By Use of Model Potentials

By Use of Model Potentials. Most of the calculations performed on diatomic molecules have preferred the approach of fitting the calculated points to some model curve, and extracting the spectroscopic constants from the known relationships between them and the parameters of the model potentials. Most frequently used has been the polynomial in R. Generally an expression in positive powers of R is used, 

Polynomial fit, positive powers of R Polynomial fit, including if-1 term Numerical solution (Cooley) Experiment 

Basic to much of the work in which calculated points are fitted to a polynomial is the Dunham analysis. Dunham31-32 showed that if a potential energy curve is expressible as a power series, 

Possible solutions to this were given by Nesbet,29 30 36 who performed calculations on distances selected according to the roots of a Chebyshev polynomial, thus facilitating interpolation, and by Peyerimhoff55 and Mehler et a/.,53 who have fitted several different sets of points to different degrees of polynomial and averaged the results. 

Commonly used polynomials are the cubic56-58 and, most usually, the quartic.29 30-30 59-76 The quartic has the advantage that it gives analytical expressions for the most commonly recorded constants, toe, x0a c, Be, o. Probably a high proportion of the work quoting a Dunham analysis uses this function too. 

---