I Rotational-Vibrational Energy state equation derivation

[EphemeralMurex]

I am having a mental block at the moment and for some reason I can't seem to derive these two equations:

From this equation:

Any assistance would be much appreciated!

 

[DrClaude]

What have you tried? Do you understand what $\tilde{\nu}_0$ stands for? Otherwise, it's just $T(\nu', J'' \pm 1) - T(\nu'', J'')$.

 

[EphemeralMurex]

I guess what confuses me is the $\nu', J''$ and $\nu''$. I don't know what they mean. Is it the first and second derivative of the original equation?

 

[DrClaude]

I guess what confuses me is the $\nu', J''$ and $\nu''$. I don't know what they mean. Is it the first and second derivative of the original equation?

No, they are labels to differentiate the different levels involved in the transition, with the double prime indicating the initial state and the prime the final state:
$$
\nu', J' \leftarrow \nu'', J''
$$

 

[EphemeralMurex]

Thank you for that explanation! So to begin deriving the equations, would I set initial values for $v''$ and $J''$ equal 0 and $J'$ and $v'$ to 1 for the $\tilde\nu$(R)? I also don't understand how to eliminate ($\nu$ + $1/2$) from the equation and bring in the (2$\beta$ - 3$\alpha$) and (2$\beta$ - 4$\alpha$) values