Anharmonicity and the Morse Potential

Modern Fourier-transform infrared and Raman spectrometers routinely resolve overtone and combination bands to fractions of a wavenumber, and anharmonicity constants extracted from these spectra feed directly into force-field and potential-energy-surface refinement for computational chemistry — a calculated harmonic frequency from density functional theory is systematically scaled or explicitly anharmonically corrected (VPT2, vibrational perturbation theory to second order) precisely because raw harmonic DFT frequencies overestimate observed fundamentals by an amount closely related to $\omega_e x_e$. Atmospheric and combustion chemistry rely on Morse-type anharmonic data for CO$_2$, CO, H$_2$O, and OH radical bands to model infrared absorption accurately across the wide temperature ranges found in flames, engines, and planetary atmospheres, where hot bands are not a curiosity but a dominant spectral feature. In astrochemistry, the anharmonicity constants of simple diatomics and small hydrides measured in cold interstellar clouds versus hot circumstellar envelopes are used as thermometers, since hot-band intensities directly encode the local vibrational temperature. Pharmaceutical and materials laboratories use overtone spectroscopy (near-infrared, where overtones and combination bands of C–H, O–H, and N–H stretches dominate) for rapid, nondestructive quantitative analysis, and the quantitative shift of these overtones below exact harmonic multiples is a direct, routinely-exploited consequence of the anharmonicity discussed here. Finally, the same convergence-toward-dissociation picture underlies the vibrational envelopes seen in Electronic Transitions and Franck-Condon spectra, where anharmonic vibrational wavefunctions in both electronic states control which vibronic transitions carry intensity, and it underlies the fine rotational structure analyzed in Vibration-Rotation Spectra: P, Q, R Branches, where vibration–rotation coupling constants are themselves consequences of the anharmonic, non-rigid nature of the real potential curve.

Ask the tutor: How would the Birge–Sponer extrapolation change if a third-order anharmonicity constant ω_e y_e were included in the level formula?

Picture the potential energy of a diatomic molecule as a function of internuclear distance $r$. Near the equilibrium separation $r_e$, any smooth potential looks locally like a parabola — this is simply the statement that the first derivative vanishes at a minimum and the leading nonzero term in a Taylor series is quadratic. This is exactly the justification for the harmonic oscillator model used for small-amplitude vibrations. But a parabola rises without bound as $r\to\infty$, which is unphysical: real bonds have a finite dissociation energy $D_e$, beyond which the atoms simply separate and the potential becomes flat. The true potential curve is therefore asymmetric. It rises steeply on the compression side (nuclei and electron clouds pushing into each other cost much more energy than the same displacement in the stretching direction), and it rises only gently, then plateaus, on the stretching side. This asymmetry has three consequences that dominate real vibrational spectra. First, the vibrational levels are not evenly spaced — they crowd together as $v$ increases, converging on the dissociation limit. Second, the average bond length increases with vibrational quantum number (because the potential is "softer" on the outside), which shows up as a temperature-dependent effective bond length in rotational spectra. Third, the wavefunctions become skewed toward large $r$ for higher $v$, changing intensity patterns for hot bands and overtones. The Morse function is the simplest analytic curve that captures all of this while remaining exactly solvable by the same operator methods used for the harmonic oscillator. It is built from three physically motivated ingredients: a finite well depth $D_e$, a curvature parameter $a$ that controls how sharply the well is shaped (and is fixed by matching the second derivative at the minimum to the harmonic force constant $k$), and the correct qualitative shape — steep on the inner wall, flat on the outer wall, exactly one minimum.

Notice how the green level spacings shrink as $v$ increases — that convergence is the direct visual signature of anharmonicity, and it is completely absent in the dashed harmonic parabola, whose levels (not drawn, but implied) would stay evenly spaced all the way up.

The Morse potential itself is defined as

where:

• $D_e$ is the well depth measured from the potential minimum (not from $v=0$),
• $r_e$ is the equilibrium bond length,
• $a$ is a range parameter (units of inverse length) controlling the width of the well,
• $r$ is the internuclear separation.

