The Chemical Bond and Molecular Potential Energy Curves

A chemical bond is, in the end, a shape. Push two atoms together and their total energy — computed with the nuclei frozen and the electrons allowed to settle — traces out a curve: high on the left where the nuclei jam together, dropping into a valley, flattening to the right toward two free atoms. That curve $U(R)$ is the master object of molecular physics. Its valley floor is the bond length, its depth is the bond strength, the stiffness of its walls is the vibrational frequency, its plateau the energy to break the molecule. Nearly every number a spectroscopist quotes for a diatomic is a property of this landscape. That such a curve exists at all is the gift of the Born-Oppenheimer approximation, taken from aq-optics-born-oppenheimer-approximation: because a nucleus is thousands of times heavier than an electron, the electrons readjust essentially instantly to any nuclear position, so one clamps the nuclei, solves the electronic Schrödinger equation once for each separation $R$ , adds the bare nucleus-nucleus repulsion, and calls the result the potential in which the slow nuclei then move. The why of the valley — that electrons shared between two nuclei are attracted to both and lower the energy — belongs to aq-optics-molecular-orbital-theory; this article takes the electronic energy as given and studies the curve it produces. The payoff is that one function forecasts three spectra at once. Expand $U(R)$ about its minimum: the quadratic term is a harmonic oscillator, higher terms are anharmonicity, and curvature and minimum together fix the moment of inertia. Different electronic states carry different curves, some with no valley at all, and transitions hop a molecule between them. Rotation, vibration, electronic bands, dissociation — all of it is geometry read off $U(R)$ .

Key takeaways

A bond forms when the electronic energy of two atoms at fixed separation $R$ (including nuclear repulsion) lies below the separated-atom energy; plotting it against $R$ gives the potential energy curve $U(R)$ .
The curve is a steep repulsive wall at small $R$ , a minimum at the bond length $R_e$ , and a plateau at large $R$ that defines the dissociation limit.
Depth from the floor is $D_e$ ; depth from the lowest level $v=0$ is the spectroscopic $D_0=D_e-\tfrac12\hbar\omega_e$ — and $D_0$ is what an experiment actually breaks.
Curvature at the bottom sets the force constant $k$ and hence $\omega_e=\sqrt{k/\mu}$ ; $R_e$ sets the rotational constant $B_e$ . Depth and stiffness are independent.
The picture rests on the Born-Oppenheimer separation: the nuclei glide on one electronic surface that plays the role of their potential.
Key idea: clamp the nuclei, solve for the electrons, add nuclear repulsion — the resulting curve $U(R)$ has a minimum $R_e$ and depth $D_e$ whose position, depth, and curvature fix bond length, dissociation energy, and vibrational frequency.

The core insight

Clamp the nuclei, solve for the electrons, add nuclear repulsion: the resulting energy $U(R)$ is a potential energy curve whose minimum $R_e$ and depth $D_e$ , together with its curvature, determine bond length, dissociation energy, and vibrational frequency.

Where this lives today

The potential energy curve is no longer a textbook schematic — it is measured and computed to spectacular precision. Frequency-comb-referenced laser spectroscopy now pins the dissociation energy $D_0$ of H₂ to better than a thousandth of a wavenumber, turning the simplest molecule into a testbed for quantum electrodynamics: any disagreement between the measured $D_0$ and the ab-initio curve would be a signal of new physics, so the curve is literally an instrument of fundamental physics. On the computational side, quantum-chemistry packages generate potential energy surfaces — the polyatomic generalization — from coupled-cluster and density-functional theory, driving everything from combustion modelling to drug-binding free energies. The curve is also the blueprint for building molecules atom by atom. In ultracold-molecule laboratories, pairs of laser-cooled atoms are associated into a high vibrational level and then transferred, by a two-photon STIRAP pulse, down to the very bottom of the ground-state well — impossible without knowing the curves in detail. From testing QED to assembling designer molecules for quantum simulation, the object doing the work is the same landscape $U(R)$ this article defined.

