Rotational-vibrational Eigen Values and Eigen Functions for Diatomic Molecules

Abstract

AS is well known, the Schrödinger equation for diatomic molecules can be solved exactly only for rotationless states in very limited cases as, for example, by Morse¹ and myself². The purpose of this communication is to give a potential curve, which is more accurate than the Morse curve and has the advantage that the Schrödinger equation can be solved exactly for vibrational-rotational states. The potential curve considered here has the form: where D_e is the dissociation energy, r_e is the equilibrium distance and γ = (2µr_e²/ħ²), where ħ is the Planck constant and µ is the reduced mass of the nuclei. In equation (1) a, b, c and n are constants, which can be determined from the three following conditions³: U(0) = + ∞, U′(r_e) = U(r_e) = 0, U″(r_e) = k_e, as also from the formula for the vibrational-rotational coupling constant α_e (ref. 4). Here k_e denotes the force constant. After simple calculation we obtain: and where Δ is the Sutherland parameter and Γ is expressed by the rotational constant B_e, the vibrational frequency ω_e and α_e as follows