# Incompatibility between the Bohr-Sommerfeld quantization condition and the Dunham expansion

I'm trying to apply the RKR (Rydberg-Klein-Rees) method which computes the classical turning points, $$a(E,J)$$ and $$b(E,J)$$ of a diatomic molecule for a rotational-vibrational energy value E, and quantum rotational number $$J$$. I'm following the chapter VI of the book of M. S. Child, to know more about this method.

My doubt is about a incompatibility that I found between the Bohr-Sommerfeld quantization condition and the Dunham expansion that the RKR considers.

Basically, the Bohr-Sommerfeld quantization can be expressed as:

$$v(E,J)=\frac{\sqrt{2\mu}}{\pi\hbar}\tag{1}\int_{a(E,J)}^{b(E,J)}\sqrt{E-V_J(R)}\,dR-\frac{1}{2}$$

where

$$v:=$$ vibrational quantum number

$$V_J(R):=$$ centrifugally corrected potential between the two atoms of the molecule

$$R:=$$ internuclear distance

The rotational-vibrational energy can be expressed through which is called a "Dunham expansion":

$$E(v,J)=\sum_{i,j=0}^\infty Y_{ij}\left(v+\frac{1}{2}\right)^i\left[J\left(J+1\right)\right]^j \tag{2}$$

where $$Y_{ij}$$ are some constants designated by Dunham parameters.

Consider now the following image from the Child's book:

which describes the curve $$V_J(R)$$ and the classical turning points. My doubt is about the value $$v$$ for which, the energy $$E(v,J)$$ is minimal. From the figure, one can easily know that in the minimum, one has $$a(E,J)=b(E,J)$$, which means that the respective $$v$$ value is, accordingly to $$(1)$$:

$$v_{min}=-\frac{1}{2}\tag{3}$$

This value is constant, so it would not depend on $$J$$. For the case $$J=0$$, the energy minimum value would (classically) be $$E_{min}=0$$ (corresponding to no vibration). One would get from $$(2)$$:

$$E(v,0)=\sum_{i=0}Y_{i0}\left(v+\frac{1}{2}\right)^i:=G(v) \tag{4}$$

If $$Y_{00}=0$$, then that would mean from $$(4)$$ that $$v=-\frac{1}{2}$$ is the minimizer of $$E$$. In that case the Bohr-Sommerfeld condition and the Dunham expansion are compatible. But for the case of $$Y_{00}\neq 0$$, then different values would be achieved for $$v_{min}$$. And if a non-null $$J$$ is considered, then $$v_{min}$$ would also depend on $$J$$. So, for these cases Bohr-Sommerfeld condition and Dunham expansion turn to be incompatible. M. S. Child says (without explaining) that $$v_{min}$$ should be chosen so that $$G(v_{min})=0$$, independently of the $$J$$ value. But is this right? Is it better than considering $$v_{min}=-\frac{1}{2}?$$