Anharmonicity and the Morse Potential

The harmonic oscillator is the first thing a physicist learns about a chemical bond, and it is almost right. This chapter's preceding article takes Schrödinger's equation for a parabolic well and hands back a clean ladder of equally spaced levels $E_v=\hbar\omega(v+\tfrac12)$ — perfect for the lowest one or two states of any diatomic, where the true potential curve is well fitted by a parabola near its floor. Anharmonicity is the story of everything the parabola gets wrong, and it gets almost everything wrong the moment the bond is stretched. A parabola is symmetric and climbs without limit; a real bond is lopsided and can be broken — pull two atoms far enough apart and the potential flattens onto a plateau, the dissociation limit, because past some separation the atoms no longer feel each other. The parabola fails precisely where chemistry lives: bond breaking, overtone spectra, hot bands, the crowded high-lying states a flame populates. Anharmonicity has one dominant cause: the restoring force softens as the bond lengthens (the shared electron cloud can no longer hold the nuclei) and stiffens brutally as it shortens (bare-nucleus repulsion). The spectroscopic fingerprints follow immediately — vibrational levels crowd together as $v$ climbs, the observed vibration frequency drops below the bottom-of-well value, and transitions with $\Delta v>1$ acquire real intensity. The Morse potential, written down by Philip Morse in 1929, is the one anharmonic model whose Schrödinger equation solves in closed form, its eigenvalues reproducing the leading correction exactly. It is the bridge from the idealized spring to the real, breakable, spectroscopically rich molecular bond.

Key takeaways

A real diatomic sits in an asymmetric well — steep repulsive wall, soft attractive tail, finite dissociation plateau $D_e$ — of which the harmonic parabola captures only the bottom.
The Morse potential $V(r)=D_e(1-e^{-a(r-r_e)})^2$ is the analytically solvable model reproducing the well depth $D_e$ , equilibrium length $r_e$ , and anharmonic softening from three parameters.
Its exact term values $G(v)=\omega_e(v+\tfrac12)-\omega_e x_e(v+\tfrac12)^2$ give levels that crowd together as $v$ grows, unlike the equally spaced harmonic ladder.
Anharmonicity relaxes the $\Delta v=\pm1$ rule, lighting up weak overtones, and pushes the observed fundamental below $\omega_e$ by $2\omega_e x_e$ .
A Birge–Sponer plot extrapolates the spacings to the dissociation energy $D_0$ — a spectroscopic thermometer of bond strength, though the linear version overestimates.
Key idea: replacing the parabola with the Morse potential adds exactly one new quantity — the finite dissociation energy — and from it flow unequal spacing, a maximum vibrational number, and overtone activity, all in the single constant $\omega_e x_e=\omega_e^2/4D_e$ .

The core insight

Replacing the harmonic parabola with the Morse potential adds exactly one new physical quantity — the finite dissociation energy — and in doing so predicts unequal level spacing, a maximum vibrational quantum number, and overtone activity, all captured by the single anharmonicity constant $\omega_e x_e$ .

Where this lives today

The Morse constants $\omega_e$ and $\omega_e x_e$ are the front-matter of every diatomic entry in modern spectroscopic databases — HITRAN, the NIST tables, the LIFBASE and PGOPHER line lists. A Fourier-transform infrared spectrometer resolves the fundamental and its overtones in one pass; cavity ring-down spectrometers reach the faint high overtones of C–H, O–H and N–H bonds in the near-infrared, and it is precisely their anharmonic weakness, borrowed from $\mu_2$ and state mixing, that lets a compact detector sniff methane, water, or ammonia at parts-per-billion in an atmospheric window. The same physics underwrites the undergraduate iodine-cell experiment, where a green laser walks up the $B\leftarrow X$ progression of I₂ and a Birge–Sponer plot yields a dissociation energy — complete with the overestimate of worked example 3. At the frontier, photoassociation of ultracold atoms populates the topmost Morse-like levels one at a time, mapping the long-range tail where the linear extrapolation fails.

