Diatomic Vibrational Potential Functions from Integration of a Poisson Equation

It is shown how to generate a diatomic vibrational potential W(R) from a knowledge of the effective electron density function F(R) which enters the Poisson equation ∇ 2W=F(R). Namely, W(R)=W(Re)+ ∫ ReR S(1−SR) F(S)dS.Using the fact that F(R) is at least roughly interpretable as proportional to the electron density at one nucleus due to the electron density following the other nucleus during vibration [A. B. Anderson and R. G. Parr, J. Chem. Phys. 53, 3375 (1970)], various reasonable forms for F(R) are proposed and tested. F(R) proportional to R−4 leads to a Fues potential. F(R) proportional to R−1exp(−η R) leads to a Hellmann potential. The simple assumption that F(R) is proportional to exp (−ζR) leads to a new potential which is a linear combination of terms R−1, exp(−ζ R), and R−1 exp(−ζ R). Predictions of these three potentials are compared.