Anharmonicity and Overtones

Take HCl gas and shine infrared light through it. The fundamental absorption near 2886 cm⁻¹ is strong and obvious. But look carefully at higher energy, near 5668 cm⁻¹, and a much weaker band appears — an overtone. A perfect harmonic oscillator forbids this transition entirely. Its existence tells us bonds are not perfect springs. Real chemical bonds resist compression far more than they resist stretching, and they eventually break. The harmonic oscillator model, built on The Harmonic Oscillator and Vibrational Levels, cannot describe bond dissociation because a parabolic potential never flattens out. Anharmonicity is the correction that lets vibrational spectroscopy actually measure bond strength. This article builds the anharmonic oscillator from the Morse potential upward. You will see how energy levels converge toward a dissociation limit. You will learn where overtones and hot bands come from. You will extract real dissociation energies from real spectra, using HCl and CO as worked examples.

Key takeaways

Real potential energy curves are asymmetric and flatten at large bond length, unlike the harmonic parabola.
The anharmonicity constant $x_e$ quantifies how vibrational levels crowd together as $v$ increases.
Overtones ( $\Delta v = 2, 3, \ldots$ ) are only observable because anharmonicity relaxes the strict harmonic selection rule.
The Birge–Sponer extrapolation turns a series of vibrational spacings into a dissociation energy.
Hot bands arise from transitions starting in $v = 1$ or higher, visible only when thermal population allows.
Key idea: Anharmonicity bends the vibrational ladder so that levels converge toward dissociation, and that same distortion is what makes overtones spectroscopically visible.

The core insight

A real bond's potential energy curve is not a symmetric parabola but a Morse-like curve that softens on stretching and stiffens on compression, so its quantized vibrational levels crowd closer together as $v$ increases, converge at the dissociation limit, and — because the wavefunctions are no longer purely harmonic — permit weak $\Delta v = 2, 3, \ldots$ overtone transitions that a strict harmonic oscillator would forbid.

The physical picture

In the harmonic approximation, the potential energy is $V(x) = \tfrac{1}{2}kx^2$ , a perfectly symmetric parabola. Its eigenvalues,

E_v = \left(v + \tfrac{1}{2}\right)h\nu, \qquad v = 0,1,2,\ldots

are evenly spaced forever — an infinite ladder with no top. That cannot be physically right: enough vibrational energy must eventually break the bond. The harmonic model is only the lowest-order approximation to the true potential near its minimum. The Morse potential fixes this while staying analytically solvable:

V(r) = D_e\left[1 - e^{-a(r-r_e)}\right]^2

Here $D_e$ is the depth of the well measured from the minimum, $r_e$ is the equilibrium bond length, and $a$ controls the curvature (bond stiffness). Unlike the parabola, the Morse curve rises steeply for compression ( $r < r_e$ ) but flattens to a plateau at $D_e$ for large $r$ , correctly describing a bond that dissociates. Expanding $V(r)$ in a Taylor series around $r_e$ reproduces the harmonic term $\tfrac{1}{2}kx^2$ plus cubic and quartic corrections — anharmonicity is literally the next term in that expansion. Solving the Schrödinger equation for the Morse potential gives quantized levels that are no longer evenly spaced:

G(v) = \tilde\nu_e\left(v+\tfrac12\right) - \tilde\nu_e x_e\left(v+\tfrac12\right)^2

with $x_e$ a small positive number (the anharmonicity constant), typically $0.01$ – $0.03$ . As $v$ increases, the negative quadratic term grows and eats into the spacing, so successive levels get closer together — the ladder converges rather than continuing uniformly, exactly as the flattening Morse curve demands.

Figure

Morse potential versus harmonic parabola, showing converging vibrational levels

The second figure shows why overtones matter experimentally: in a harmonic oscillator the selection rule $\Delta v = \pm 1$ is exact, so only the fundamental (0→1) absorbs. Anharmonic mixing of wavefunctions weakens that rule, allowing weak $\Delta v = 2$ (first overtone), $\Delta v = 3$ (second overtone), and hot bands starting from thermally populated $v=1$ .

Figure

Stick spectrum: fundamental, overtones, and a hot band for HCl-like molecule

The core relations

V(r) = D_e\left[1 - e^{-a(r-r_e)}\right]^2

where: $D_e$ is well depth from the minimum, $a$ sets curvature, $r_e$ is equilibrium length.