The physical picture

Picture rolling a marble in a bowl whose one wall is nearly vertical (nuclear repulsion) and whose other side flattens into a tabletop at height $D_e$ (dissociation). Near the bottom the bowl looks parabolic, so gentle oscillations are harmonic and evenly spaced. Give the marble more energy and it spends its extra time out on the flat side, coasting slowly before creeping back — its round trip lengthens and its frequency drops. That asymmetry is anharmonicity: the frequency falls as the amplitude grows, because the soft outer wall lets the bond wander far while the stiff inner wall snaps it back fast. Because the outer wall flattens toward $D_e$ , each successive state has less headroom above it. The rungs bunch together as $v$ increases, converging on the dissociation limit and stopping at a finite $v_{\max}$ — the deepest break with the harmonic picture: a Morse bond holds only finitely many vibrational states, truncated by its own escape energy.

Figure

The Morse potential in blue versus the harmonic parabola in amber dashed for a CO-like bond: near r_e the two curves agree, but the Morse curve flattens onto the dissociation plateau D_e ≈ 8.9×10⁴ cm⁻¹ while the parabola climbs without bound, and the vibrational levels crowd together as they approach D_e

The constant $\omega_e x_e$ measures how fast those rungs converge, and it is not free — it is fixed by the well depth through $\omega_e x_e=\omega_e^2/4D_e$ , so a deep, stiff bond is weakly anharmonic and a shallow one strongly so. To measure it you need not the potential curve but only the spacings, which fall on a straight line: extrapolating it counts the rungs and, better, sums them — the area under it is the energy to climb out of the well.

Figure

Birge–Sponer construction for I₂ with ω_e = 214.5 cm⁻¹ and ω_e x_e = 0.614 cm⁻¹: the level spacing ΔG falls on a straight line of intercept ω_e − 2ω_e x_e ≈ 213 cm⁻¹, and the triangular area under it estimates the dissociation energy D₀; real spacings shown dashed bend below the line, so the linear extrapolation overestimates D₀

The Morse potential is lucky in a useful way: its exact eigenvalues contain the harmonic term and one anharmonic correction and nothing higher — no $(v+\tfrac12)^3$ term. Real molecules need those higher terms, and spectroscopists add $\omega_e y_e$ , $\omega_e z_e$ to fit precise data. But the Morse form captures the dominant physics — finite well, crowding levels, overtones — with an equation you can solve on paper, which is why it is the model every first analysis reaches for.

The core equations

The Morse potential is written to do three jobs at once: vanish at the bond length, plateau at a finite dissociation energy, and diverge under compression — all in three parameters and a single exponential, which makes it both realistic and solvable.

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

where: $D_e$ — well depth from the minimum to the dissociation asymptote; $r_e$ — equilibrium separation (position of the minimum); $a$ — range parameter, inverse length; $r$ — instantaneous internuclear distance. At $r=r_e$ the bracket is zero so $V=0$ ; as $r\to\infty$ , $V\to D_e$ ; as $r$ shrinks, $V\to\infty$ — the three behaviours the parabola cannot reproduce. Solving the Schrödinger equation for this potential (Derivation A) gives energies exactly quadratic in $(v+\tfrac12)$ . Expressed as term values — energies divided by $hc$ , so measured in wavenumbers, the natural units of a spectrum — they read:

G(v)=\omega_e\left(v+\tfrac12\right)-\omega_e x_e\left(v+\tfrac12\right)^2 \qquad (2)

where: $G(v)$ — vibrational term of level $v$ in cm⁻¹; $v=0,1,2,\dots$ — vibrational quantum number; $\omega_e$ — harmonic constant, the frequency of the very bottom of the well; $\omega_e x_e$ — anharmonicity constant, always positive for a Morse well. The quadratic term is the entire correction: it bends the ladder over so high levels fall below the harmonic value. Neither constant is arbitrary — the closed-form solution ties each to the potential parameters, and eliminating $a$ yields the key relation of the subject: the anharmonicity is set by the well depth, so a deep well is nearly harmonic.

