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Sandbox Physics
Molecular electrons and the breaking of a bond

L4 · Chapter 9

A fractional-spin test for a broken bond

Sandbox Physics ·

Mixing the two degenerate spin states of a hydrogen atom should not change its exact energy. Approximate energy expressions can nevertheless raise the energy of a balanced spin mixture. That small atom exposes a difficulty also encountered when a singlet bond separates.

Identify the statement before looking at a curve

The source is Cohen, Mori-Sánchez and Yang, Fractional spins and static correlation error in density functional theory: arXiv:0805.1724v1, submitted 12 May 2008, published in Journal of Chemical Physics 129, 121104. Equation 3 states that an ensemble of the two degenerate hydrogen spin ground states has constant exact energy.

Figure 1 connects restricted molecular dissociation on the left with atomic fractional-spin errors on the right. Its caption specifies self-consistent cc-pVQZ calculations; the atomic errors are multiplied by two to compare with a two-atom molecule. The published plot includes HF, LDA and B3LYP. The calculation below instead tests HF and PBE in one fixed STO-3G orbital.

Reading: A. J. Cohen

A fractional occupation does not cut an electron in half

Prepare a whole spin-up hydrogen atom with probability 1/2+γ1/2+\gamma, or a whole spin-down atom with probability 1/2−γ1/2-\gamma, with −1/2≤γ≤1/2-1/2\leq\gamma\leq1/2. Every realization contains one electron. The spin-resolved ensemble densities have fractional occupations, and the total spatial density remains the same.

Both pure states have the same energy. The ensemble energy is the probability-weighted mean of their energies, so it must stay constant as the probabilities change. At γ=0\gamma=0, the spin density is zero everywhere, but the atom still contains one electron and has a one-electron energy.

A distant fragment of a correlated molecular singlet has this kind of mixed spin description when the other fragment is ignored. This connects the atomic test to the separated singlet without assigning a classical permanent spin to either atom.

nα=12+γ,nβ=12−γ,nα+nβ=1n_\alpha=\frac12+\gamma,\qquad n_\beta=\frac12-\gamma,\qquad n_\alpha+n_\beta=1
The numbers are ensemble occupations of a normalized spatial orbital. The electron count stays one.
Eensemble=(12+γ)E0+(12−γ)E0=E0E_{\mathrm{ensemble}}=(\tfrac12+\gamma)E_0+(\tfrac12-\gamma)E_0=E_0
Linearity of the statistical expectation explains the exact constancy in this degenerate one-electron case.

Reading: Sandbox Physics

Calculate the extra HF term directly

Fix the normalized hydrogen STO-3G contraction. Its one-electron energy is h11=−0.46658184955727544h_{11}=-0.46658184955727544 hartree. There is only one spatial function, so no remaining orbital-shape parameter can relax. The Coulomb integral is J=(11∣11)=0.7746059439198982J=(11|11)=0.7746059439198982 hartree.

Evaluate the usual fractional-occupation HF density expression. Same-spin self-interaction cancels, but an opposite-spin term nαnβJn_\alpha n_\beta J remains. In an exact ensemble there is only one electron in each member, so this term describes a spurious interaction between mutually exclusive ensemble alternatives.

At balanced spin, nαnβ=1/4n_\alpha n_\beta=1/4. The extra energy is therefore J/4=0.19365148598J/4=0.19365148598 hartree, and the HF density expression gives −0.27293036358-0.27293036358 hartree. At either pure-spin endpoint the extra term vanishes.

EHF(γ)=h11+(14−γ2)J,EHF(0)−EHF(12)=J4E_{\mathrm{HF}}(\gamma)=h_{11}+(\tfrac14-\gamma^2)J,\qquad E_{\mathrm{HF}}(0)-E_{\mathrm{HF}}(\tfrac12)=\frac J4
This analytic result applies to the stated fixed one-orbital fractional-occupation expression. Averaging the complete HF ensemble members gives the constant one-electron energy; the fractional density expression here yields the extra term.

Reading: Sandbox Physics

Read a newly computed constancy-error curve

The numerical PBE calculation uses the same fixed spatial density, PySCF 2.14.0, Libxc 7.1.2 and the unpruned 99-by-590 atomic grid. At each fractional spin it evaluates the spin-density functional. Its endpoint energy already differs slightly from the exact one-electron energy within this basis.

