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:



$v:=$ 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 $(1)$:


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 $v_{min}=-\frac{1}{2}?$


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