Vibration-Rotation Coupling

Vibration-rotation coupling is not a historical curiosity confined to textbook HCl spectra — it is a working analytical tool. High-resolution infrared and microwave spectroscopy of interstellar and cometary molecules (CO, HCN, CS, and dozens of other species) relies on precisely tabulated $B_e$, $\alpha_e$ values to assign rotational lines seen in radio-telescope surveys, since a line's exact frequency depends on which vibrational state (and hence which $B_v$) is populated. Atmospheric remote sensing — retrieving CO$_2$, CH$_4$, H$_2$O, and CO concentrations from satellite infrared spectra (e.g., for climate monitoring) — depends on databases like HITRAN, which tabulate $B_v$ for every relevant vibrational band precisely because unresolved vibration-rotation coupling would bias retrieved gas concentrations. In combustion diagnostics, coherent anti-Stokes Raman spectroscopy (CARS) thermometry extracts flame temperatures from the relative populations of rotational lines within specific vibrational bands, a measurement whose accuracy depends on knowing $B_v$ (not just $B_e$) for the vibrational state being probed. In fundamental physics, precision measurements of $\alpha_e$ and higher Dunham coefficients in simple hydrides (HD, HeH$^+$) constrain tests of fundamental constants and provide benchmarks for ab initio potential energy surfaces computed with post-Hartree–Fock and coupled-cluster methods, connecting this humble correction term to modern quantum chemistry validation pipelines. Finally, the entire fine structure of P, Q, and R branches in vibration-rotation spectra — the shifting spacing and eventual "band head" formation seen in real IR spectra — is a direct, visually striking consequence of $B_v$ differing between the upper and lower vibrational states, which is the natural next topic once this coupling term is understood.

Ask the tutor: How does knowing $B_v$ for two vibrational states let you predict exactly where the P and R branch lines fall in a real infrared spectrum?

Picture a diatomic molecule as two point masses connected by a spring that is not perfectly harmonic — it is easier to stretch than to compress, because at short range the internuclear repulsion rises steeply while at long range the bond simply dissociates. This asymmetry is the essential physics: a Morse-like potential $V(r)$ has a shallow, slightly right-leaning shape around its minimum at $r_e$. In the ground vibrational state $v=0$, the molecule oscillates with a small amplitude and the vibrational wavefunction is nearly Gaussian, symmetric enough that $\langle r\rangle \approx r_e$. But as $v$ increases, the anharmonic potential's asymmetry means the classical turning points are not symmetric about $r_e$: the outer turning point moves out faster than the inner one moves in, so the vibrationally-averaged bond length $\langle r\rangle_v$ increases with $v$. Now bring in rotation. The rigid-rotor rotational constant is $B = \hbar^2/(2\mu r^2)$ (in angular frequency units) or $B = h/(8\pi^2 \mu c r^2)$ in wavenumbers, and it depends on $r$ through $1/r^2$. Because the molecule vibrates roughly $10$–$100$ times faster than it rotates (vibrational frequencies are typically $10^{13}$–$10^{14}$ Hz, rotational frequencies $10^{10}$–$10^{11}$ Hz), a rotating molecule does not sense the instantaneous $r(t)$ but rather the time-average, or equivalently the quantum expectation value $\langle 1/r^2\rangle_v$ over the vibrational wavefunction in level $v$. Since $\langle r\rangle_v$ grows with $v$ due to anharmonicity, $\langle 1/r^2\rangle_v$ shrinks, and so does the effective rotational constant $B_v$. This is the entire physical content of vibration-rotation coupling: larger $v$ → larger average bond length → smaller rotational constant → smaller $\alpha_e$-scaled correction subtracted from $B_e$. There is a second, smaller contribution that acts in the same direction: even for a purely harmonic oscillator (no anharmonicity at all), the finite-amplitude vibration causes $\langle 1/r^2 \rangle \neq 1/r_e^2$ purely from the convexity of $1/r^2$ as a function of displacement (Jensen's-inequality-type curvature effect). Because $1/r^2$ is a convex function of $r$ near $r_e$, averaging it over any symmetric vibrational distribution gives $\langle 1/r^2\rangle > 1/r_e^2$ — this alone would make $B_v$ larger than the naive $B_e$ estimate based on equilibrium geometry for both harmonic and anharmonic oscillators. But the anharmonic asymmetry (the cubic term in $V(r)$) dominates and pushes $\langle r \rangle_v$ outward as $v$ grows, and this effect wins, producing the experimentally universal result $\alpha_e > 0$: $B_v$ decreases with increasing $v$.

