FR13-B · Two electrons, several answers

Why Hartree–Fock Struggles to Break a Bond

Why can a good description near equilibrium fail as a bond stretches? Stretching hydrogen makes two electronic arrangements nearly equally important. Restricted Hartree–Fock keeps both electrons in one spatial orbital and misses that change. Mixing configurations recovers the singlet; unrestricted Hartree–Fock lowers the energy by allowing different spin orbitals.

Drag a hydrogen nucleus and compare restricted Hartree–Fock, unrestricted Hartree–Fock and full configuration interaction in the same basis. Follow the density, spin contamination and natural occupations as the bond stretches.

FCI total energy $-0.944$

RHF energy above FCI $0.183$

UHF spin contamination $0.964$

Second natural occupation $0.644$

Hydrogen · STO-3G

Pull a bond apart

Nuclear separation

R/a_0

4.00

Density in the scene

Drag either nucleus apart. UHF colors reveal where each spin density is larger.

What the cloud means

$n(\mathbf r)=\sum_{\mu\nu}P_{\mu\nu}\chi_\mu(\mathbf r)\chi_\nu(\mathbf r)$

The cloud uses the same density matrix as the calculation. Its translucent surfaces join points of equal density. Gold spheres mark enlarged nuclei. Cyan marks total density; in UHF, coral marks alpha-spin excess and cyan marks beta-spin excess.

Total-density surfaces use the two levels below. In UHF, the inner surfaces show the signed spin-density difference. The section view uses continuously shaded density.

$n=0.017,\ 0.075\;a_0^{-3}$

$n_\alpha-n_\beta=\pm0.020\;a_0^{-3}$

$a_0=0.529177210903\,\mathrm{\mathring A}$ $E_{\rm h}=27.211386246\,\mathrm{eV}$

Model assumptions

This is the Born–Oppenheimer ground state of hydrogen in the minimal STO-3G basis. FCI is exact within that finite basis. UHF may lower its energy by breaking spin symmetry. The cloud shows the selected method’s one-electron density; it does not show electron trajectories or the joint two-electron probability. Nuclear motion is excluded.

Calculated in your browser · STO-3G

One molecule, three energy curves

RHFUHFFCI

The second orbital becomes occupied

Near dissociation, two natural orbitals each approach one electron. The singlet needs more than one determinant.

Units, density and calculation details

\operatorname{Tr}(PS)=2

All total energies include nuclear repulsion. FCI solves the same finite Hamiltonian as HF. The saved calculation includes its geometry, method, density matrices and numerical diagnostics.

Learn how the calculation works, from orbitals to a broken bond

Start with the essentials

Focus question: Why can a good description near equilibrium fail as a bond stretches?
One-sentence intuition: As hydrogen dissociates, two configurations become important and the second natural orbital gains occupation.

Core mathematical model

Mix the two closed-shell configurations

|\Psi\rangle=c_g|\sigma_g^2\rangle+c_u|\sigma_u^2\rangle

For the minimal-basis ground-state symmetry of hydrogen, the two configurations supply the essential static correlation. The browser diagonalizes the full two-electron Hamiltonian.

Read natural occupations and spin contamination

n_1+n_2=2,\qquad 0\le n_i\le2,\qquad \langle\hat S^2\rangle_{\rm singlet}=0

Two natural occupations approaching one signal the growing importance of two configurations. A nonzero unrestricted spin expectation means its determinant is not a pure singlet.

Common difficulties

Lower energy and the right spin

Typical misconceptionThe lowest single-determinant energy must have the exact spin symmetry.

Better mental modelUHF can lower its energy by mixing spin sectors. Compare energy and the expectation of spin squared together.

Run the experiment

01
Predict before stretching
Choose Near the energy minimum, then Correlation grows. Say what you expect the density and the energy difference to do before changing the preset.
What to observe: The molecular density separates into two lobes, and the restricted description loses accuracy as the bond stretches.
02
Follow the second diagnostic
Select Almost two atoms. Compare FCI and UHF density, then read the second natural occupation and spin contamination.
What to observe: FCI retains a pure singlet. UHF gains a lower energy by allowing spin symmetry to break.
03
Save a reproducible comparison
Use Save calculation at two separations. Compare the saved basis, geometry, density matrices and total energies.
What to observe: Every record identifies the finite electronic model. Comparing it with a larger basis tests a separate source of error.

Why Hartree–Fock Struggles to Break a Bond

One molecule, three energy curves