The physical picture

Slide two hydrogen atoms slowly together and watch the energy. Far apart they ignore one another — the energy is flat. As the electron clouds overlap, each electron can sit between the two nuclei, attracted to both at once; this shared occupancy lowers the energy and draws the atoms in. Keep pushing and two effects fight back: the bare nuclei feel each other's Coulomb repulsion, and the Pauli principle forbids the electrons from all crowding into the same low-energy region. The energy turns around and climbs a wall. The bond length is the truce between the long-range pull and the short-range shove.

Figure

The bond as a compromise a steep short-range repulsion from the nuclei and the Pauli principle in red opposes the longer-range attraction of electrons shared between the nuclei in blue their sum in black is a well whose floor marks the equilibrium separation R_e where the net force vanishes

Now treat $U(R)$ as a valley and the nuclei as a ball rolling in it. Where the ball rests is the bond length; how steeply the walls rise near the floor sets how fast the molecule vibrates; how deep the valley is sets the energy to dissociate. Crucially, depth and stiffness are independent — a valley can be deep yet soft, or shallow yet stiff. Quantum mechanics adds one twist the classical ball lacks: the molecule can never rest on the floor. Its lowest state floats a zero-point energy $\tfrac12\hbar\omega_e$ above the bottom, so the usable binding energy $D_0$ is always a little less than the geometric depth $D_e$ . The second figure shows the real H₂ ground-state curve with every feature carrying its measured value.

Figure

The ground-state potential energy curve of H₂ from the Morse model the well bottom sits at R_e = 0.741 Å the depth from the minimum is D_e = 4.75 eV and the vibrational ladder v = 0, 1, 2, … converges toward the H + H dissociation limit because the lowest level v = 0 lies a zero-point energy ½ħω_e = 0.27 eV above the floor the measurable bond energy is D_0 = 4.48 eV

The core equations

The starting point is the electronic problem at a single frozen geometry. Fix the nuclei at separation $R$ , keep every Coulomb interaction plus the constant nucleus-nucleus repulsion, and solve the clamped-nuclei Schrödinger equation; the result is a set of electronic energies, one per state, each depending on $R$ as a parameter. Traced out over $R$ , each eigenvalue is a potential energy curve.

\hat{H}_{\text{el}}(R)\,\psi_n(\mathbf{r};R)=U_n(R)\,\psi_n(\mathbf{r};R) \qquad (1)

where: $\hat{H}_{\text{el}}(R)$ — electronic Hamiltonian with nuclei clamped at $R$ , including nucleus-nucleus repulsion; $\mathbf{r}$ — collective electron coordinates; $U_n(R)$ — the $n$ -th potential energy curve (electronic energy plus nuclear repulsion); $\psi_n$ — electronic wavefunction, depending on $R$ only parametrically. Physical reading: solving once per geometry turns a coupled problem into a family of ordinary curves. Once $U_n(R)$ is in hand, the nuclei obey a conventional Schrödinger equation in which that curve is the potential energy, and their bound solutions are the vibrational and rotational states of the molecule.

\big[\hat{T}_{\text{nuc}}+U_n(R)\big]\,\chi(R)=E\,\chi(R) \qquad (2)

where: $\hat{T}_{\text{nuc}}$ — nuclear kinetic-energy operator; $\chi(R)$ — nuclear (vibration-rotation) wavefunction; $E$ — total molecular energy; $U_n(R)$ — the curve, as potential. Physical reading: the electrons become a landscape, and vibration and rotation follow from its shape. For a bonding state the curve has a minimum. The equilibrium bond length $R_e$ is where the net force vanishes (first derivative zero) and the curve turns upward (second derivative positive) — the point that anchors every other molecular constant.

R_e:\quad \left.\frac{\mathrm{d}U}{\mathrm{d}R}\right|_{R_e}=0,\qquad \left.\frac{\mathrm{d}^2U}{\mathrm{d}R^2}\right|_{R_e}>0 \qquad (3)

where: $R_e$ — equilibrium internuclear separation, the floor of the well. Physical reading: $R_e$ is purely geometric, read off the curve, and it alone fixes the moment of inertia. The well has two depths, and confusing them is the classic error. The geometric depth $D_e$ runs from the very bottom to the dissociation plateau; the spectroscopic depth $D_0$ runs from the lowest level $v=0$ , which floats a zero-point energy above the floor, so $D_0$ is smaller and is what a dissociation experiment reports.