The starting point is the Dunham-type expansion of the total vibration-rotation term value $G(v) + F_v(J)$, in which the rotational constant is allowed to depend on $v$:

where:

• $F_v(J)$ — the rotational term value (wavenumbers) for vibrational state $v$, rotational quantum number $J$
• $B_v$ — the effective rotational constant in vibrational level $v$
• $D_v$ — the centrifugal distortion constant in level $v$ (small, usually treated as $v$-independent for this purpose)

The vibration-dependence is packaged into the linear expansion

where:

• $B_e$ — the equilibrium rotational constant, defined using the equilibrium bond length $r_e$ at the potential minimum: $B_e = h/(8\pi^2 c \mu r_e^2)$
• $\alpha_e$ — the vibration-rotation coupling constant (wavenumbers), always positive for typical anharmonic diatomics
• $\gamma_e$ — a second-order correction, orders of magnitude smaller than $\alpha_e$, usually negligible

Reading: $B_e$ is the rotational constant the molecule would have if it sat motionless at the bottom of the well; it is never directly observable because $v=0$ already carries zero-point motion. $\alpha_e$ tells you how fast $B_v$ falls as vibrational energy increases, and it is precisely the bridge between the vibrational anharmonicity and the rotational spectrum's fine structure. The microscopic definition of $B_v$ that makes the averaging explicit is

where:

• $\mu$ — the reduced mass of the diatomic
• $\langle 1/r^2\rangle_v$ — the expectation value of $1/r^2$ in the vibrational eigenstate $v$
• $c$ — speed of light (converting to wavenumber units)

Reading: this equation is the whole physics — $B_v$ is not evaluated at a fixed geometry but is a quantum-mechanical average over the vibrational probability density $|\psi_v(r)|^2$, and everything that follows is simply computing that average perturbatively for an anharmonic (Morse-like) potential.

Example 1 — Foundational: computing $\alpha_e$ and $B_v$ for HCl from equilibrium constants. Given for $^1$H$^{35}$Cl: $B_e = 10.593\ \text{cm}^{-1}$, $\omega_e = 2990.9\ \text{cm}^{-1}$, $\omega_e x_e = 52.82\ \text{cm}^{-1}$. Step 1: Apply the Pekeris-derived working formula,

Step 2: Compute $B_e^3 = 10.593^3 = 1188.6\ \text{cm}^{-3}$ (since $10.593^2 = 112.21$, and $112.21 \times 10.593 = 1188.6$). Step 3: Multiply by $\omega_e x_e$: $1188.6 \times 52.82 = 62{,}780\ \text{cm}^{-4}$ (to 4 sig figs: $1188.6 \times 52.82 = 62{,}780.2$). Step 4: Take the square root: $\sqrt{62{,}780.2} = 250.56\ \text{cm}^{-2}$. Step 5: First term: $6 \times 250.56 / 2990.9 = 1503.4/2990.9 = 0.5028\ \text{cm}^{-1}$. Step 6: Second term: $6 B_e^2/\omega_e = 6 \times 112.21/2990.9 = 673.26/2990.9 = 0.2251\ \text{cm}^{-1}$. Step 7: Subtract:

This compares well with the experimental HCl value $\alpha_e = 0.3019\ \text{cm}^{-1}$; the ~8% discrepancy is expected because the working formula truncates higher-order Dunham terms and because tabulated $\omega_e x_e$ values themselves carry some fitting-model dependence — the qualitative and semi-quantitative agreement is what the approximate formula is designed to deliver. Step 8: Using the experimental $\alpha_e = 0.3019\ \text{cm}^{-1}$, compute $B_0$ and $B_1$:

These match the experimentally tabulated values ($B_0 = 10.4400$, $B_1 = 10.1366\ \text{cm}^{-1}$) to within the third significant figure — the small residual is due to the neglected $\gamma_e(v+\tfrac12)^2$ term.

Limiting case check: if anharmonicity vanished entirely ($\omega_e x_e \to 0$), the working formula gives $\alpha_e \to -6B_e^2/\omega_e$, a small negative number — reflecting only the harmonic zero-point-motion (convexity) contribution described physically above, confirming that anharmonicity is required to produce the experimentally observed sign and magnitude of $\alpha_e$ in real molecules.