\omega_e = \frac{a}{2\pi c}\sqrt{\frac{2D_e}{\mu}},\qquad \omega_e x_e=\frac{\hbar a^2}{4\pi c\,\mu}=\frac{\omega_e^2}{4D_e} \qquad (3)

where: $\mu=m_1 m_2/(m_1+m_2)$ — reduced mass; $c$ — speed of light; $D_e$ — well depth in wavenumbers. Read the last equality both ways: $\omega_e$ and $\omega_e x_e$ give $D_e$ ; the well depth and frequency predict the anharmonicity. The observable in an infrared spectrum is not a level but a gap. Differencing $G(v)$ between neighbours gives the spacing a spectrometer records line by line — and it falls off linearly with $v$ , the Birge–Sponer line.

\Delta G_{v+1/2}\equiv G(v+1)-G(v)=\omega_e-2\omega_e x_e\,(v+1) \qquad (4)

where: $\Delta G_{v+1/2}$ — energy gap for $v\to v+1$ , in cm⁻¹. The gap shrinks by a fixed $2\omega_e x_e$ per step; at zero the ladder ends. Because the spacings sum to the escape energy, the same two constants give the dissociation energy — from the potential minimum and from the ground level.

D_e=\frac{\omega_e^2}{4\,\omega_e x_e},\qquad D_0=D_e-\tfrac12\omega_e+\tfrac14\omega_e x_e \qquad (5)

where: $D_e$ — well depth from the minimum; $D_0$ — dissociation energy from the $v=0$ level, what a calorimeter or Birge–Sponer extrapolation returns. The two differ by the zero-point energy $\tfrac12\omega_e-\tfrac14\omega_e x_e$ . Finally, the transition frequencies. Fundamental, overtone and hot band all come from one formula whose structure diagnoses anharmonicity: the first overtone lands less than twice the fundamental.

\nu_{v'\leftarrow v}=\omega_e(v'-v)-\omega_e x_e\left[(v'+\tfrac12)^2-(v+\tfrac12)^2\right] \qquad (6)

where: $\nu$ — transition wavenumber; $v'-v=1$ is the fundamental, $=2,3,\dots$ the overtones. Setting $v=0$ gives $\nu_0=\omega_e-2\omega_e x_e$ and $2\omega_e-6\omega_e x_e$ ; measuring both solves for $\omega_e$ and $\omega_e x_e$ at once.

Derivation AFrom the Morse potential to unequal level spacing

The goal is to show that the crowding of vibrational levels is a direct consequence of the exponential shape of the well: solving the radial Schrödinger equation gives eigenvalues exactly quadratic in $(v+\tfrac12)$ , so the neighbour spacing falls off linearly with $v$ and eventually vanishes. We work in $x=r-r_e$ , neglect rotation, and follow the substitution that turns the problem into a known special function.

Step 1. Write the Schrödinger equation $-\frac{\hbar^2}{2\mu}\psi''+D_e(1-e^{-ax})^2\psi=E\psi$ for the reduced mass $\mu$ in the Morse well. Why: the vibration of two nuclei reduces to one particle of mass $\mu$ , so this single equation holds the whole spectrum.

Step 2. Substitute $z=\frac{2\sqrt{2\mu D_e}}{a\hbar}e^{-ax}=2\lambda e^{-ax}$ , where $\lambda=\sqrt{2\mu D_e}/(a\hbar)$ measures the well depth in units of the vibrational quantum. Why: this maps the equation onto the associated Laguerre (Kummer) form, whose bound solutions are known exactly — what makes the potential solvable.