Reading: at $r = r_e$, $V=0$ (the minimum); as $r\to\infty$, $V\to D_e$ (dissociation limit, flat); as $r\to 0$, $V\to D_e(1-e^{ar_e})^2$, a large finite number — the Morse potential is actually too soft at very short range compared to real repulsive walls, but this defect rarely matters for the low-lying levels chemists probe. The curvature parameter is fixed by matching the harmonic force constant:

where:

• $k$ is the harmonic force constant obtained from $\left.\dfrac{d^2V}{dr^2}\right|_{r_e}$,
• $\mu$ is the reduced mass of the diatomic,
• $\omega_e$ is the harmonic vibrational wavenumber (cm$^{-1}$) that would be observed if the well were purely quadratic.

Reading: $a$ is not an independent free parameter once $D_e$ and $\omega_e$ (equivalently $k$) are known — the shape of the well near the bottom is tied to the same force constant that governs the harmonic oscillator, while $D_e$ controls how far out the well extends before flattening. The exact bound-state vibrational term values (in wavenumbers) are

where:

• $v = 0, 1, 2, \dots$ is the vibrational quantum number, restricted to a finite set of values,
• $\omega_e$ is the harmonic wavenumber (the spacing the levels would have at the very bottom of the well),
• $\omega_e x_e$ is the anharmonicity constant (also in cm$^{-1}$), always positive for a Morse-like well, and related to $\omega_e$ and $D_e$ by

Reading: the correction term is quadratic in $(v+\tfrac12)$, so successive level spacings decrease linearly with $v$ — this single fact is what makes the Birge–Sponer plot a straight line. The spacing between adjacent levels — the quantity directly measured in a spectrum as a "hot band" or overtone spacing — is

where:

• $\Delta G_{v+1/2}$ is the energy gap between levels $v$ and $v+1$, indexed at the half-integer midpoint by convention,
• the right-hand side is linear in $(v+1)$, decreasing steadily as $v$ increases.

Reading: plotting $\Delta G_{v+1/2}$ against $(v+1)$ gives a straight line of slope $-2\omega_e x_e$ and intercept $\omega_e$ — exactly the Birge–Sponer extrapolation used to estimate dissociation energies from a handful of measured overtone spacings.

Example 1 — Foundational: level spacings and dissociation for a textbook diatomic. Consider a diatomic (asymmetric numbers, not $^1$H$^{35}$Cl exactly, to keep the arithmetic transparent but realistic) with

(these are, in fact, close to real CO values, chosen because CO is a workhorse in vibrational spectroscopy). Step 1: Zero-point energy.

Step 2: Fundamental transition $v=0\to1$.

This is the observed IR fundamental — already 26.6 cm$^{-1}$ below the harmonic value $\omega_e$, a directly measurable anharmonic shift. Step 3: First overtone $v=0\to2$.

Compare to twice the fundamental: $2\times 2117.12 = 4234.24\ \text{cm}^{-1}$. The overtone falls 26.6 cm$^{-1}$ below twice the fundamental — the signature "less than double" pattern of real overtones, not an artifact of rounding. Step 4: Limiting case check. As $\omega_e x_e \to 0$ (approaching the pure harmonic limit), $\Delta G_{1/2}\to \omega_e$ and the overtone $\to 2\omega_e$ exactly — recovering the evenly-spaced harmonic ladder, confirming the formula collapses correctly onto the known limiting behavior. Step 5: Maximum vibrational quantum number and dissociation energy estimate.

Converting to more familiar units ($1$ cm$^{-1}$ = $0.01196$ kJ/mol):

This overshoots the true $D_0$ of CO (about $1072$ kJ/mol from $D_e$, actually — the Morse model is not perfect but lands in the right ballpark), which is the expected behavior: the Morse potential's outer wall is somewhat too soft/hard compared to the true curve at very large $v$, so $D_e$ from the simple formula is an approximation, refined in practice by higher-order terms ($\omega_e y_e$, etc.) and by the Birge–Sponer sum described next.

Example 2 — Applied: distinguishing hot bands from overtones in a heated IR cell. A vibrational spectroscopy experiment heats a sample of HCl-like gas (using the same illustrative constants as Example 1's molecule for continuity, applied here to a different diatomic scenario with $\omega_e = 2990.9\ \text{cm}^{-1}$, $\omega_e x_e = 52.8\ \text{cm}^{-1}$, values close to real HCl) to a temperature high enough to populate $v=1$ appreciably. Two new lines appear near the fundamental: a hot band ($v=1\to2$) and the first overtone ($v=0\to2$). The task is to predict both positions and show they are not degenerate. Step 1: Fundamental $v=0\to1$.

