Vibration-Rotation Coupling and Band Structure

A free rotating, vibrating diatomic is often modeled as a rotor whose stiffness depends on how far it currently is stretched. To a good first approximation, however, vibration and rotation are independent, so the total internal energy separates additively into vibrational and rotational parts. This is the essence of the Born-Oppenheimer-based separation of vibration and rotation, valid because the bond oscillates roughly 10¹²–10¹⁴ times faster than the molecule tumbles, so the rotational motion "sees" a time-averaged bond length for each vibrational state. That averaging is the whole story. In $v=0$ the molecule spends more time near its equilibrium separation $r_e$; in $v=1$, because the potential is anharmonic (steeper on the compression side, softer on the extension side — see Anharmonicity and Overtones), the average bond length $\langle r\rangle_v$ increases with $v$. A longer average bond means a larger moment of inertia $I_v = \mu\langle r^2\rangle_v$, hence a smaller rotational constant $B_v = h/(8\pi^2 c I_v)$. Every vibrational level therefore carries its own, slightly different, ladder of rotational sublevels — recall from Bond Lengths from Rotational Spectra that $B$ converts directly into $r$. When infrared light drives a transition from $v=0$ to $v=1$, it must simultaneously change $J$, because photons carry one unit of angular momentum and the vibrating bond (for a $\Sigma$ electronic state, zero electronic orbital angular momentum along the axis) has no other angular momentum reservoir to absorb it. The result is a strict rotational selection rule $\Delta J = \pm 1$, splitting the single vibrational transition into two combs of lines — the R branch ($\Delta J=+1$, higher frequency) and the P branch ($\Delta J=-1$, lower frequency) — with a gap where the forbidden $\Delta J=0$ Q branch would sit for a $^{1}\Sigma$ molecule. Molecules with unpaired electronic angular momentum (like NO) or bending vibrations in polyatomics can show an allowed, often intense, Q branch.

where $\tilde\nu_e$ is the equilibrium vibrational wavenumber, $x_e$ the anharmonicity constant, $B_v$ the vibrational-state-dependent rotational constant, and $D_v$ the centrifugal distortion constant.

where $B_e$ is the equilibrium rotational constant (at $r_e$) and $\alpha_e$ is the vibration-rotation coupling constant.

where $\tilde\nu_0$ is the pure vibrational band origin and $J$ labels the rotational level in the lower ($v=0$) state.

where the same $J$ convention applies, now for $\Delta J=-1$ transitions.

where $m = J+1$ for the R branch and $m = -J$ for the P branch, unifying both branches into one polynomial in $m$ (with $m\neq 0$).

where this isolates $B_0$, the rotational constant of the lower vibrational state, independent of band origin.

Example 1 — Foundational: predicting P(2) and R(2) for HCl. Given for H³⁵Cl: $\tilde\nu_0 = 2886.0\ \text{cm}^{-1}$, $B_0 = 10.440\ \text{cm}^{-1}$, $B_1 = 10.136\ \text{cm}^{-1}$. R(2) means the R-branch line originating from lower state $J=2$:

Compute each piece.

Summing,

P(2) means the P-branch line from lower state $J=2$:

Sanity check: both lines lie within about 45–60 cm⁻¹ of the 2886 cm⁻¹ origin, consistent with real HCl spectra where the branches span roughly 2600–3100 cm⁻¹, and R(2) > P(2) as required since the R branch lies at higher wavenumber.

Example 2 — Applied: extracting $B_0$, $B_1$, $B_e$, $\alpha_e$, and bond length from measured lines. Suppose careful measurement of the HCl (H³⁵Cl, $\mu = 1.6266\times10^{-27}$ kg) fundamental gives these observed line positions (cm⁻¹): R(0) = 2906.25, R(1) = 2925.78, P(1) = 2865.09, P(2) = 2843.56. Step 1 — combination differences to isolate $B_0$. Lines sharing the same upper state combine as

The standard, robust combination is

Using $J=1$: this needs R(0) and P(2), both reaching the same upper level $J'=1$.

Step 2 — combination differences to isolate $B_1$. Lines sharing the same lower state combine as

Using $J=1$ (R(1) and P(1) both start from lower level $J''=1$):

Step 3 — $B_e$ and $\alpha_e$. From $B_v = B_e - \alpha_e(v+\tfrac12)$,

Subtracting,

Adding appropriately,

Step 4 — bond length from $B_e$. Convert $B_e$ (cm⁻¹) to SI using

So $r_e = 127.4\ \text{pm}$, matching the accepted HCl equilibrium bond length (127.5 pm) to within rounding — an excellent sanity check, and noticeably shorter than the $v=0$ vibrationally-averaged bond length (about 128.8 pm) obtained from $B_0$ alone, exactly as expected since $B_e$ corrects for zero-point stretching.

The systematic interpretation of rotational fine structure in vibrational bands owes much to Gerhard Herzberg, whose multi-volume treatises on molecular spectra and molecular structure, published from the 1930s through the 1960s, became the standard reference that trained generations of spectroscopists to read P, Q, and R branches quantitatively. Herzberg combined meticulous high-resolution measurements with the term-value formalism to extract bond lengths, force constants, and dissociation energies for hundreds of diatomic and small polyatomic molecules, turning infrared and visible spectroscopy into a precision structural tool rather than a qualitative fingerprinting method.

Ask the tutor: How does this P/R branch analysis change once we move to the normal modes of a bent triatomic like water?