How to Read the Potential Energy Graph of the Hydrogen Molecule

The potential energy curve for H₂ is the standard picture of why a covalent bond forms and why it has a definite length. The x-axis is the internuclear distance $r$ and the y-axis is the energy of the two-atom system.

The four regions of the curve

1. Large $r$, energy ≈ 0. Two separate hydrogen atoms don’t interact. This is the reference level, so energies below it are negative.
2. $r$ decreasing, energy falling. Each electron is attracted to both nuclei, and the electron density between the nuclei holds them together. The system becomes more stable, so the energy drops.
3. The minimum, at $r_e \approx 74\ \text{pm}$. Here attraction and repulsion balance. This distance is the bond length, and the molecule vibrates around it.
4. Very small $r$, energy rising steeply. Nucleus–nucleus repulsion (and electron–electron repulsion) dominates, so pushing the atoms closer costs a lot of energy.

What the numbers mean

Feature on the graph	Value for H₂	Meaning
$r$ at the minimum	≈ 74 pm (0.74 Å)	equilibrium bond length
Depth of the well from the bottom, $D_e$	≈ 458 kJ/mol (4.75 eV)	bond energy if the molecule could sit at the very bottom
Energy to separate from the lowest vibrational level, $D_0$	≈ 432 kJ/mol (4.48 eV)	bond dissociation energy at 0 K
Bond enthalpy at 298 K	≈ 436 kJ/mol	the value in most data tables

The depths differ because a molecule can’t sit exactly at the bottom of the well. Quantum mechanics requires a zero-point vibrational energy, about $26\ \text{kJ/mol}$ for H₂, so the lowest real energy level is slightly above the minimum. Textbook diagrams usually label the well “bond energy ≈ 436 kJ/mol”. That is the room-temperature bond enthalpy, a few kJ/mol above $D_0$ because of thermal energy.

Why the curve has this shape

A simple model captures it: the Morse potential

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

which has its minimum $-D_e$ at $r = r_e$ and tends to $0$ as $r \to \infty$. Near the minimum it looks like a parabola (a harmonic oscillator), which is why bond vibrations are roughly harmonic for small amplitudes. Further out it flattens, which shows that a bond can break.

Quantum chemically, the shape comes from the bonding molecular orbital: as $r$ shrinks, the two 1s orbitals overlap more and the electron energy drops, until nuclear repulsion takes over.

Comparing curves

• A deeper well means a stronger bond.
• A minimum at larger $r$ means a longer bond. For example, the bond length in Cl₂ is about 199 pm.
• A narrower well means a stiffer bond, with a higher vibrational frequency.
• For two helium atoms there is essentially no well, which is why He₂ is not a stable molecule under normal conditions.

Common mistakes

• Reading the bond energy as positive at the minimum. The minimum is below zero. The bond energy is the energy you must add to climb back to zero.
• Thinking the atoms are at rest at $r_e$. They vibrate about it, even at absolute zero.
• Saying repulsion “starts” at the minimum. Attraction and repulsion act at all distances. The minimum is simply where they balance.