---
title: Anharmonicity and Rovibrational Structure
module: Molecular Spectra
moduleNumber: 2
lessonNumber: 2
order: 202
summary: >
  The rigid rotor and harmonic oscillator are first approximations. A real bond
  follows the Morse potential, whose levels converge toward dissociation; a real
  rotor stretches centrifugally; and vibration couples to rotation, so the
  rotational constant depends on the vibrational level. This lesson works out the
  anharmonic and centrifugal corrections, the Birge-Sponer route to the
  dissociation energy, the isotope shift, and the thermal band envelope.
topics: [Molecular Spectra]
draft: false
sources:
  - book: Tipler & Llewellyn
    ref: "Ch. 9 — Molecular Structure and Spectra; §9-4 Energy Levels and Spectra of Diatomic Molecules"
  - book: Kittel
    ref: "Ch. 4 — Phonons I: Crystal Vibrations; anharmonic potential"
  - book: Hook & Hall
    ref: "Ch. 2 — background on the anharmonic interatomic potential"
---

The [rigid rotor and harmonic oscillator](/condensed-matter/molecular-spectra/rotational-vibrational-spectra)
give equally spaced vibrational levels and a rotational ladder of constant line
spacing. A measured band departs from both: the vibrational overtones crowd
together, the rotational lines are not quite evenly spaced, and the two branches
turn back on themselves at high rotational quantum number. Each departure is a
small correction to the leading model, and each carries a physical constant. This
lesson computes those corrections and reads the molecular parameters they encode.
Throughout, energies are quoted as **term values** $G(v)$ and $F(J)$ in the
spectroscopist's units of wavenumber (inverse centimeters), where a term value
$T$ and an energy $E$ are related by $E = hcT$.

## The Morse potential and anharmonic levels

The parabolic approximation fails away from the bottom of the well. A real bond
softens as it stretches and dissociates at large separation, so the potential
climbs steeply on the inner (repulsive) side and levels off to a finite
dissociation asymptote on the outer side. The **Morse potential** captures both
features in a form that is still exactly solvable:

$$
U(r) = D_e\Bigl[1 - e^{-a(r - r_0)}\Bigr]^2,
$$

where $D_e$ is the well depth measured from the minimum, $r_0$ the equilibrium
separation, and $a$ sets the curvature. Expanding about $r_0$ recovers a
harmonic term $U \approx D_e a^2 (r-r_0)^2$, so the small-oscillation frequency is

$$
\omega_e = a\sqrt{\frac{2D_e}{\mu}}, \qquad a = \omega_e\sqrt{\frac{\mu}{2D_e}},
$$

with $\mu$ the [reduced mass](/condensed-matter/molecular-spectra/rotational-vibrational-spectra).
The Schrödinger equation for this potential has the closed-form eigenvalues

$$
G(v) = \omega_e\left(v + \tfrac12\right) - \omega_e x_e\left(v + \tfrac12\right)^2,
\qquad \omega_e x_e = \frac{\hbar\omega_e}{4D_e} \cdot \frac{1}{hc},
$$

exact with only two terms and no higher powers. The **anharmonicity constant**
$\omega_e x_e$ is positive, so each successive level sits slightly below where the
harmonic ladder would place it.

$$
% caption: The Morse potential rises steeply on the repulsive side and flattens to
% the dissociation asymptote D_e; its vibrational levels are spaced by omega_e at
% the bottom but converge as v grows, ending at the dissociation limit.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,-0.3) -- (0,4.6) node[left] {$U(r)$};
  \draw[->, black] (0,0) -- (7.4,0) node[below] {$r$};
  % Morse curve: De=3.6, a via factor 0.9, r0 at x=1.5
  \draw[acc, very thick, domain=0.72:7.1, samples=160, variable=\x]
    plot ({\x},{3.6*(1 - exp(-(\x-1.5)*0.62))*(1 - exp(-(\x-1.5)*0.62))});
  % dissociation asymptote
  \draw[black, dashed] (0,3.6) -- (7.1,3.6);
  \node[black, anchor=west, font=\scriptsize] at (5.4,3.75) {dissociation limit};
  % converging levels: heights approach 3.6
  \foreach \y/\xa/\xb in {0.55/1.02/2.32, 1.35/0.95/2.9, 2.0/0.9/3.55, 2.5/0.87/4.2, 2.86/0.85/4.85, 3.12/0.84/5.5, 3.3/0.83/6.1}{
    \draw[black] (\xa,\y) -- (\xb,\y);
  }
  \node[acc, anchor=west] at (4.0,1.3) {Morse potential};
  \draw[black, <->] (3.7,0.02) -- (3.7,3.58);
  \node[black, anchor=west, font=\scriptsize] at (3.75,1.9) {$D_e$};
\end{tikzpicture}
$$