D_e=U(\infty)-U(R_e),\qquad D_0=D_e-\tfrac{1}{2}\hbar\omega_e \qquad (4)

where: $D_e$ — depth from the minimum; $U(\infty)$ — dissociation asymptote; $D_0$ — binding energy from $v=0$ ; $\tfrac12\hbar\omega_e$ — zero-point energy. Physical reading: zero-point motion always makes a molecule slightly easier to break than the raw depth suggests, so $D_0<D_e$ without exception. The stiffness of the well, not its depth, sets the vibrational frequency. Taylor-expand $U(R)$ about $R_e$ : the leading nontrivial term is quadratic, its coefficient is the force constant $k$ , and the reduced mass turns stiffness into a frequency exactly as for a mass on a spring.

k=\left.\frac{\mathrm{d}^2U}{\mathrm{d}R^2}\right|_{R_e},\qquad \omega_e=\sqrt{\frac{k}{\mu}} \qquad (5)

where: $k$ — force constant, the curvature at $R_e$ ; $\mu=m_Am_B/(m_A+m_B)$ — reduced mass; $\omega_e$ — harmonic vibrational frequency. Physical reading: a sharply curved well vibrates fast and a gently curved one slowly, whatever the depth. The bond length reappears in rotation through the moment of inertia $I=\mu R_e^2$ . The rotational constant is inversely proportional to that inertia, so the spacing of rotational lines is a direct measurement of $R_e$ .

B_e=\frac{\hbar}{4\pi c\,\mu R_e^2} \qquad (6)

where: $B_e$ — equilibrium rotational constant in wavenumbers; $\mu R_e^2=I$ — moment of inertia; $c$ — speed of light. Physical reading: rotation weighs the molecule and measures its length; a longer or heavier bond gives closer-spaced levels. Finally, one analytic curve captures the real shape astonishingly well: the Morse potential. It is zero at $R_e$ , climbs to $D_e$ as $R\to\infty$ , carries the correct curvature at the bottom, and — unlike a parabola — turns over into a finite plateau, building in anharmonicity too.

U(R)=D_e\Big[1-e^{-a(R-R_e)}\Big]^2,\qquad a=\sqrt{\frac{k}{2D_e}} \qquad (7)

where: $a$ — range parameter tying stiffness to depth; $D_e$ — well depth; $R_e$ — equilibrium separation. Physical reading: three numbers ( $D_e$ , $R_e$ , $a$ ) reproduce bond length, dissociation energy, frequency, and leading anharmonicity at once.

Derivation AWhere the curve comes from: separating electrons and nuclei

The goal is to show why a molecule — whose true Hamiltonian tangles electronic and nuclear coordinates with no exact separation — can still be described by a single function $U(R)$ that the nuclei feel as a potential. The engine is the huge nuclear-to-electron mass ratio: the nuclei lumber while the electrons dart, so the electrons stay in an eigenstate of the instantaneous nuclear configuration, and that eigenvalue becomes the potential governing the slow nuclear drift. We build this in three moves and name what is thrown away.

Step 1. Split the full Hamiltonian into nuclear kinetic energy and everything else: $\hat H=\hat T_{\text{nuc}}+\hat H_{\text{el}}(R)$ , where $\hat H_{\text{el}}$ holds the electronic kinetic energy and all Coulomb terms, including nuclear repulsion, at fixed $R$ . Why: $\hat T_{\text{nuc}}$ is the one term coupling different nuclear geometries; setting it aside leaves a problem in the electrons alone.

Step 2. Write the total wavefunction as a product, $\Psi(\mathbf r,R)\approx\psi_n(\mathbf r;R)\,\chi(R)$ , with the electronic part solving equation (1) at each $R$ . Why: this Born-Oppenheimer ansatz says the electrons occupy a definite eigenstate of the current nuclear positions — their state is slaved to $R$ .

