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Sandbox Physics
Molecular electrons and the breaking of a bond

L3 · Chapter 5

Mixing configurations to keep the electrons apart

Sandbox Physics ·

A correlated singlet can keep one electron on each distant atom by mixing two doubly occupied molecular configurations. Their amplitudes cancel the unwanted ionic terms while preserving spin symmetry.

Let the state use more than one configuration

RHF chooses one determinant and optimizes its orbital. Configuration interaction also varies the amplitudes of several determinants. Full configuration interaction, or FCI, includes every permitted determinant in the chosen finite orbital space. Diagonalizing the Hamiltonian then gives the exact eigenstates of that finite-basis problem.

In minimal-basis hydrogen, the even singlet ground state lives in the space spanned by a doubly occupied bonding orbital and a doubly occupied antibonding orbital. Call these ∣g2⟩|g^2\rangle and ∣u2⟩|u^2\rangle. Other symmetry and spin sectors remain part of a full construction, but they do not couple to this ground-state block.

The diagonal entries below are the energies of those configurations. The off-diagonal entry measures their coupling. The eigenvalue equation allows nonzero coefficients only when the determinant of the shifted matrix vanishes. For this two-by-two matrix that condition is a quadratic equation, shown below. Varying both amplitudes is already enough to give the ground state an important freedom that neither restricted orbital optimization nor one spin-broken determinant fully supplies.

∣Ψ⟩=cg∣g2⟩+cu∣u2⟩,∣cg∣2+∣cu∣2=1|\Psi\rangle=c_g|g^2\rangle+c_u|u^2\rangle,\qquad |c_g|^2+|c_u|^2=1
The two configurations are orthonormal when the molecular orbitals are orthonormal.
(A−E)(D−E)−V2=0(A-E)(D-E)-V^2=0
Expanding the shifted two-by-two determinant gives the quadratic equation for both eigenvalues.
Hsinglet=(AVVD),E−=A+D−(A−D)2+4V22H_{\mathrm{singlet}}=\begin{pmatrix}A&V\\V&D\end{pmatrix},\qquad E_-=\frac{A+D-\sqrt{(A-D)^2+4V^2}}2
This real two-by-two matrix is written for electronic energies. Add the same nuclear repulsion to each eigenvalue.

Reading: Sandbox Physics · Psi4 manual

Subtract the antibonding pair

At zero overlap, expand both doubly occupied spatial functions. The bonding pair contains the sum of the four atomic products. The antibonding pair has the same same-atom terms, but opposite signs on the two different-atom terms. Subtracting the pair functions cancels the same-atom terms.

With equal-magnitude, opposite-sign coefficients, the normalized result is proportional to χA(1)χB(2)+χB(1)χA(2)\chi_A(1)\chi_B(2)+\chi_B(1)\chi_A(2). Combine it with the antisymmetric singlet spin function. The full wavefunction is antisymmetric, both nuclei retain one electron at separation, and total spin remains zero.

The antibonding orbital contributes to a better ground state here because a configuration amplitude is part of a many-electron wavefunction. It cannot be interpreted as independently promoting a fixed fraction of molecules to an excited state.

∣g2⟩−∣u2⟩2⟶χA(1)χB(2)+χB(1)χA(2)2  α(1)β(2)−β(1)α(2)2\frac{|g^2\rangle-|u^2\rangle}{\sqrt2}\longrightarrow\frac{\chi_A(1)\chi_B(2)+\chi_B(1)\chi_A(2)}{\sqrt2}\;\frac{\alpha(1)\beta(2)-\beta(1)\alpha(2)}{\sqrt2}
The right side assumes orthonormal separated atomic functions and explicitly includes the singlet spin factor.

Reading: Sandbox Physics · Psi4 manual · A. J. Cohen

Occupations reveal when one pair is no longer enough

The one-electron reduced density matrix collects all predictions for one-electron observables after the other electron has been summed out. Its eigenvectors are natural orbitals. Its spin-summed eigenvalues, called natural occupations, lie between zero and two and add to the electron count.

For this two-configuration singlet, symmetry makes the bonding and antibonding orbitals natural orbitals, with occupations ng=2∣cg∣2n_g=2|c_g|^2 and nu=2∣cu∣2n_u=2|c_u|^2. Near equilibrium the first is close to two and the second close to zero. At separation both tend to one. The second configuration now carries a substantial share of the state.

At R=1.4R=1.4 bohr, STO-3G FCI gives occupations about 1.974590411.97459041 and 0.025409590.02540959. At four bohr they are about 1.356078991.35607899 and 0.643921010.64392101. These are properties of the computed correlated state, not counts of electrons that can be tracked through individual orbitals.

ng+nu=2,ng=2∣cg∣2,nu=2∣cu∣2n_g+n_u=2,\qquad n_g=2|c_g|^2,\qquad n_u=2|c_u|^2
The simple coefficient formula uses the two-determinant even-singlet structure. In a general molecule the reduced density matrix must be diagonalized.

Reading: Sandbox Physics · Psi4 manual

Exact within a basis still leaves spatial error

Near equilibrium, many small configuration contributions describe how electrons avoid one another on short length scales; this is often called dynamic correlation. When two or more configurations are needed with substantial weights, as in the stretched bond, the difficulty is called static correlation. The labels describe useful physical situations, not a unique measurable partition of correlation energy.

FCI removes the determinant-truncation error for a fixed basis. A minimal basis still cannot express all radial flexibility or the sharp short-distance structure of an electron pair. Larger-basis calculations remain necessary for quantitative molecular energies. The small FCI model is valuable because its precise answer lets us isolate method failure from basis limitations.

Reading: Sandbox Physics · Psi4 manual · A. J. Cohen

Try the prediction

Follow the second natural occupation

  1. Compare the equilibrium and stretched presets. Predict whether the second FCI occupation increases or decreases.
  2. Watch the FCI occupations together with the RHF–FCI energy difference.
  3. Compare the UHF and FCI states at a long bond. Explain how both improve the energy through different wavefunction freedoms.

Exercises and explanations

  1. Diagonalize the illustrative electronic block A=−1.8A=-1.8, D=−1.2D=-1.2, V=0.1V=0.1 hartree.

    Read the explanation

    The lower eigenvalue is (−3−0.4)/2≈−1.81622777(-3-\sqrt{0.4})/2\approx-1.81622777 hartree. It is below the better diagonal entry because the state can mix both configurations. A geometry-specific nuclear repulsion must still be added.

  2. If ∣cu∣2=0.1|c_u|^2=0.1, what are the two natural occupations in this block?

    Read the explanation

    They are nu=0.2n_u=0.2 and ng=1.8n_g=1.8. The coherent amplitudes specify one quantum state and encode the joint probabilities of the two electrons.

  3. Can a natural occupation near one by itself prove that a computed state is a singlet?

    Read the explanation

    No. Natural occupations summarize one-electron information and do not determine the complete spin structure. Inspect the state symmetry or squared-spin expectation separately.

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. Psi4 manual, CI: Configuration Interaction

    CI expansion and full configuration interaction within a chosen orbital space. Here FCI is exact only for the stated finite-basis Hamiltonian.

  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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