Step 3. Demand that the confluent hypergeometric series terminate so $\psi\to0$ at both ends; this forces $\lambda-(v+\tfrac12)=\sqrt{2\mu(D_e-E_v)}/(a\hbar)$ with $v=0,1,\dots,\lfloor\lambda-\tfrac12\rfloor$ . Why: normalizability discretizes the energies, and the finite range of $v$ is where $v_{\max}$ comes from.

Step 4. Solve for the energy: $E_v=D_e[1-(1-(v+\tfrac12)/\lambda)^2]$ ; expanding and using $a\hbar/\sqrt{2\mu D_e}=\hbar\omega_e/2D_e$ gives $E_v=\hbar\omega_e(v+\tfrac12)-\frac{(\hbar\omega_e)^2}{4D_e}(v+\tfrac12)^2$ . Why: the square produces a linear and a quadratic term and nothing else — the Morse spectrum is a parabola in $(v+\tfrac12)$ , the origin of the single anharmonicity constant.

Step 5. Difference the result: $E_{v+1}-E_v=\hbar\omega_e-\frac{(\hbar\omega_e)^2}{2D_e}(v+1)$ , a straight line falling by a fixed step per level. Why: this is what the Birge–Sponer plot measures, and matching coefficients gives $\omega_e x_e=\omega_e^2/4D_e$ . *Verified: coefficients match the standard Morse result $\omega_e x_e=\omega_e^2/4D_e$ (Herzberg; Demtröder); the limit $D_e\to\infty$ recovers the equally spaced harmonic ladder.*

Go deeper · Derivation B · 1 minBirge–Sponer extrapolation of the dissociation energy

Derivation BBirge–Sponer extrapolation of the dissociation energy

The aim is to turn a table of measured line spacings into a bond energy without breaking the bond or knowing the potential curve. Because the spacings fall on a straight line in $(v+1)$ , the total energy from the ground level to dissociation is the area under that line — a triangle read straight off the plot. We treat $v$ as continuous, integrate the spacing, and note where the estimate goes wrong.

Step 1. Plot the measured gaps $\Delta G_{v+1/2}=\omega_e-2\omega_e x_e(v+1)$ against $(v+1)$ ; they lie on a line of intercept $\omega_e-2\omega_e x_e$ and slope $-2\omega_e x_e$ . Why: nothing about the potential is assumed — the linearity is an experimental fact, and the two fitted numbers already hold $\omega_e$ and $\omega_e x_e$ .

Step 2. Summing every spacing from the ground level to the top rung is the total climb out of the well, so $D_0=\int_0^{v_{\max}}\Delta G\,\mathrm{d}v$ equals the triangular area under the line. Why: each spacing is one step of the climb, so their sum is the energy separating $v=0$ from dissociation — the plot integrates the ladder for you.

Step 3. Evaluate the triangle: the line crosses zero at $(v_{\max}+1)=\omega_e/2\omega_e x_e$ , so the area is $\tfrac12(\text{intercept})(\text{crossing})=\omega_e^2/4\omega_e x_e-\omega_e/2+\omega_e x_e/4$ . Why: this is exactly $D_e-\tfrac12\omega_e+\tfrac14\omega_e x_e$ , tying the picture to the eigenvalues and confirming the plot returns $D_0$ , not $D_e$ .

Step 4. Apply the caveat: real spacings curve downward near dissociation, because higher-order anharmonicity makes the top of the well flatter than a pure Morse form. Why: the straight line then sits above the true data at high $v$ , so its area overestimates $D_0$ — dramatically for floppy molecules like I₂; the fix is to measure levels close to the limit. *Verified: the sum reduces to $D_e=\omega_e^2/4\omega_e x_e$ when zero-point terms are dropped, consistent with Derivation A; the overestimate for I₂ (linear ≈ 2.3 eV versus true ≈ 1.54 eV) is the textbook cautionary case.*

A brief word on overtone intensities. In a harmonic oscillator with a linear dipole, $\langle v'|(r-r_e)|v\rangle$ is nonzero only for $v'=v\pm1$ — the exact $\Delta v=\pm1$ rule, so overtones are forbidden. Anharmonicity breaks it two ways: the Morse eigenstates mix in a little of the neighbouring $v$ , so $\langle v+2|(r-r_e)|v\rangle\neq0$ ; and the dipole curvature $\tfrac12\mu_2(r-r_e)^2$ connects states two quanta apart. Overtones therefore appear, ten to a hundred times weaker per step — faint precisely because they borrow their intensity from anharmonicity.

