A new functional form for representing vibrational eigenenergies of diatomic molecules. II. Application to H2 ground state

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Abstract
An expression previously proposed for representing vibrational eigenenergies Ev as a function of quantum number v is applied to the H2 ground electronic state. The expression is Ev=D−(vD−v)m[L/N], where D is the dissociation limit energy, vD and m are parameters, and [L/N] is a rational fraction in (vD−v). We integrate the vibrational Schrödinger equation to obtain 15 Born–Oppenheimer (BO) Ev with reduced mass μ0=918.048 electron masses, and 77 BO Ev with 25 μ0; error in Ev attributable to the potential is estimated to be 0.02 cm−1. The proposed functional form with a variety of [L/N] is fitted to the 14 BO μ0 first differences ΔE (v+1/2); with the exception of [4/0] all L+N=4 fits have an rms error in calculated ΔE of 0.02 cm−1. The proposed formula is then fitted to 15 BO differences including D−E14. The error in calculated D−E14 can be made negligible while the rms error in calculated energy differences remains 0.02 cm−1. Mass 25 μ0 is used to check applicability of the proposed functional form to a large number of levels; several fits yield all 76 ΔE with an rms error of only 0.04 cm−1. Finally, the proposed expression is applied to experimental vibrational energies. For these, better fits have an rms error of ∼0.07 cm−1, presumably due primarily to experimental errors.