G(v) = \tilde\nu_e\left(v+\tfrac12\right) - \tilde\nu_e x_e\left(v+\tfrac12\right)^2

where: $\tilde\nu_e$ is the harmonic wavenumber, $x_e$ the (dimensionless) anharmonicity constant.

\Delta G_{v+\tfrac12} = G(v+1) - G(v) = \tilde\nu_e - 2\tilde\nu_e x_e (v+1)

where: this is the observed spacing between adjacent levels, linear in $v$ .

D_0 = \sum_{v=0}^{v_{\max}} \Delta G_{v+\tfrac12} \approx \frac{\tilde\nu_e}{4x_e} - \frac{\tilde\nu_e}{2}

where: $D_0$ is the dissociation energy from the ground vibrational state (Birge–Sponer sum/extrapolation).

D_e = D_0 + \tfrac12 h c\,\tilde\nu_e

where: $D_e$ is the well depth relative to the potential minimum, the zero-point correction bridges $D_0$ and $D_e$ .

a = \tilde\nu_e\left(2\pi^2 c\mu/D_e\right)^{1/2}

where: $\mu$ is reduced mass, $c$ speed of light; this links Morse curvature to observed spectroscopic constants.

Derivation AFrom the Morse potential to the Birge–Sponer dissociation energy

Goal: show how measured line spacings $\Delta G_{v+\tfrac12}$ yield the dissociation energy $D_0$ without ever solving the Schrödinger equation exactly.

Step 1. Start from the anharmonic term series. Solving the vibrational Schrödinger equation for the Morse potential exactly gives quantized terms

G(v) = \tilde\nu_e\left(v+\tfrac12\right) - \tilde\nu_e x_e\left(v+\tfrac12\right)^2

Because $x_e > 0$ and small, the quadratic term is a small negative correction that grows with $v$ , bending the ladder downward relative to the harmonic case.

Step 2. Take the difference between successive levels. The energy gap between $v$ and $v+1$ is what a spectrometer actually measures as a series of absorption lines:

\Delta G_{v+\tfrac12} = G(v+1)-G(v)

= \tilde\nu_e\left[(v+\tfrac32) - (v+\tfrac12)\right] - \tilde\nu_e x_e\left[(v+\tfrac32)^2 - (v+\tfrac12)^2\right]

= \tilde\nu_e - \tilde\nu_e x_e (2v+2)

\Delta G_{v+\tfrac12} = \tilde\nu_e - 2\tilde\nu_e x_e(v+1)

This is a straight line if you plot $\Delta G_{v+\tfrac12}$ against $(v+1)$ : intercept $\tilde\nu_e$ , slope $-2\tilde\nu_e x_e$ .

Step 3. Recognize the physical meaning of the plot's zero-crossing. Dissociation occurs at the $v_{\max}$ where adjacent levels merge, i.e. $\Delta G_{v+\tfrac12} \to 0$ . Beyond that point the "vibrational levels" become a continuum — the atoms are unbound. Setting the linear expression to zero locates this limit:

0 = \tilde\nu_e - 2\tilde\nu_e x_e(v_{\max}+1) \;\Rightarrow\; v_{\max}+1 = \frac{1}{2x_e}

Step 4. Sum all the spacings up to that limit. The total energy needed to climb from $v=0$ to dissociation is the sum of every rung, which — because the plot is linear — equals the area under the $\Delta G$ vs. $v$ line from $v=0$ to $v_{\max}$ :

D_0 = \sum_{v=0}^{v_{\max}} \Delta G_{v+\tfrac12}

Evaluating this sum (or equivalently integrating the linear trend and using the intercept/slope from Step 2) gives the compact working formula:

D_0 \approx \frac{\tilde\nu_e}{4x_e} - \frac{\tilde\nu_e}{2}

In practice, chemists rarely trust the extrapolated tail of the plot (real levels deviate from strict linearity near dissociation), so $D_0$ is most reliably obtained from the graphical area under measured $\Delta G$ points, extrapolated only over the last, most uncertain stretch.

Step 5. Connect to the well depth. Since $D_0$ is measured from the zero-point level ( $v=0$ ), while $D_e$ is measured from the true potential minimum, the correction is exactly the zero-point energy:

\boxed{D_e = D_0 + \tfrac12 h c\,\tilde\nu_e}

This closes the loop: from a handful of measured line positions, you obtain $\tilde\nu_e$ , $x_e$ , then $D_0$ , then the fundamental bond dissociation energy $D_e$ .