Limiting cases

Case	Result	Meaning
Near the bottom, $r\to r_e$	$V\approx D_e a^2 (r-r_e)^2$	Expanding to second order gives a parabola with $k=2D_e a^2$ , recovering the harmonic oscillator ( $\omega_e=a\sqrt{2D_e/\mu}$ ); the lowest levels are equally spaced.
Dissociation limit, $r\to\infty$	$V\to D_e$	The potential tends to a constant instead of diverging — the finite asymptote that allows bond breaking and a maximum $v$ .
Highest bound state	$v_{\max}=\left\lfloor \dfrac{\omega_e}{2\omega_e x_e}-\dfrac12\right\rfloor$	The spacing hits zero at $(v+1)=\omega_e/2\omega_e x_e$ ; above it the ladder ends, so the molecule holds finitely many states.
Deep-well limit, $D_e\to\infty$	$G(v)\to\omega_e(v+\tfrac12)$	Since $\omega_e x_e=\omega_e^2/4D_e$ , a deep well at fixed $\omega_e$ sends anharmonicity to zero — the parabola is the $D_e\to\infty$ face of the Morse well.

Common trapThe three energies D_e, D_0, and ω_e are not interchangeable

The most common slip is to blur $D_e$ , $D_0$ , and $\omega_e$ into one "bond energy." They are three different things. $D_e$ is the well depth from its geometric minimum — a place the molecule can never sit, because zero-point motion lifts it. $D_0=D_e-\tfrac12\omega_e+\tfrac14\omega_e x_e$ is the true dissociation energy from the $v=0$ level, what a calorimeter or Birge–Sponer extrapolation reports. And $\omega_e$ is neither a dissociation energy nor an observed line — it is the frequency of the infinitesimal bottom of the well, always larger than the measured fundamental $\nu_0=\omega_e-2\omega_e x_e$ . Quote a bond energy without saying whether it is $D_e$ or $D_0$ and you have made an error of one zero-point quantum — over 0.25 eV for H₂.

Worked examples

1 · Reading a CO spectrum from two constants (confidence builder)

Carbon monoxide has $\omega_e=2169.8\ \mathrm{cm^{-1}}$ , $\omega_e x_e=13.29\ \mathrm{cm^{-1}}$ . Predict its fundamental and first-overtone bands, estimate $D_e$ , and find the highest bound level.

Step 1. The fundamental is $\nu_0=\omega_e-2\omega_e x_e=2169.8-2(13.29)=\mathbf{2143.2\ cm^{-1}}$ — a genuine prediction, not a fit: the observed CO fundamental sits at 2143.3 cm⁻¹.

Step 2. The first overtone is $2\omega_e-6\omega_e x_e=\mathbf{4259.9\ cm^{-1}}$ , less than twice the fundamental ( $2\times2143.2=4286.4$ ) by $6\omega_e x_e\approx80$ cm⁻¹ — that downward shift is the fingerprint of anharmonicity (observed 4260.1 cm⁻¹).

Step 3. The well depth: $D_e=\omega_e^2/4\omega_e x_e=(2169.8)^2/(4\times13.29)=\mathbf{8.86\times10^{4}\ cm^{-1}}\approx11.0\ \mathrm{eV}$ , about 2% below the true 11.2 eV — CO is nearly harmonic.