Subtract each method’s own pure-spin endpoint when testing constancy. For PBE the endpoint is about −0.46437566325-0.46437566325 hartree, and balanced spin gives −0.40932512759-0.40932512759. The constancy error is 0.055050535660.05505053566 hartree. Measuring against the exact finite-basis level would include a separate endpoint offset.

0.000.050.100.150.20-0.500.5
Horizontal: spin imbalance from −0.5 to 0.5Vertical: constancy error in hartree
  • Exact ensemble
  • HF
  • PBE
Original calculation for one fixed normalized STO-3G hydrogen orbital. Each curve is measured from its own pure-spin endpoint; the exact line follows ensemble linearity. These are new finite-basis results with different conditions from the published figure 1.
Original fixed-orbital numerical calculation, energy in hartree. Constancy error is measured from each method’s pure-spin endpoint.
Energy expressionPure-spin endpointBalanced spinConstancy error
Exact finite-basis ensemble−0.466581850-0.466581850−0.466581850-0.4665818500
Fractional-occupation HF−0.466581850-0.466581850−0.272930364-0.2729303640.1936514860.193651486
PBE on the fixed density−0.464375663-0.464375663−0.409325128-0.4093251280.0550505360.055050536

Reading: Sandbox Physics · J. P. Perdew

Separate a reproduced condition from a reproduced figure

The new calculation reproduces the constancy test and its finite-orbital HF mechanism. It does not reproduce the numerical values of the paper’s self-consistent cc-pVQZ figure. The basis, orbital relaxation and selected approximate functional differ. The saved records specify those changes and preserve all calculated points under a CC0 license.

The exact constant energy is an analytic result. The PBE curve is numerical evidence for the specified functional and density. Choosing an approximation that behaves well for larger correlated systems remains a research problem; this one-atom diagnostic does not rank every functional or certify a method for arbitrary bond breaking.

The course began by allowing a single orbital to adjust its shape. The stretched bond then required a richer many-electron state. Fractional spin supplies a precise further question for an approximate functional: does it preserve the energy of degenerate alternatives when their ensemble weights change?

Reading: Sandbox Physics

Try the prediction

Connect the atomic test to the molecular comparison

  1. Read the fractional-spin curve above and identify the two pure-spin endpoints and the balanced midpoint.
  2. In the method-error experiment, compare restricted PBE and RHF errors as the bond stretches.
  3. Explain why the one-atom constancy error is a dissociation diagnostic while the finite-distance molecular error also depends on overlap, orbital relaxation and basis.

Exercises and explanations

  1. At γ=1/4\gamma=1/4, calculate the occupations and the extra HF energy as a fraction of JJ.

    Read the explanation

    The occupations are 3/43/4 and 1/41/4. Their product is 3/163/16, so the spurious opposite-spin term is 3J/163J/16, smaller than the balanced value J/4J/4. The exact one-electron ensemble remains constant.

  2. Why subtract PBE’s own endpoint when plotting its constancy error?

    Read the explanation

    Constancy asks whether changing only the ensemble weights changes the energy. Subtracting the same method’s endpoint isolates that change. Comparing with the exact energy would combine this error with the method’s pure-spin endpoint error.

  3. What would be required to reproduce the original figure quantitatively?

    Read the explanation

    Use the locked paper’s basis, functionals, spin restrictions, self-consistent fractional occupations, geometry units and energy reference, including the factor of two on atomic errors. Match numerical integration and convergence, then compare the actual curves with a documented uncertainty. The one-orbital calculation changes several of those conditions.

Sources and calculation scope

  1. Sandbox Physics, FR13 reference inputs, raw numerical records and reproduction guide (2026)

    Original generated data: H2/STO-3G method energies, forces and occupations; independent Psi4 checks; basis and DFT-grid checks; fixed-orbital fractional-spin data. The guide links scripts, raw outputs, runtime versions and data licenses.

  2. J. P. Perdew, K. Burke and M. Ernzerhof, Generalized Gradient Approximation Made Simple (1996)

    Published PBE generalized-gradient functional, with the 1997 erratum. The method-error experiment selects PBE; its saved numerical record fixes the library, grid and spin restriction.

  3. A. J. Cohen, P. Mori-Sánchez and W. Yang, Fractional spins and static correlation error in density functional theory (2008)

    Locked preprint v1, 12 May 2008; published as J. Chem. Phys. 129, 121104, DOI 10.1063/1.2987202. The final chapter studies the exact constancy condition and a newly computed finite-basis illustration of the hydrogen-atom figure.

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