---
title: "Why Hartree–Fock Struggles to Break a Bond Interactive Lab | Sandbox Physics"
description: "Why can a good description near equilibrium fail as a bond stretches? Stretching hydrogen makes two electronic arrangements nearly equally important."
canonical_url: "https://sandboxphysics.com/labs/h2-bond-breaking/"
language: "en"
page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/h2-bond-breaking/"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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

RHF UHF FCI

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](https://sandboxphysics.com/stories/molecular-electronic/)

Physics tutorial

## Why Hartree–Fock Struggles to Break a Bond

**Background** At short separation, two hydrogen nuclei share a compact electronic state. At large separation, each neutral atom should retain one electron, and their spins form a singlet in the exact finite-basis ground state.

**Why it matters** The smallest molecule already shows why single-determinant approximations can struggle with chemical change.

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 misconception** The lowest single-determinant energy must have the exact spin symmetry.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

Physical conclusion

A stretched bond tests both electronic correlation and the symmetry of the chosen state.

## Related experiments and articles

Molecular electrons and breaking bonds · Reading path

### Orbitals, self-consistency, correlation and dissociation

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Why a shared orbital struggles when the bond stretches**](https://sandboxphysics.com/stories/molecular-electronic/restricted-and-unrestricted/)
2. [Article **Mixing configurations to keep the electrons apart**](https://sandboxphysics.com/stories/molecular-electronic/configuration-interaction/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)