Why does band heads arise in vibrational-rotational spectra?

I learn that a band head is a region of maximum intensity due to many lines falling together. It can be determined by differentiating the expression for the R branch.


To no avail, I cannot find any explanation on why band heads arise after going through a few textbooks and Internet resources.


My guess: It might be due to the breakdown of Born-Oppenheimer Approximation which explain the unequal spacing in vib-rot spectra. The rotational constants decreases at higher vibrational states, to a limit?


Why does the absorption lines of R branch seem to 'U-turn' at higher J'' value?


Attached is the vib-rot spectra. This is one of the materials I searched from the Internet.(<https://cefrc.princeton.edu/sites/g/files/toruqf1071/files/Files/2015%20Lecture%20Notes/Hanson/pLecture2.pdf> , slide 28)[![enter image description here](https://i.stack.imgur.com/P1lgP.png)](https://i.stack.imgur.com/P1lgP.png)


FYI, I am at the entry-level of learning molecular spectroscopy. Any explanation is really appreciated.




It's been a long time since I did this, but from my old notes... this is caused by the rotational constant $B$ being different in the two vibrational states $v = 0$ and $v = 1$. Generally we have that $B\_1 < B\_0$ (where $B\_n$ denotes $B$ in the vibrational state $v = n$). This makes sense because $B = h/8\pi^2cI$ (expressed in wavenumber units) is inversely proportional to the moment of inertia $I$, and we expect $I$ to increase with vibrational excitation.


So, if $v\_0$ is the vibrational frequency, then the *R*-branch transition frequencies (for the $(J, v) \to (J + 1, v+1)$ transition) are given by


$$\begin{align}
R(J) &= v\_0 + B\_1(J+1)(J+2) - B\_0J(J+1) \\
 &= v\_0 + J^2(B\_1 - B\_0) + J(3B\_1 - B\_0) + 2B\_1
\end{align}$$


and the spacing between successive transitions is


$$\begin{align}
R(J + 1) - R(J) &= [(J+1)^2 - J^2](B\_1 - B\_0) + [(J+1) - J](3B\_1 - B\_0) \\
 &= (2J + 1)(B\_1 - B\_0) + 3B\_1 - B\_0 \\
 &= 2J(B\_1 - B\_0) + 4B\_1 - 2B\_0
\end{align}$$


Since $B\_1 - B\_0 < 0$, this separation decreases with increasing $J$ as observed. (Notice that if we let $B\_1 = B\_0 = B$ then we recover the "simple" behaviour where the spacing between lines is $2B$.) Naturally, exactly the opposite scenario is seen in the *P*-branch; I'll leave the maths to the reader.


A very thorough book for spectroscopy is [Hollas, *Modern Spectroscopy* (4th ed.)](https://www.wiley.com/en-gb/Modern+Spectroscopy%2C+4th+Edition-p-9780470844168), which explains all of this in detail.


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