Step 3. Insert the product into the full Schrödinger equation and drop the derivative-coupling terms in which $\hat T_{\text{nuc}}$ acts on the $R$ -dependence of $\psi_n$ . What survives is $[\hat T_{\text{nuc}}+U_n(R)]\chi=E\chi$ — equation (2). Why: the neglected terms are smaller by roughly the mass ratio, so the nuclei move in the single potential $U_n(R)$ — the exact sense in which the electronic eigenvalue becomes a curve. *Why the approximation is so good: the discarded non-adiabatic couplings scale as $m_e/M_{\text{nuc}}\sim10^{-4}$ , making them negligible in almost all situations. Where two curves approach, however, these terms revive and the single-surface picture breaks down — the origin of predissociation.*

Go deeper · Derivation B · 1 minHarmonic frequency from the curvature of the well

Derivation BHarmonic frequency from the curvature of the well

The goal is to connect a measured infrared line to a geometric property of the curve — its curvature at the bottom — by showing that small-amplitude nuclear motion in any smooth well is, to leading order, a harmonic oscillator whose frequency depends only on curvature and reduced mass. This is why one vibrational line reports the second derivative of $U$ at $R_e$ , and why the same expansion underlies the harmonic-oscillator and rigid-rotor articles downstream.

Step 1. Taylor-expand the curve about its minimum: $U(R)=U(R_e)+\tfrac12 U''(R_e)(R-R_e)^2+\cdots$ , with no linear term. Why: $R_e$ is a stationary point, so $U'(R_e)=0$ kills the linear term and the quadratic term leads — the well looks parabolic close enough to the bottom.

Step 2. Define the displacement $x\equiv R-R_e$ and force constant $k\equiv U''(R_e)$ , so near the floor $U\approx U(R_e)+\tfrac12 kx^2$ . Why: naming the curvature $k$ recasts the molecular potential as an ordinary spring, exact at leading order.

Step 3. The nuclear equation (2) becomes $\big[-\tfrac{\hbar^2}{2\mu}\tfrac{\mathrm d^2}{\mathrm dx^2}+\tfrac12 kx^2\big]\chi=E_{\text{vib}}\chi$ , the textbook one-dimensional oscillator with reduced mass $\mu$ . Why: every harmonic-oscillator result now transfers verbatim to molecular vibration.

Step 4. Read off the spectrum $E_v=\hbar\omega_e(v+\tfrac12)$ with $\omega_e=\sqrt{k/\mu}$ , so the level spacing measures the curvature. Why: the lowest level $v=0$ retains $\tfrac12\hbar\omega_e$ — the zero-point energy responsible for $D_0<D_e$ — and the uniform spacing $\hbar\omega_e$ is what an infrared fundamental reports.

Go deeper · Derivation C · 1 minThe Morse curve: bond parameters in closed form

Derivation CThe Morse curve: bond parameters in closed form

The goal is to exhibit one analytic function that reproduces the near-equilibrium behaviour and the dissociation plateau together, with parameters mapping one-to-one onto the measurable constants $D_e$ , $R_e$ , $k$ , and the anharmonicity — so that a handful of spectral lines can be inverted for the whole curve, including a dissociation energy obtained without ever breaking the molecule. The Morse potential does this and, remarkably, has an exact quantum spectrum.

Step 1. Check the anchors: at $R=R_e$ the bracket vanishes so $U(R_e)=0$ , and as $R\to\infty$ the exponential dies so $U\to D_e$ . Why: this fixes $D_e$ as the depth and gives the correct flat dissociation asymptote, which a parabola never has.

Step 2. Differentiate twice at the minimum to get $U''(R_e)=2D_e a^2=k$ , hence $a=\sqrt{k/2D_e}$ . Why: the range parameter is not free — it is locked to stiffness and depth, so fitting curvature and depth fixes the whole curve.

Step 3. The exact Morse levels are $E_v=\hbar\omega_e(v+\tfrac12)-\hbar\omega_e x_e(v+\tfrac12)^2$ with $\omega_e x_e=\hbar\omega_e^2/4D_e$ . Why: the negative quadratic term crowds successive levels together as $v$ grows — the hallmark of a real bond — and shows deeper wells are more harmonic.

