Orthogonal Valence Bond as a Bridge Between the Valence Bond and Molecular Orbital Representations of the Wave Function

The wave function of a N-electron system is a remarkably complex object, being a function defined in a 3N space. This complexity puts us through the wringer and our ability to catch even the main features of the wave function is sometimes limited, in particular in non standard situations (where standard means a closed shell system in its ground state at the equilibrium geometry). With the aim to compute and analyze the wave function, two main strategies (valence bond and molecular orbitals) have been developed, both presenting issues and merits and often being complementary. In this chapter, after a short description of the two approaches and an analysis of their relative merits, we try to bridge these two worlds in the frame of the orthogonal valence bond strategy. A few examples of the use of this approach are reported. They show how orthogonal valence bond can help in the rationalization of many properties of quantum systems, ranging from small diatomic molecules to solid state materials. Most of the model Hamiltonians, crucial for understanding of electronic properties, from Hückel to Hubbard and all spin Hamiltonians, are based on an Orthogonal VB description.