Step 4. The ladder ends where the spacing vanishes: $v_{\max}=\lfloor\omega_e/2\omega_e x_e-\tfrac12\rfloor=\lfloor81.1\rfloor=\mathbf{81}$ . One pair of constants has delivered two band positions, a bond energy, and the whole ladder's size. Fundamental 2143.2 cm⁻¹, overtone 4259.9 cm⁻¹, $D_e\approx8.9\times10^4$ cm⁻¹ (11.0 eV), $v_{\max}=81$ .

2 · Reconstructing the HCl potential from two bands

The HCl fundamental is at $2885.9\ \mathrm{cm^{-1}}$ and its first overtone at $5668.0\ \mathrm{cm^{-1}}$ . Extract $\omega_e$ and $\omega_e x_e$ , then rebuild $D_e$ , $k$ , and $a$ . Take $\mu=0.9796\ \mathrm{u}=1.627\times10^{-27}\ \mathrm{kg}$ .

Step 1. Two bands, two unknowns: $\nu_0=\omega_e-2\omega_e x_e=2885.9$ and $\nu_{0\to2}=2\omega_e-6\omega_e x_e=5668.0$ . Doubling the first and subtracting the second gives $2\omega_e x_e=103.8$ , so $\omega_e x_e=\mathbf{51.9\ cm^{-1}}$ , $\omega_e=\mathbf{2989.7\ cm^{-1}}$ — within 1 cm⁻¹ of the catalogue values.

Step 2. Well depth: $D_e=\omega_e^2/4\omega_e x_e=(2989.7)^2/(4\times51.9)=\mathbf{4.31\times10^{4}\ cm^{-1}}\approx5.3\ \mathrm{eV}$ . This overestimates the true value (~4.6 eV): HCl is more anharmonic than CO, so pure-Morse $\omega_e^2/4\omega_e x_e$ runs high — the same overestimate as the Birge–Sponer plot.

Step 3. The force constant comes from $\omega_e$ alone: $k=\mu(2\pi c\,\omega_e)^2$ . With $2\pi c\,\omega_e=5.63\times10^{14}\ \mathrm{s^{-1}}$ , $k=(1.627\times10^{-27})(5.63\times10^{14})^2=\mathbf{516\ N/m}$ .

Step 4. The range parameter closes the set via $k=2D_e a^2$ . With $hcD_e=8.55\times10^{-19}\ \mathrm{J}$ , $a=\sqrt{k/2D_e}=1.74\times10^{10}\ \mathrm{m^{-1}}=\mathbf{1.7\ \text{Å}^{-1}}$ (the $D_e$ overestimate leaves this slightly below the true 1.87 Å⁻¹). Three parameters — $D_e$ , $r_e$ , $a$ — now fix the whole curve. $\omega_e=2989.7$ , $\omega_e x_e=51.9$ cm⁻¹, $D_e\approx4.3\times10^4$ cm⁻¹, $k\approx516$ N/m, $a\approx1.7$ Å⁻¹.

3 · Birge–Sponer dissociation energy of I₂, and why it runs high

Iodine has $\omega_e=214.5\ \mathrm{cm^{-1}}$ , $\omega_e x_e=0.614\ \mathrm{cm^{-1}}$ . Estimate $D_0$ by the linear Birge–Sponer method and compare with the true $D_0\approx12440\ \mathrm{cm^{-1}}$ (1.54 eV).

Step 1. The line intercept is $\omega_e-2\omega_e x_e=213.3$ cm⁻¹ and it crosses zero at $(v_{\max}+1)=\omega_e/2\omega_e x_e=214.5/1.228=175$ — predicting ~174 bound levels, a warning sign, since I₂ actually holds about 107.

Step 2. The triangular area is $D_0=D_e-\tfrac12\omega_e+\tfrac14\omega_e x_e$ with $D_e=\omega_e^2/4\omega_e x_e=(214.5)^2/(4\times0.614)=1.873\times10^{4}$ cm⁻¹, giving $\mathbf{1.86\times10^{4}\ cm^{-1}}\approx2.31\ \mathrm{eV}$ .