Example 2 — Applied: extracting bond length change between $v=0$ and $v=3$ in CO from spectroscopic constants. Carbon monoxide has $B_e = 1.9313\ \text{cm}^{-1}$, $\alpha_e = 0.01750\ \text{cm}^{-1}$ (standard tabulated values), reduced mass $\mu = \dfrac{m_C m_O}{m_C+m_O}$ with $m_C = 12.000\ \text{u}$, $m_O = 15.9949\ \text{u}$. Step 1: Compute $\mu$ in atomic mass units:

Step 2: Convert to kg: $\mu = 6.8562 \times 1.66054\times10^{-27}\ \text{kg} = 1.1387\times10^{-26}\ \text{kg}$. Step 3: Compute $B_0$ and $B_3$:

Step 4: Convert $B_0$ to SI (angular-frequency-based) to extract $r_0$. Using

with $h = 6.62607\times10^{-34}\ \text{J s}$, $c = 2.99792\times10^{10}\ \text{cm/s}$ (so that $B_v$ in cm$^{-1}$ combines correctly). Step 5: Compute the denominator for $v=0$:

First, $8\pi^2 = 78.957$. Then $78.957 \times 2.99792\times10^{10} = 2.3671\times10^{12}$. Multiply by $\mu$: $2.3671\times10^{12} \times 1.1387\times10^{-26} = 2.6957\times10^{-14}$. Multiply by $B_0 = 1.92255$: $2.6957\times10^{-14} \times 1.92255 = 5.1836\times10^{-14}\ \text{J s cm}^{-1}\!\cdot\!\text{s}^{-1}$ — carrying units carefully, this product has units of J·s² (the cm$^{-1}$ and cm/s cancel a length, leaving inverse-time-squared structure consistent with $r^2$ after dividing by $h$). Step 6: Divide by $h$:

Step 7: Take the square root:

Step 8: Repeat with $B_3 = 1.87005\ \text{cm}^{-1}$ in place of $B_0$. Since $r_v^2 \propto 1/B_v$,

Step 9: The bond stretches by

over three vibrational quanta — about 1.4% of the bond length, consistent with the intuitive picture that higher vibrational excitation systematically lengthens the average bond.

Limiting case check: as $v \to 0$ with $\alpha_e$ artificially set to zero, $B_v \to B_e$ for all $v$, and $r_v \to r_e$ independent of vibrational state — recovering the rigid-rotor/harmonic-oscillator (uncoupled) limit exactly, as it must.

Example 3 — Competition level: given only measured line positions, extract $\alpha_e$ and $B_e$ without prior knowledge of tabulated constants. Suppose (asymmetric, invented but physically reasonable numbers modeled on a light hydride) rotational combination differences from the $v=0\to1$ fundamental band of a diatomic give the ground-state rotational term differences:

(this combination difference, $R(J{=}0)-P(J{=}1)$, isolates $B_0$ exactly regardless of $B_1$ — a standard identity used in branch analysis) and the excited-state combination difference

Step 1: Apply the two-point solution from the second derivation:

Step 2: Compute $B_e$:

Step 3: Sanity-check by reconstructing $B_0$ and $B_1$ from $B_e$ and $\alpha_e$:

Step 4: Use $B_e$ to find $r_e$. Assume this is a hypothetical hydride with $\mu = 0.9922\ \text{u} = 1.6479\times10^{-27}\ \text{kg}$ (close to but not identical to HCl's $\mu$, to keep numbers asymmetric and independently checkable). Using

compute the denominator: $8\pi^2 c = 78.957 \times 2.99792\times10^{10} = 2.3671\times10^{12}\ \text{cm/s}$. Multiply by $\mu$: $2.3671\times10^{12}\times1.6479\times10^{-27} = 3.9016\times10^{-15}$. Multiply by $B_e = 10.688$: $3.9016\times10^{-15}\times10.688 = 4.1698\times10^{-14}$. Step 5: Divide $h$ by this: $r_e^2 = 6.62607\times10^{-34}/4.1698\times10^{-14} = 1.5889\times10^{-20}\ \text{m}^2$. Step 6: Square root: $r_e = 1.2605\times10^{-10}\ \text{m} = 126.05\ \text{pm}$.

Limiting case check: note that the ratio $\alpha_e/B_e = 0.442/10.688 = 0.0414$, a small number ($\ll 1$) — this is a necessary self-consistency check, because the entire perturbative derivation assumed $\alpha_e \ll B_e$ (equivalently $\hbar\omega \gg \hbar B$, fast vibration compared to rotation); had the extracted ratio come out order-unity, the linear expansion of $B_v$ itself would be invalid and higher Dunham terms ($\gamma_e$) would be mandatory.