Step 4. The levels converge and pile up where the spacing vanishes, giving $D_e\approx\omega_e^2/(4\,\omega_e x_e)$ in consistent units. Why: measuring $\omega_e$ and $\omega_e x_e$ near the bottom extrapolates to the dissociation energy — the Birge-Sponer method. *Why the Morse level formula is useful in practice: the level formula is exact for the Morse potential. For H₂ the Birge-Sponer extrapolation gives $D_e\approx4.95$ eV against the measured $4.75$ eV — a 4% overestimate typical of Morse, because the real curve deviates from the Morse shape at higher levels.*

Limiting cases

The one curve interpolates between four regimes, each set by a different effect:

Case	Result	Meaning
Small separation $R\to0$	$U(R)\to +\dfrac{Z_AZ_B e^2}{4\pi\varepsilon_0 R}\to+\infty$	bare nuclear Coulomb repulsion dominates; the curve is a near-vertical wall
Large separation $R\to\infty$	$U(R)\to E_A+E_B$	the curve flattens to the sum of separated-atom energies; this plateau defines the dissociation limit
Near equilibrium	$U(R)\approx U(R_e)+\tfrac12 k(R-R_e)^2$	a parabolic well; curvature $k$ gives harmonic vibrations at $\omega_e=\sqrt{k/\mu}$
Purely repulsive state	$\mathrm{d}U/\mathrm{d}R<0$ for all $R$	no minimum: the electronic state is unbound and the atoms simply fly apart (antibonding)

Worked examples

1 · Zero-point energy and the true bond strength of H₂ (confidence builder)

The ground state of ¹H₂ has $D_e=4.75$ eV and $\tilde\omega_e=4401\ \mathrm{cm^{-1}}$ . Find the zero-point energy and $D_0$ , and compare with the measured value.

Step 1. Convert with $1\ \mathrm{cm^{-1}}=1.2398\times10^{-4}$ eV: the full vibrational quantum is $\tilde\omega_e=4401\ \mathrm{cm^{-1}}=0.5457$ eV.

Step 2. The zero-point energy is half of that: $\tfrac12\hbar\omega_e=\mathbf{0.273\ eV}$ (2200 cm⁻¹) — the energy the molecule keeps even at absolute zero, unable to sit on the well floor.

Step 3. Subtract from the geometric depth: $D_0=D_e-\tfrac12\hbar\omega_e=4.75-0.273=\mathbf{4.48\ eV}$ , matching the measured $D_0=4.478$ eV (36118 cm⁻¹). The bond you actually break is 0.27 eV weaker than the raw well depth, and D_0 = 4.48 eV is the number to compare with experiment.

2 · Force constant of H₂ from its vibrational frequency

Using $\tilde\omega_e=4401\ \mathrm{cm^{-1}}$ and the reduced mass of ¹H₂, find the force constant $k$ of the bond.

Step 1. Reduced mass: for two identical ¹H atoms, $\mu=m_H/2=(1.008\ \mathrm{u})/2=0.5039\ \mathrm{u}=8.37\times10^{-28}$ kg.

Step 2. Angular frequency: $\omega_e=2\pi c\,\tilde\omega_e=2\pi(2.998\times10^{10}\ \mathrm{cm/s})(4401\ \mathrm{cm^{-1}})=8.29\times10^{14}\ \mathrm{rad/s}$ .

Step 3. Invert $\omega_e=\sqrt{k/\mu}$ : $k=\mu\omega_e^2=(8.37\times10^{-28})(8.29\times10^{14})^2=\mathbf{5.8\times10^{2}\ N/m}$ , about 575 N/m. The H₂ bond acts like a spring of stiffness ≈ 570 N/m over a 0.7 Å bond. This used only the curvature of the well, never its depth — illustrating that $k$ and $D_e$ are independent.

3 · Bond length of CO from its rotational constant

Carbon monoxide has $B_e=1.931\ \mathrm{cm^{-1}}$ . Find the moment of inertia and bond length $R_e$ , using $\mu(\mathrm{^{12}C^{16}O})=6.856\ \mathrm{u}$ .

