A Simple Potential Function for Diatomic Molecules

A Simple Potential Function for Diatomic Molecules.— The simplest assumption which can be made concerning the force between the atoms of a diatomic molecule is that it is proportional to the displacement of the internuclear distance from its equilibrium value re. This corresponds to the potential function 

Comparison with Equation 35-14 shows that the force constant k is equal to 

Insertion of this form for f/(r) into the radial equation 35-13 yields the equation 

Since the approximation which we have used for U(r) is good only for p small compared to r it is legitimate to introduce the expansion 

By making a suitable transformation p = f -(- a, we can eliminate the term containing the first power in the independent variable, obtaining thereby an equation of the same form as Equation 11-1, the wave equation for the harmonic oscillator, which we have previously solved. It is easily verified that the proper value for a is 

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