Step 2: Hot band $v=1\to2$.

This line sits $2885.3-2779.7=105.6\ \text{cm}^{-1}$ below the fundamental — always shifted to lower wavenumber (red of the fundamental) because deeper anharmonic levels have smaller spacings, and the hot band starts from an already-excited, more weakly bound level. Step 3: First overtone $v=0\to2$.

which is close to, but measurably different from, twice the fundamental ($2\times 2885.3 = 5770.6\ \text{cm}^{-1}$), a gap of $105.6\ \text{cm}^{-1}$ — again exactly $2\omega_e x_e$, consistent with the general spacing formula. Step 4: Relative intensity reasoning (qualitative but essential for interpretation). At room temperature the Boltzmann population of $v=1$ relative to $v=0$ is governed by

Using $\Delta G_{1/2}=2885.3\ \text{cm}^{-1}$ and $T=298$ K, with $hc/k_B = 1.4388$ cm·K,

This vanishingly small population explains why hot bands are normally negligible at room temperature and only become visible upon heating (or in vibrationally hot flames/plasmas) — a point that connects directly to why hot-band analysis is an experimental technique for high-temperature or laser-pumped samples, not routine cold-gas IR. Step 5: Limiting-case check. If $T\to 0$, $N_1/N_0\to 0$ exactly, the hot band vanishes entirely, and only the fundamental and (weak) overtone survive — consistent with the physical picture that hot bands require thermal population of excited vibrational states.

Example 3 — Competition-level: Birge–Sponer extrapolation from four measured overtone spacings. A student measures four successive vibrational spacings for a diatomic (constructed with deliberately non-round, unequal numbers to test the method honestly) in an emission spectrum:

Task: extract $\omega_e$, $\omega_e x_e$, and estimate $D_0$ by Birge–Sponer summation, without assuming these are round textbook constants. Step 1: Confirm linearity and extract the slope. The formula predicts $\Delta G_{v+1/2} = \omega_e - 2\omega_e x_e(v+1)$, linear in $(v+1)$. Taking successive differences:

The constant difference of $-47.6$ cm$^{-1}$ per step confirms the linear Morse prediction and directly gives

Step 2: Extract $\omega_e$ from the intercept. Using $\Delta G_{1/2} = \omega_e - 2\omega_e x_e(1)$:

Step 3: Find where the extrapolated line hits zero — this locates $v_{\max}$.

so $v_{\max} \approx 32.70$ (the last bound level is near $v=32$). Step 4: Birge–Sponer sum for $D_0$. The dissociation energy measured from $v=0$ (i.e., $D_0$, not $D_e$) is the sum of all level spacings from $v=0$ up to $v_{\max}$, which — because the spacings form an arithmetic sequence — is well approximated by the area of the triangle in the Birge–Sponer plot:

Converting to kJ/mol:

Step 5: Cross-check using the closed-form Morse relation and compare. Independently, $D_e = \omega_e^2/(4\omega_e x_e) = (1603.9)^2/(4\times23.8) = 2{,}572{,}495/95.2 = 27{,}022.6\ \text{cm}^{-1}$, in close agreement (within rounding) with the triangular Birge–Sponer sum — the tiny discrepancy of about $3$ cm$^{-1}$ arises because the triangle approximation slightly overcounts compared to the exact discrete sum, and this is the well-known systematic tendency of Birge–Sponer plots to overestimate $D_0$ if extrapolated using only low-$v$ data, since real potentials deviate from the pure Morse shape near dissociation. Step 6: Limiting-case sanity check. If the four measured spacings had instead been equal rather than linearly decreasing, the extracted $\omega_e x_e$ would be exactly zero, $v_{\max}\to\infty$, and the Birge–Sponer sum would diverge — correctly signaling that a genuinely harmonic (never-dissociating) oscillator has no finite dissociation energy, an internally consistent limiting behavior of the method.