Step 1. Rearrange equation (6) for the moment of inertia: $I=\hbar/(4\pi c\,B_e)$ . With $\hbar=1.055\times10^{-34}$ J·s and $c=2.998\times10^{10}$ cm/s, the denominator is $4\pi(2.998\times10^{10})(1.931)=7.28\times10^{11}$ , so $I=1.055\times10^{-34}/7.28\times10^{11}=\mathbf{1.45\times10^{-46}\ kg\,m^2}$ .

Step 2. With $\mu=6.856\ \mathrm{u}=1.139\times10^{-26}$ kg, invert $I=\mu R_e^2$ : $R_e=\sqrt{I/\mu}=\sqrt{1.45\times10^{-46}/1.139\times10^{-26}}=1.128\times10^{-10}$ m. R_e = 1.128 Å, the accepted CO bond length — recovered from nothing but the spacing of its microwave rotational lines.

Physics of the portrait

The potential energy curve is where quantum mechanics, spectroscopy, and chemistry meet, and no one did more to make it a tabulated object than the spectroscopist who measured the curves of diatomic molecules line by line.

Origins · 1 minGerhard Herzberg and how this was discovered

Gerhard Herzberg

1904-1999 · molecular spectroscopy and the structure of diatomic molecules

Gerhard Herzberg was the experimentalist who turned the abstract potential energy curve into a working tool. Born in Hamburg in 1904 and forced to leave Germany in 1935, he built at the National Research Council in Ottawa the leading molecular-spectroscopy laboratory of its era. His three-volume Molecular Spectra and Molecular Structure — beginning with Spectra of Diatomic Molecules — became the reference generations of physicists and chemists reach for, tabulating $R_e$ , $D_e$ , $\omega_e$ , $\omega_e x_e$ , and $B_e$ for molecule after molecule, all read from the geometry of measured curves. Herzberg understood the curve as something you reconstruct from data: vibrational spacing gives the curvature, anharmonicity extrapolates to the dissociation energy, rotational lines give the bond length, and band systems reveal the excited-state curves above the ground state. He detected molecular hydrogen's quadrupole spectrum and identified free radicals and molecular ions central to astrophysics — work for which he received the 1971 Nobel Prize in Chemistry, an unusual honour for someone who always thought of himself as a physicist. His legacy is precisely the object of this article: the curve as a catalogued fact about every bond.

History

1916

— Lewis introduces the shared electron-pair picture of the chemical bond

1927

— Heitler and London derive the first quantum potential curve, for H₂

1927

— Born and Oppenheimer justify separating electronic from nuclear motion

1929

— Morse publishes his analytic potential with closed-form vibrational levels

1939

— Herzberg's Spectra of Diatomic Molecules codifies R_e, D_e, ω_e, and B_e

Going further

Texts. Foot, Atomic Physics (2005), Ch. 8-9 · Demtröder, Laser Spectroscopy, Vol. 1 · Herzberg, Spectra of Diatomic Molecules (the definitive catalogue of curves and constants) · Atkins and Friedman, Molecular Quantum Mechanics. The key relations used throughout are: the harmonic expansion $\omega_e=\sqrt{k/\mu}$ , the rotational constant $B_e=\hbar/(4\pi c\mu R_e^2)$ , and the Morse relations $k=2D_e a^2$ and $\omega_e x_e=\hbar\omega_e^2/4D_e$ . The tabulated constants for the worked examples are: H₂ has $R_e=0.741$ Å, $D_e=4.75$ eV, $\tilde\omega_e=4401\ \mathrm{cm^{-1}}$ , $\tilde\omega_e x_e=121\ \mathrm{cm^{-1}}$ , giving zero-point 0.27 eV, $D_0=4.48$ eV (measured 4.478 eV), $k\approx575$ N/m, and Morse $D_e\approx4.95$ eV; CO has $B_e=1.931\ \mathrm{cm^{-1}}\Rightarrow R_e=1.128$ Å.

Prerequisites. The Born-Oppenheimer Approximation · Molecular Orbital Theory and Bonding/Antibonding States. Leads to. The Harmonic Oscillator Model of Vibration · Anharmonicity and the Morse Potential · The Rigid Rotor Model · The Franck-Condon Principle.