Worked examples

Example 1 — Foundational: locating the anharmonicity constant and overtone of HCl. The fundamental absorption of ¹H³⁵Cl is observed at $\tilde\nu_{0\to1} = 2886$ cm⁻¹, and the first overtone at $\tilde\nu_{0\to2} = 5668$ cm⁻¹. Find $\tilde\nu_e$ and $x_e$ . Using the term expression with $v=0$ as origin:

\tilde\nu_{0\to1} = G(1)-G(0) = \tilde\nu_e - 2\tilde\nu_e x_e

\tilde\nu_{0\to2} = G(2)-G(0) = 2\tilde\nu_e - 6\tilde\nu_e x_e

Two equations, two unknowns. Multiply the first by 2:

2\tilde\nu_{0\to1} = 2\tilde\nu_e - 4\tilde\nu_e x_e = 5772 \text{ cm}^{-1}

Subtract the second equation:

2\tilde\nu_{0\to1} - \tilde\nu_{0\to2} = (2\tilde\nu_e - 4\tilde\nu_e x_e) - (2\tilde\nu_e - 6\tilde\nu_e x_e) = 2\tilde\nu_e x_e

2\tilde\nu_e x_e = 5772 - 5668 = 104 \text{ cm}^{-1}

So $\tilde\nu_e x_e = 52$ cm⁻¹. Substitute back into the first equation:

\tilde\nu_e = 2886 + 2(52) = 2886 + 104 = 2990 \text{ cm}^{-1}

x_e = \frac{52}{2990} = 0.0174

Sanity check: literature values for H³⁵Cl are $\tilde\nu_e = 2990.9$ cm⁻¹ and $x_e = 0.0174$ — an excellent match. The small, positive, dimensionless $x_e$ confirms weak anharmonicity, consistent with HCl being a stiff, strongly bound diatomic.

Example 2 — Applied: Birge–Sponer dissociation energy of CO. Carbon monoxide is one of the most strongly bound diatomics known, useful for testing the Birge–Sponer method at a large $D_0$ . Spectroscopic constants for CO are $\tilde\nu_e = 2170.2$ cm⁻¹ and $x_e = 0.00612$ (i.e. $\tilde\nu_e x_e = 13.3$ cm⁻¹).

Step 1 — dissociation vibrational quantum number.

v_{\max} + 1 = \frac{1}{2x_e} = \frac{1}{2(0.00612)} = 81.7

Step 2 — Birge–Sponer sum for $D_0$ .

D_0 \approx \frac{\tilde\nu_e}{4x_e} - \frac{\tilde\nu_e}{2}

\frac{\tilde\nu_e}{4x_e} = \frac{2170.2}{4(0.00612)} = \frac{2170.2}{0.02448} = 88{,}652 \text{ cm}^{-1}

D_0 \approx 88{,}652 - \frac{2170.2}{2} = 88{,}652 - 1085 = 87{,}567 \text{ cm}^{-1}

Step 3 — convert to molar energy. Using $1$ cm⁻¹ corresponds to $hc \tilde\nu \times N_A = (6.626\times10^{-34}\,\text{J s})(2.998\times10^{10}\,\text{cm s}^{-1})(6.022\times10^{23}\,\text{mol}^{-1}) = 11.96$ J mol⁻¹ per cm⁻¹:

D_0 = 87{,}567 \text{ cm}^{-1} \times 11.96 \text{ J mol}^{-1}\text{cm} = 1.047\times10^{6} \text{ J mol}^{-1} = 1047 \text{ kJ mol}^{-1}

Step 4 — well depth $D_e$ .

D_e = D_0 + \tfrac12 hc\tilde\nu_e = 87{,}567 + 1085 = 88{,}652 \text{ cm}^{-1} = 1060 \text{ kJ mol}^{-1}

Sanity check: the experimental bond dissociation energy of CO is about $1071$ kJ mol⁻¹, among the strongest known diatomic bonds. Our Birge–Sponer estimate (1047–1060 kJ mol⁻¹) is close but slightly low — a well-known feature of the method, since real levels curve away from the idealized straight-line extrapolation near dissociation, making the simple sum a systematic underestimate of a few percent.