The spacing between adjacent levels is the first difference of the term values,

$$
\Delta G_{v+1/2} \equiv G(v+1) - G(v) = \omega_e - 2\omega_e x_e (v + 1),
$$

a straight line that decreases with $v$. The ladder terminates when the spacing
reaches zero, at the largest bound quantum number

$$
v_{\max} = \frac{\omega_e}{2\omega_e x_e} - 1 = \frac{2 D_e/hc}{\omega_e} - 1.
$$

Beyond $v_{\max}$ the molecule is unbound. Summing the spacings up to that point,
or equivalently taking the area under the $\Delta G$ line, gives the dissociation
energy measured from the ground vibrational level,

$$
D_0 = D_e - G(0) = \sum_{v=0}^{v_{\max}-1} \Delta G_{v+1/2}.
$$

> **Definition (Spectroscopic vs true dissociation energy).** $D_e$ is the depth
> of the potential well measured from its minimum; $D_0$ is measured from the
> zero-point level $G(0) = \tfrac12\omega_e - \tfrac14\omega_e x_e$. The
> difference is the zero-point energy, so $D_0 < D_e$ always, and it is $D_0$ that
> a bond-breaking experiment reports.

The linearity of $\Delta G$ is the basis of the **Birge-Sponer extrapolation**:
plot the measured level spacings against $v$, fit a straight line, and read off
both $\omega_e$ (the intercept) and $\omega_e x_e$ (half the slope). The total
area under the line, out to where it crosses zero, is $D_0$. Real potentials fall
below the Morse form near dissociation, so the linear extrapolation slightly
overestimates $D_0$, but it fixes the constants from just the first few overtone
bands.

$$
% caption: Birge-Sponer plot: the vibrational level spacing falls linearly with v,
% the intercept gives omega_e and the slope the anharmonicity; the shaded area out
% to the zero crossing equals the dissociation energy D_0.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (7.6,0) node[below] {$v$};
  \draw[->, black] (0,0) -- (0,4.2) node[left] {spacing};
  % line from (0,3.7) to (6.8,0)
  \fill[acc!12] (0,0) -- (0,3.7) -- (6.8,0) -- cycle;
  \draw[acc, very thick] (0,3.7) -- (6.8,0);
  % measured points
  \foreach \x/\y in {0.3/3.54,1.3/3.0,2.3/2.46,3.3/1.92}
    \fill[black] (\x,\y) circle (1.6pt);
  \node[black, anchor=west, font=\scriptsize] at (0.15,0.55) {area $=D_0$};
  \node[acc, anchor=south west, font=\scriptsize] at (3.6,1.6) {straight line};
  \node[black, anchor=north, font=\scriptsize] at (6.8,-0.08) {$v_{\max}$};
\end{tikzpicture}
$$

**Worked example — dissociation energy of H₂ from its constants.** For the
hydrogen molecule $\omega_e = 4401\ \text{cm}^{-1}$ and $\omega_e x_e = 121\
\text{cm}^{-1}$. The largest bound level is $v_{\max} = 4401/(2\cdot 121) - 1
\approx 17$, and the depth is $D_e = \omega_e^2/(4\omega_e x_e) = 4401^2/(484)
\approx 4.00\times10^4\ \text{cm}^{-1}$, or $4.96\ \text{eV}$ using $1\ \text{eV}
= 8065\ \text{cm}^{-1}$. Subtracting the zero-point term $G(0) = 2170\
\text{cm}^{-1}$ gives $D_0 \approx 3.78\times10^4\ \text{cm}^{-1} = 4.69\
\text{eV}$, within a few percent of the measured $4.48\ \text{eV}$; the small
excess is the Morse overestimate near dissociation.

## Centrifugal distortion of the rotor

A rotating molecule is not perfectly rigid. As it spins, the bond stretches under
the centrifugal load until the restoring force balances it, so the moment of
inertia grows with the rotational quantum number and the levels fall below the
rigid-rotor prediction. The radial motion sees an **effective potential** that
adds the rotational barrier to the vibrational well,

$$
U_{\text{eff}}(r) = U(r) + \frac{\hbar^2 J(J+1)}{2\mu r^2},
$$

whose minimum shifts outward from $r_0$ as $J$ increases.