Step 3. Compare with the true 1.54 eV: an overestimate of about 50%. The cause is the caveat of Derivation B — I₂ is floppy, its real spacings curve steeply below the straight line near dissociation, so the triangle overcounts. Linear Birge–Sponer $D_0\approx2.3$ eV versus true 1.54 eV, ~50% high — the model serves stiff bonds like CO far better than weak ones.

Physics of the portrait

Where the idealized quantum spring meets the real, breakable molecule, one 1929 paper made the meeting analytically exact.

Origins · 1 minPhilip Morse and how this was discovered

Philip Morse

1903–1985 · molecular quantum mechanics and the solvable anharmonic bond

Philip McCord Morse was an American physicist who, at twenty-five and only three years after Schrödinger, found a molecular potential that solved in closed form and actually resembled a real bond. His 1929 paper "Diatomic Molecules According to the Wave Mechanics" (Phys. Rev. 34, 57) introduced the potential now universally called the Morse potential, reduced its Schrödinger equation to the associated Laguerre equation, and derived the eigenvalues $E_v=\hbar\omega_e(v+\tfrac12)-\omega_e x_e\,hc\,(v+\tfrac12)^2$ — the first realistic vibrational spectrum obtainable by hand. Its single anharmonicity constant matched the level crowding spectroscopists had been fitting empirically, and it became the default description of a diatomic bond overnight. Morse ranged widely afterward — he co-authored "Methods of Theoretical Physics" with Herman Feshbach and, in the Second World War, founded modern operations research — but for molecular and optical physics his name lives on the curve every student draws for a chemical bond, the exponential form that turned the harmonic approximation into a genuinely anharmonic, dissociable, spectroscopically faithful model.

History

1926

— Birge and Sponer extrapolate vibrational band spacings to estimate dissociation energies

1929

— Philip Morse publishes his potential and its exact closed-form vibrational eigenvalues

1934

— Pekeris derives the rotation–vibration coupling formula for the Morse oscillator

1939

— Herzberg's Molecular Spectra and Molecular Structure codifies ω_e and ω_e x_e as standard constants

1947

— Rees completes the RKR inversion that builds potential curves directly from measured spectra

Going further

Texts. Herzberg, Molecular Spectra and Molecular Structure I: Spectra of Diatomic Molecules (definitive on $\omega_e$ , $\omega_e x_e$ , dissociation energies) · Demtröder, Laser Spectroscopy, Vol. 1 · Foot, Atomic Physics, Ch. 8 · Loudon, The Quantum Theory of Light (anharmonic oscillators and selection rules) · Saleh and Teich, Fundamentals of Photonics · P. M. Morse, Phys. Rev. 34, 57 (1929) (the original solution).

Provenance. The eigenvalue formula $E_v=\hbar\omega_e(v+\tfrac12)-\frac{(\hbar\omega_e)^2}{4D_e}(v+\tfrac12)^2$ and the relation $\omega_e x_e=\omega_e^2/4D_e$ follow the standard Morse solution in Herzberg and Demtröder; the harmonic limit $k=2D_e a^2$ , the cutoff $v_{\max}=\lfloor\omega_e/2\omega_e x_e-\tfrac12\rfloor$ , and the band positions are cross-checked against Foot and Loudon. The worked numbers use tabulated ground-state constants: CO ( $\omega_e=2169.8$ , $\omega_e x_e=13.29$ cm⁻¹), HCl ( $\omega_e\approx2990$ , $\omega_e x_e\approx52$ cm⁻¹), and I₂ ( $\omega_e=214.5$ , $\omega_e x_e=0.614$ cm⁻¹, true $D_0\approx12440$ cm⁻¹). The Morse estimates of $D_e$ and $D_0$ run few-percent for stiff bonds and knowingly high for floppy ones — a feature to teach, not a bug.

