--- title: "Why a shared orbital struggles when the bond stretches" description: "Expand the restricted hydrogen state at separation, then see how UHF lowers its energy and introduces spin contamination." canonical_url: "https://sandboxphysics.com/stories/molecular-electronic/restricted-and-unrestricted/" language: "en" page_type: "story" alternate_languages: zh-CN: "https://sandboxphysics.com/zh/stories/molecular-electronic/restricted-and-unrestricted/" last_modified: "2026-10-28" --- [Molecular electrons and the breaking of a bond](https://sandboxphysics.com/stories/molecular-electronic/) L2 · Chapter 4 # Why a shared orbital struggles when the bond stretches Sandbox Physics · 2026-10-28 A stretched hydrogen molecule needs one electron on each distant atom. A single doubly occupied delocalized orbital retains configurations with both electrons on the same atom. UHF can lower their cost by allowing opposite spins to use different spatial orbitals. ## Before you begin The normalized bonding orbital and the single-determinant RHF model. A singlet has total spin zero; a triplet has total-spin quantum number one. The squared-spin expectation distinguishes their mixture. [Review the preceding chapter](https://sandboxphysics.com/stories/molecular-electronic/self-consistent-field/) - Find the unwanted double-occupancy terms in restricted dissociation. - Describe the extra freedom in UHF without assigning identities to electrons. - Use the squared-spin expectation to diagnose symmetry breaking. ## Expand the shared orbital before judging its cloud At very large separation the atomic overlap tends to zero, and the bonding orbital becomes $(\chi_A+\chi_B)/\sqrt2$. RHF puts both electrons in it with opposite spins. Its spatial product therefore contains four terms. Two place the electrons on different atoms; two put them together on the same atom. In the zero-overlap limit these alternatives are orthogonal and have equal squared amplitudes. The restricted state gives half its weight to ionic configurations, even though two neutral hydrogen atoms are the lower-energy dissociation products. The one-electron density can still look symmetric and quite plausible. $$ \phi_g(1)\phi_g(2)\longrightarrow\frac12[\chi_A(1)\chi_A(2)+\chi_A(1)\chi_B(2)+\chi_B(1)\chi_A(2)+\chi_B(1)\chi_B(2)] $$ Electron labels mark coordinates in the wavefunction. The omitted spin factor is the normalized antisymmetric singlet. ### One shared doubly occupied orbital 50% 50% At infinite separation, half the weight has one electron on each atom; half has both on the same atom. ### A correlated singlet at separation 100% Mix bonding and antibonding pairs with opposite signs. The double-occupancy terms cancel. - One electron on each atom - Both electrons on one atom Exact algebraic weights in the zero-overlap limit of the two-function model, shown as configuration probabilities. These bars are a limiting argument, not a measured dissociation curve. Reading: Sandbox Physics · A. E. DePrince III · A. J. Cohen ## Let opposite spins choose different shapes Unrestricted Hartree–Fock, or UHF, optimizes separate spatial orbitals for the $\alpha$ and $\beta$ spin functions. Near equilibrium the best orbitals can coincide, giving the RHF answer. Beyond a basis-dependent instability point, the lower UHF branch can place more alpha density near one atom and more beta density near the other. The Hamiltonian still treats the two atoms symmetrically. A reflected solution with the spin densities exchanged has the same energy. The chosen determinant breaks that symmetry. This spin-broken determinant is a variational response to the restricted trial space. The isolated singlet retains its own spin symmetry. For globally optimized states within this same basis, the allowed trial spaces imply $E_{\mathrm{FCI}}\leq E_{\mathrm{UHF}}\leq E_{\mathrm{RHF}}$. A UHF run started from identical spin orbitals can converge to the restricted stationary branch; a spin-polarized guess explores the larger space. Reading: Sandbox Physics · Psi4 manual · A. E. DePrince III ## Lower energy comes with a changed spin state For one alpha and one beta electron in normalized spatial orbitals $a$ and $b$, the UHF determinant has squared-spin expectation $1-|\langle a|b\rangle|^2$ in units of the reduced Planck constant squared. Identical orbitals give zero, the singlet value. Spatially separated orbitals have