Limiting cases & common traps

Situation	What happens	Common mistake
$x_e \to 0$	Recovers exact harmonic oscillator, evenly spaced levels, no overtones allowed	Forgetting that overtones vanish exactly in this limit, not just weaken
Low $v$ (near bottom of well)	Morse potential well approximated by parabola; harmonic formula nearly exact	Using $x_e$ -corrected formula where harmonic suffices, adding needless complexity
$v \to v_{\max}$	Levels crowd together; spacing $\Delta G \to 0$ ; continuum begins	Assuming the vibrational ladder is infinite, as in strict harmonic treatment
Room-temperature IR spectrum	Hot bands from $v=1$ are weak but visible for low-frequency modes; negligible for stiff, high-frequency bonds like C–H	Confusing a hot band (starts above $v=0$ ) with an overtone (starts at $v=0$ , $\Delta v \geq 2$ )
Overtone intensity	Weak, roughly 10–100× less intense than fundamental	Assuming overtones are forbidden outright, rather than merely weak

The chemist behind it

The dissociation-energy extrapolation method used throughout this article is named for Raymond Thayer Birge and Hertha Sponer, who in 1926 showed that a simple plot of vibrational term differences against quantum number could be extrapolated to extract bond dissociation energies directly from band spectra — years before anyone could measure such energies by calorimetry for many diatomic gases. Their method turned routine absorption and emission spectra into a quantitative tool for thermochemistry, linking spectroscopy directly to bond energetics.

Origins · 1 minRaymond Thayer Birge and how this was discovered

Raymond Thayer Birge

1887–1980 · American physicist who, with Hertha Sponer, developed the extrapolation method for obtaining dissociation energies from vibrational spectra, and who was also a leading figure in establishing precise values of fundamental physical constants

Ask the tutor: How does anharmonicity change when rotation is added, and what do the resulting P and R branch spacings reveal about bond length?

Why is x_e almost always positive?

A positive $x_e$ reflects the physical fact that real bonds soften on stretching and cannot be compressed indefinitely — the Morse potential's asymmetric shape. This asymmetry makes the quadratic correction term negative in the term expression, which is written with an explicit minus sign, so $x_e$ itself comes out positive for essentially all normal chemical bonds. A negative $x_e$ would imply levels spreading apart with $v$ , which is unphysical for a bound potential approaching a finite dissociation limit.

How is D_e different from D_0, and why does it matter?

$D_e$ is the well depth measured from the bottom of the potential curve, a purely theoretical reference point. $D_0$ is the energy actually needed to dissociate the molecule starting from its real ground vibrational state, $v=0$ , which sits above the bottom by the zero-point energy $\tfrac12 hc\tilde\nu_e$ . Experimentally measured bond dissociation energies (from thermochemistry or spectroscopy) correspond to $D_0$ , so converting between the two using the zero-point correction is essential for comparing spectroscopic and thermochemical bond energies.

Why does the Birge–Sponer method tend to underestimate D0?

The method assumes $\Delta G_{v+\tfrac12}$ decreases perfectly linearly with $v$ , based on truncating the Morse expansion at the quadratic anharmonicity term. Near the dissociation limit, higher-order anharmonic terms (cubic, quartic in the Dunham expansion) become important and the real curve bends away from that straight line, typically dropping faster. Summing the idealized linear trend overestimates the number of accessible levels beyond where data exist, but overall the truncated linear extrapolation usually slightly underestimates the true $D_0$ found by direct thermochemical measurement.

Are overtone bands useful for anything beyond confirming anharmonicity?

Yes — overtone spectroscopy is the basis of near-infrared (NIR) analytical chemistry, widely used for rapid, nondestructive analysis of water, C–H, N–H, and O–H content in food, pharmaceuticals, and agriculture. Because overtone intensities are weak, samples do not need to be diluted as they would for strong fundamental IR absorption, allowing direct measurement of bulk or turbid samples. The overtone wavenumbers also give an independent route to $\tilde\nu_e$ and $x_e$ without needing the fundamental transition at all.

How does anharmonicity connect to what comes after vibrational levels?

Once you accept that vibrational levels are unevenly spaced, the next natural question is how rotation modifies each vibrational level — leading to vibration-rotation coupling and the fine rotational structure (P and R branches) seen within each vibrational band. Anharmonicity and rotation both perturb the simple picture in complementary ways, and real high-resolution spectra require both corrections simultaneously.