The energy eigenvalues of hyperbolical potential functions

Abstract

The energy eigenvalues E(υ) of the hyperbolical potential functions V(r)_± = hcA{δ - σ(coth α(r - r ₀))^±1}² were determined with a semiclassical procedure (the Bohr-Sommerfeld quantization condition) and a quantum-mechanical method (the Schrödinger equation). The resulting term values G(υ) = E(υ)/hc are different: G(υ) = D _e - A{δσ(s + υ + 1/2)^-1 - (s + υ + 1/2)}² (for υ = 0, 1, 2, 3, …). D _e is the spectroscopic dissociation energy in cm^-1. s = σ for the semiclassical and s = (σ² + ¼)^1/2 for the quantum-mechanical solutions. The energy eigenvalues of V(r)_± include all known energy functions which one obtains from exact quantum-mechanical or semi-classical solutions. Hence V(r)_± is not only a better description for the potential energy of a molecular ‘vibration’ than the Morse function, it represents also perfectly intermolecular interactions and includes with the Kratzer energy eigenvalues the Rydberg terms of an electron in an atom. The ‘vibration’ of a two atomic molecule is discussed in space and in a plane. The semiclassically calculated term values represent the pure vibrational (= radial) energies, they are equivalent with the energy values for ‘vibrations’ in a plane. The quantum-mechanical calculated term values include these radial terms and zero-point energies of a ‘rotation’ in space.