$$
% caption: The rotational term hbar^2 J(J+1)/2mu r^2 adds to the vibrational well,
% pushing the minimum of the effective potential outward; the bond stretches and
% the moment of inertia grows with J.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,-0.3) -- (0,4.4) node[left] {$U$};
  \draw[->, black] (0,0) -- (6.6,0) node[below] {$r$};
  % base vibrational well (J=0)
  \draw[black, very thick, domain=0.95:5.9, samples=140, variable=\x]
    plot ({\x},{2.6*(1 - exp(-(\x-1.7)*0.7))*(1 - exp(-(\x-1.7)*0.7))});
  \node[black, anchor=west, font=\scriptsize] at (4.6,1.7) {$J=0$};
  % effective well with centrifugal term (adds c/x^2)
  \draw[acc, very thick, domain=1.02:5.9, samples=160, variable=\x]
    plot ({\x},{2.6*(1 - exp(-(\x-1.7)*0.7))*(1 - exp(-(\x-1.7)*0.7)) + 1.6/(\x*\x)});
  \node[acc, anchor=west, font=\scriptsize] at (4.4,3.0) {large $J$};
  % markers of the two minima
  \fill[black] (1.7,0.0) circle (1.6pt);
  \fill[acc] (2.15,0.36) circle (1.8pt);
  \draw[black, dashed] (1.7,0) -- (1.7,-0.18);
  \node[black, anchor=north east, font=\scriptsize] at (1.74,-0.16) {$r_0$};
  \draw[black, dashed] (2.15,0.36) -- (2.15,-0.18);
  \node[black, anchor=north west, font=\scriptsize] at (2.12,-0.16) {stretched};
\end{tikzpicture}
$$

Perturbing the rigid rotor to first order in the stretch gives the rotational
term with a quartic correction,

$$
F(J) = B_v\, J(J+1) - D_J\, [J(J+1)]^2,
\qquad B_v = \frac{\hbar}{4\pi c\,\mu r_v^2},
$$

where $D_J$ is the **centrifugal distortion constant** (positive, so it lowers
the levels). The Kratzer relation ties $D_J$ to the vibrational frequency,

$$
D_J = \frac{4 B_e^3}{\omega_e^2},
$$

so a stiffer bond (larger $\omega_e$) distorts less. For a light hydride $D_J$ is
of order $10^{-3}\ \text{cm}^{-1}$ against a rotational constant $B$ of order
$10\ \text{cm}^{-1}$, a part in $10^4$ per unit of $J(J+1)$ — negligible at low
$J$, but growing as $J^4$ it becomes visible by $J \sim 20$, where it pulls the
high-$J$ lines closer together.

## Vibration-rotation coupling

The rotational constant is not a single number but depends on the vibrational
state, because a higher vibrational level has a larger mean-square bond length and
therefore a larger moment of inertia. Averaging $1/r^2$ over the anharmonic
vibrational wavefunction gives a linear dependence,

$$
B_v = B_e - \alpha_e\left(v + \tfrac12\right),
$$

where $B_e$ is the constant at the potential minimum and $\alpha_e > 0$ the
**vibration-rotation coupling constant**, typically a percent of $B_e$. The
constant governing an observed band is not $B_e$ but the pair $B_0$ and $B_1$ for
the two vibrational levels involved.

This coupling reshapes the P and R branches. In the fundamental $v: 0 \to 1$
band with $\Delta J = \pm 1$, the line positions relative to the band origin
$\nu_0$ are

$$
\nu_R(J) = \nu_0 + (B_1 + B_0)(J+1) + (B_1 - B_0)(J+1)^2, \qquad J = 0,1,2,\ldots
$$

$$
\nu_P(J) = \nu_0 - (B_1 + B_0)\,J + (B_1 - B_0)\,J^2, \qquad J = 1,2,3,\ldots
$$

Because $B_1 < B_0$ the quadratic coefficient $B_1 - B_0$ is negative. In the R
branch it opposes the linear growth, so the lines spread more slowly, stop, and
turn back at a **band head**; in the P branch it reinforces the spreading, so the
P lines fan out ever wider. The single combination-difference
$\nu_R(J-1) - \nu_P(J+1) = 4B_v(J + \tfrac12)$ isolates one vibrational level's
rotational constant from the band alone, so $B_0$ and $B_1$, and thus $\alpha_e$,
are extracted without knowing $\nu_0$.

## The isotope effect

Replacing an atom by a heavier isotope changes the reduced mass but not the
electronic potential $U(r)$, since the potential is set by the electrons. Every
constant that depends on $\mu$ therefore shifts by a predictable ratio. Writing
$\rho = \sqrt{\mu/\mu'}$ for the two isotopologues,

- vibrational frequency $\omega_e \propto \mu^{-1/2}$, so $\omega_e' = \rho\,\omega_e$;
- anharmonicity $\omega_e x_e \propto \mu^{-1}$, so $\omega_e' x_e' = \rho^2\,\omega_e x_e$;
- rotational constant $B_e \propto \mu^{-1}$, so $B_e' = \rho^2 B_e$;
- coupling $\alpha_e \propto \mu^{-3/2}$, so $\alpha_e' = \rho^3 \alpha_e$.

