The Hydrogen Molecular Ion $H_2^+$ with Overlap: Symmetric and Antisymmetric States via the LCAO Method

Abstract:
In this note, we present a derivation of the symmetric and antisymmetric states of the hydrogen molecular ion $H_2^+$ with overlap. Employing the Born-Oppenheimer approximation and the LCAO method, we construct the generalized eigenvalue problem. In addition, we provide a physical interpretation of the hopping term, a parameter that couples the time evolution of the atomic orbitals. Finally, we determine the eigenenergies and the eigenstates of the system.