Ask the tutor: Ask me to build the H₂ curve number by number — zero-point energy, force constant, Birge-Sponer dissociation — or to show how one landscape sets rotation, vibration, and electronic bands at once.

Why can a molecule be described by one potential curve when the real Hamiltonian couples electrons and nuclei?

Because the nuclei are thousands of times heavier than the electrons and move far more slowly. At any instant the fast electrons settle into an eigenstate of the current nuclear positions, so their energy depends on $R$ only as a parameter. Solving the electronic problem at each fixed $R$ gives $U(R)$ , and to excellent accuracy the nuclei then move in it as an ordinary potential — the Born-Oppenheimer approximation. The neglected coupling terms are smaller by roughly the mass ratio, about $10^{-4}$ , which is why the single-curve picture is so reliable — except where two curves nearly touch.

What is the difference between D_e, D_0, and the bond energy a chemist quotes?

$D_e$ is the depth from the very bottom of the curve — geometric. $D_0$ is measured from the lowest level $v=0$ , which floats $\tfrac12\hbar\omega_e$ above the floor, so $D_0=D_e-\tfrac12\hbar\omega_e$ is always smaller and is what a dissociation experiment reports. The bond enthalpies chemists tabulate are cousins of $D_0$ with thermal and pressure-volume corrections to a reference temperature, usually 298 K. For H₂ the three sit near 4.75 eV, 4.48 eV, and 4.52 eV — small but real differences worth keeping straight.

Can you find the dissociation energy without actually breaking the molecule?

Yes — one of the most elegant tricks in molecular spectroscopy. Because a real bond is anharmonic, its levels crowd together as you climb: $E_v=\hbar\omega_e(v+\tfrac12)-\hbar\omega_e x_e(v+\tfrac12)^2$ . Measure a few low-lying spacings for $\omega_e$ and $\omega_e x_e$ , then extrapolate to where the spacing vanishes. The Morse model gives $D_e\approx\omega_e^2/(4\omega_e x_e)$ ; the graphical version is the Birge-Sponer plot. For H₂ this yields $D_e\approx4.95$ eV against the true 4.75 eV — a few percent high, since Morse is only approximate, but remarkable for never having pulled the atoms apart.

Why doesn't a deeper well always mean a higher vibrational frequency?

Because the frequency depends on the curvature of the well at its bottom, not on how deep it is: $\omega_e=\sqrt{k/\mu}$ with $k=(\mathrm d^2U/\mathrm dR^2)_{R_e}$ . Depth $D_e$ and curvature $k$ are independent features of a shape — a well can be deep but broad (large $D_e$ , small $k$ , low frequency) or shallow but sharply pinched (small $D_e$ , large $k$ , high frequency). One molecule can be hard to break yet floppy, another easy to break yet stiff. Only in a model like Morse, which links them through $a=\sqrt{k/2D_e}$ , do depth and stiffness become correlated.

How does a potential energy curve generalize to a molecule with more than two atoms?

For a diatomic there is one internal coordinate, $R$ , so the energy is a curve. A molecule with $N$ atoms has $3N-6$ internal coordinates ( $3N-5$ if linear), so the electronic energy is a surface over that many dimensions. The same Born-Oppenheimer logic applies, but now minima are stable geometries, saddle points are reaction transition states, and the curvature matrix eigenvalues are the normal-mode frequencies. Everything here is the one-dimensional special case; the surface is where reaction dynamics and computational chemistry live.

What happens where two potential curves cross?

When two curves of the same symmetry approach, the Born-Oppenheimer neglect of coupling breaks down: they cannot actually cross but repel into an avoided crossing, and a molecule can hop from one surface to the other. This non-adiabatic behaviour drives predissociation — a bound molecule leaks onto a repulsive curve and falls apart, broadening spectral lines — and internal conversion in photochemistry. Curves of different symmetry cross freely. Conical intersections, the polyatomic version, are the funnels through which excited molecules relax, central to vision and photostability. So crossings mark exactly where the single-curve picture reaches its limit.