# 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)

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.), which explains all of this in detail.