FR13-A · Electrons determine their own field

Build a Self-Consistent Hydrogen Molecule

How can an electron density agree with the field it creates? Start with a trial density, calculate its mean field, and solve for new orbitals. Repeat until the density changes negligibly and agrees with the field. This finds a self-consistent Hartree–Fock state; its energy still depends on the allowed orbitals.

Pull the two nuclei apart and follow the density as a Hartree–Fock calculation settles. Replay its actual iterations and see how the total energy and self-consistency residual change.

Total RHF energy $-1.12$

Self-consistency residual (hartree) $5.94\times10^{-10}$

Iteration $31$

Electron number $2$

Hydrogen · STO-3G

Let the density settle

Nuclear separation

R/a_0

1.40

Actual SCF iterations1

Drag the gold nucleus to change the separation. The cloud follows the selected iteration.

Initial guess and damping

Initial guessOrbital damping

Damping mixes some of the previous orbital coefficients into the next step, then normalizes the orbital. Replay speed is a display choice and has no physical time unit.

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

The fixed-nuclei hydrogen molecule uses two contracted STO-3G atomic orbitals. Browser calculations include nuclear repulsion and solve the electronic problem at the chosen separation. Iteration playback shows a numerical algorithm, not the passage of physical time. The density cloud is a three-dimensional density visualization with enlarged nuclei; its brightness is scaled for visibility.

Calculated in your browser · STO-3G

One density, one self-consistency test

Total energyResidual

Current AO density matrix

P=\begin{pmatrix}0.603&0.603\\0.603&0.603\end{pmatrix}

A changing density changes the Fock operator. The residual measures whether the two agree.

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.

The logarithmic plot displays residuals below the following level at that floor; the numerical readout retains the calculated value. $10^{-15}$

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

Start with the essentials

Focus question: How can an electron density agree with the field it creates?
One-sentence intuition: Self-consistency means the occupied orbitals agree with the mean field generated by their own density.

Core mathematical model

Solve in a nonorthogonal atomic basis

FC=SC\varepsilon,\qquad P_{\mu\nu}=2\sum_{i\in\mathrm{occ}}C_{\mu i}C_{\nu i}

The overlap matrix accounts for atomic orbitals that overlap in space. Two electrons occupy each restricted spatial orbital.

Check whether density and field agree

r=\tfrac12\lVert FPS-SPF\rVert_{\mathrm F},\qquad N=\operatorname{Tr}(PS)=2

The residual is the root-mean-square matrix commutator for this two-orbital basis. It vanishes at self-consistency. The overlap-weighted trace counts electrons.

Common difficulties

Convergence and accuracy

Typical misconceptionA tiny residual means the molecular energy is exact.

Better mental modelThe residual checks self-consistency inside the chosen Hartree–Fock model. Missing configurations and basis functions still limit the answer.

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
Replay a numerical decision
Press Replay iterations, then move the iteration slider. Compare an uneven initial guess with a nearly symmetric guess.
What to observe: Different guesses take different paths. The converged restricted density agrees when the same solution is reached.
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.

Build a Self-Consistent Hydrogen Molecule

One density, one self-consistency test