The heavier isotopologue has the lower frequencies and the tighter rotational
spacing. For hydrogen versus deuterium the effect is large: the reduced mass of
$\text{H}_2$ is $0.504\ \text{u}$ and of $\text{D}_2$ is $1.007\ \text{u}$, so
$\rho = \sqrt{0.504/1.007} = 0.707$. The deuterium vibrational frequency is
$0.707$ of hydrogen's, and its rotational lines are spaced at half the interval.

$$
% caption: Isotope shift of a vibration band: substituting deuterium for hydrogen
% halves the reduced-mass ratio, so the heavier band origin sits at rho times the
% lighter frequency and its rotational lines pack more closely.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (10.6,0) node[below] {frequency};
  % light isotopologue band, origin near 7.2, wide spacing
  \foreach \x in {5.4,6.0,6.6}
    \draw[acc, very thick] (\x,0) -- (\x,1.5);
  \draw[black, dashed] (7.2,0) -- (7.2,1.8);
  \foreach \x in {7.8,8.4,9.0}
    \draw[acc, very thick] (\x,0) -- (\x,1.5);
  \node[acc, anchor=south] at (7.2,1.85) {light (H)};
  % heavy isotopologue band, origin near 4.0, tighter spacing
  \foreach \x in {2.9,3.3,3.7}
    \draw[black, very thick] (\x,0) -- (\x,1.1);
  \draw[black, dashed] (4.0,0) -- (4.0,1.4);
  \foreach \x in {4.3,4.7,5.1}
    \draw[black, very thick] (\x,0) -- (\x,1.1);
  \node[black, anchor=south] at (4.0,1.45) {heavy (D)};
\end{tikzpicture}
$$

The isotope shift of the zero-point energy has a chemical consequence: because
$G(0) \propto \omega_e \propto \mu^{-1/2}$, a C-D bond sits deeper in its well than
a C-H bond by the difference in zero-point energy, which is why deuterated bonds
break more slowly. Spectroscopically, resolving both isotopologues in a natural
sample fixes the atomic mass ratio from the band positions alone.

## Thermal population and the band envelope

The intensity of a rotational line is proportional to the population of its lower
level, which is the degeneracy $2J+1$ times the
[Boltzmann factor](/statistical-mechanics/foundations/classical-statistics-and-equipartition),

$$
N_J \propto (2J+1)\, e^{-hcB\,J(J+1)/k_B T}.
$$

The degeneracy raises the low levels and the exponential suppresses the high
ones, so the population peaks at an intermediate $J$. Setting $\d N_J/\d J = 0$
gives the most populated level,

$$
J_{\max} = \sqrt{\frac{k_B T}{2 hc B}} - \frac12,
$$

which for a typical light molecule at room temperature is near $J \approx 7$-$10$.
The band intensity therefore rises from the origin, reaches a maximum a few lines
out in each branch, and falls off again, tracing a characteristic double-humped
**envelope** over the P and R branches.

$$
% caption: The thermal envelope of a rovibrational band: line intensities follow
% the (2J+1) Boltzmann population, rising to a peak a few lines from the origin in
% each branch and decaying at high J, giving the two-humped band contour.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (11.0,0) node[below] {frequency};
  \draw[->, black] (0,0) -- (0,3.4) node[left] {intensity};
  % P branch heights (rise then fall), left of gap at 5.6
  \foreach \x/\h in {1.0/0.6,1.7/1.3,2.4/2.0,3.1/2.5,3.8/2.3,4.5/1.6}
    \draw[acc, very thick] (\x,0) -- (\x,\h);
  \draw[black, dashed] (5.6,0) -- (5.6,2.9) node[above, font=\scriptsize] {origin};
  % R branch heights
  \foreach \x/\h in {6.7/1.6,7.4/2.3,8.1/2.5,8.8/2.0,9.5/1.3,10.2/0.6}
    \draw[acc, very thick] (\x,0) -- (\x,\h);
  % dashed envelope
  \draw[black, densely dotted] (1.0,0.6) .. controls (2.6,3.0) and (4.0,2.9) .. (4.5,1.6);
  \draw[black, densely dotted] (6.7,1.6) .. controls (7.2,2.9) and (8.6,3.0) .. (10.2,0.6);
  \node[acc, anchor=north, font=\scriptsize] at (2.8,-0.1) {P branch};
  \node[acc, anchor=north, font=\scriptsize] at (8.4,-0.1) {R branch};
\end{tikzpicture}
$$

The envelope is a thermometer. Because $J_{\max}$ grows as $\sqrt{T}$, the
separation of the two intensity peaks widens with temperature in a known way, so
fitting the contour of an unresolved band returns the gas temperature. In
astrophysical and combustion spectra, where individual lines may blur together,
the band shape alone measures both the rotational constant and the temperature of
the emitting gas.

With the anharmonic, centrifugally distorted, temperature-broadened band in hand,
the [next lesson](/condensed-matter/molecular-spectra/raman-and-electronic-bands)
turns to the transitions the infrared dipole rule forbids, reached instead by
scattering and by electronic excitation.