant and configuration interaction), and the generalized valence bond approach.