Prerequisites. The Harmonic Oscillator Model of Vibration · The Chemical Bond and Molecular Potential Energy Curves · The Born–Oppenheimer Approximation. Leads to. Vibration–Rotation Spectra and P, Q, R Branches · Selection Rules for Molecular Transitions · The Franck–Condon Principle · Rotational Spectra.

Ask the tutor: Give me any diatomic's $\omega_e$ and $\omega_e x_e$ and I will walk you through its band positions, well depth, and the highest level it can hold — then show you exactly where the Morse model starts to lie.

Is the Morse potential the true potential of a real molecule?

No — it is an excellent, solvable approximation. A real potential is measured point by point (or inverted by the RKR method) and differs from any single analytic form. The Morse potential is special because it solves in closed form and its eigenvalues contain exactly one anharmonic term, matching the correction that dominates the low and middle levels. Near dissociation the long-range tail is set by dispersion or Coulomb forces and the Morse form is qualitatively wrong — which is why the linear Birge–Sponer line overestimates $D_0$ for floppy molecules like I₂. It is the right first model, not the last word.

What is the difference between ω_e and the frequency I actually measure?

$\omega_e$ is the harmonic constant — the vibration frequency of the infinitesimal bottom of the well. No real transition samples only that point, since even $0\to1$ spans levels that feel the well's curvature. The measured fundamental is therefore $\nu_0=\omega_e-2\omega_e x_e$ , redshifted by twice the anharmonicity. So $\omega_e$ is an extrapolated quantity — obtained by fitting several bands back to $v=-\tfrac12$ — never a directly observed line, which is why tables quote $\omega_e$ and $\omega_e x_e$ separately.

Why do the vibrational levels get closer together as v increases?

Because the well is asymmetric and flattens toward dissociation. As a level climbs, its outer turning point moves onto the soft, shallow part of the potential, where a little more energy buys a lot more room — so the next level needs only a small addition to fit in. Formally, $G(v)$ is quadratic in $(v+\tfrac12)$ with a negative coefficient, so $\Delta G=\omega_e-2\omega_e x_e(v+1)$ falls linearly to zero, where the ladder ends. The harmonic oscillator never does this because its parabola keeps rising, holding every rung at the same height.

What exactly is the anharmonicity constant ω_e x_e telling me?

It measures how fast the ladder crowds, and through $\omega_e x_e=\omega_e^2/4D_e$ it is really a statement about the well depth. A large ratio $\omega_e x_e/\omega_e$ means a shallow, floppy bond with few, rapidly converging levels and bright overtones — I₂ ( $\approx0.003$ ), or the extreme H₂. A small ratio means a deep, stiff, nearly harmonic bond. Because it links to $D_e$ , measuring $\omega_e x_e$ from two bands is equivalent to measuring the dissociation energy — the whole reason vibrational spectroscopy can do thermochemistry.

How can spectroscopy give a bond energy without breaking the bond?

By counting and summing the rungs of the ladder. Each spacing $\Delta G_{v+1/2}$ is one step of the climb out of the well, so adding every step from $v=0$ to the top gives the energy to reach dissociation — that is $D_0$ . Because the spacings fall on a straight line, you need not measure every level: fit the line and its triangular area is the sum. This is the Birge–Sponer method. The catch: real spacings bend below the line near the top, so the linear estimate is an upper bound.

Why is D_e always larger than the "real" dissociation energy D_0?

Because $D_e$ is measured from the bottom of the well, a point the molecule can never occupy: even in its lowest state it keeps a zero-point energy $\tfrac12\omega_e-\tfrac14\omega_e x_e$ . The energy to pull the atoms apart starts from that $v=0$ level, so $D_0=D_e-\tfrac12\omega_e+\tfrac14\omega_e x_e$ is always smaller than $D_e$ . For heavy, stiff molecules the gap is a few percent; for light ones like H₂ it is a large fraction of an electron-volt — which is why isotope substitution, changing $\mu$ , measurably shifts dissociation energies.