Help with a derivation from a paper (diatomic molecular potential)

[Malamala]

Hello! I am confused about the derivation in the screenshot below. This is in the context of a diatomic molecular potential, but the question is quite general. Say that the potential describing the interaction between 2 masses, as a function of the radius between them is given by the anharmonic oscillator potential in eq 4., where $r_e$ is the equilibrium separation. What I need is to calculate the expectation value of a new variable, $X(r)$ in between 2 wavefunctions of such a potential, eq. 2 (please ignore eq. 3 and most of the comments in the paragraph after, as they are not related to my question). They Taylor expand $X(r)$ as in eq. 5 and then they claim that from there it follows that $X_\nu$ (eq. 2) is given by eq. 6. Can someone help me understand how to go from eq. 2, 4 and 5 to eq. 6? Thank you!

 

[div_grad]

Can you explain what is $X(r)$ and give a reference to the source from which you took the screenshot?

 

[Malamala]

Can you explain what is $X(r)$ and give a reference to the source from which you took the screenshot?

X(r) can be any function of r (well any function that can be written as a Taylor series around some value). The original paper is this: https://www.sciencedirect.com/science/article/abs/pii/0022285279900602