The Old Quantum Theory for One-Electron Diatomic Ions and Molecule

Abstract

The modern version of the Old Quantum Theory (OQT) is applied to one-electron diatomic systems in the clamped nucleus approximation. The first part reviews the theory, whose key feature is that only those trajectories whose classical action integrals are integer or half-integer multiples of Planck’s constant are allowed. The OQT is not correct, notoriously so in the case of the ground state of H+2 . Nonetheless, elements of the underlying classical structure have effects on quantum results, an understanding of which may aid in the interpretation of wave functions and energies. In particular, the classical separatracies provide an interpretation for the failure to describe the ground state of H+2 , for kinks in the electronic energy at small internuclear separation, and for quantum critical points. The OQT generates a ground-state Born-Oppenheimer potential energy minimum for the artificial neutral one-electron molecule; for such low quantum numbers, the accuracy is not high, emphasizing the importance of quantum concepts such as superposition of states in the real world. In addition, while accuracy of the classical results does increase with quantum number (Bohr Correspondence Principle), all OQT results show some disagreement with quantum values regardless of quantum number.