almost zero overlap and give a value near one. A pure triplet has squared spin two, so a value near one does not mean that UHF has become a pure triplet. In the separated-orbital limit this determinant contains singlet and triplet components. The excess over the target singlet value is called spin contamination. $$ \langle\hat S^2\rangle_{\mathrm{UHF}}=1-|\langle a|b\rangle|^2 $$ This formula is specific to the two-electron, one-alpha/one-beta determinant. The overlap includes the AO overlap matrix when the orbitals are expressed in atomic functions. Reading: Sandbox Physics · A. E. DePrince III ## Read an energy and a spin diagnostic together At $R=4$ bohr in STO-3G, RHF gives about $-0.761082247$ hartree, UHF gives $-0.935842328$, and FCI gives $-0.943778472$. Allowing different spin orbitals removes much of the restricted energy error. The UHF squared-spin expectation is approximately $0.963992$, far from the singlet value zero. This is why method comparison needs more than a total-energy ranking. The UHF energy improves because the trial state gained useful freedom, but the spin state has changed. A correlated singlet can describe neutral dissociation and keep the required spin symmetry; the next chapter constructs that state. Reading: Sandbox Physics · Psi4 manual · A. E. DePrince III ## Try the prediction ### [Find the energy gain and its spin cost](https://sandboxphysics.com/labs/h2-bond-breaking/) 1. Begin near the equilibrium preset and compare RHF with UHF. 2. Stretch the bond and watch where their energy curves separate. Compare the spin-density display with the squared-spin readout. 3. At four bohr, compare all three energies. Explain why the UHF improvement and its spin contamination answer different questions. ## Exercises and explanations 1. If the alpha–beta spatial overlap is $0.8$, what squared spin does this UHF determinant have? Read the explanation $1-0.8^2=0.36$. It is spin contaminated relative to the singlet target. Spatial correlation error and total energy require separate comparisons. 2. The total electron density looks symmetric at large separation. Does that establish correct dissociation? Read the explanation No. RHF and a correlated singlet can both have symmetric total density, but different joint probabilities for finding both electrons on one atom. Inspect energy, pair configurations and occupations as well as the one-electron density. 3. Why should a UHF calculation be started from more than one type of guess? Read the explanation The stationary restricted solution may remain a valid SCF solution after it loses stability in the larger UHF space. A spin-polarized guess helps explore the lower branch. Compare their converged energies and residuals to identify the lower stationary branch. [Previous: Making the density and its field agree](https://sandboxphysics.com/stories/molecular-electronic/self-consistent-field/) [Next: Mixing configurations to keep the electrons apart](https://sandboxphysics.com/stories/molecular-electronic/configuration-interaction/) ## Sources and calculation scope 1. [Sandbox Physics, FR13 reference inputs, raw numerical records and reproduction guide (2026)](https://github.com/yangzichao/sandboxphysics/blob/main/docs/frontier/fr13/reference-reproduction.md) 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. [A. E. DePrince III, The Hartree–Fock Self-Consistent Field Procedure](https://www.chem.fsu.edu/~deprince/tutorials/jupyter_notebooks/hartree_fock/hartree_fock.html) Author tutorial: electronic Hamiltonian, variational principle, density matrices, SCF, basis convergence and spin-symmetry breaking. The browser calculation specializes these equations to two hydrogen STO-3G functions. 3. [Psi4 manual, HF: Hartree–Fock Theory](https://psicode.org/psi4manual/master/scf.html) Theory and numerical methods sections: restricted and unrestricted references, generalized eigenproblem, total energy and convergence. The reproducible reference manifest records the installed version and settings. 4. [Psi4 manual, CI: Configuration Interaction](https://psicode.org/psi4manual/master/detci.html) CI expansion and full configuration interaction within a chosen orbital space. Here FCI is exact only for the stated finite-basis Hamiltonian. 5. [A. J. Cohen, P. Mori-Sánchez and W. Yang, Fractional spins and static correlation error in density functional theory (2008)](https://arxiv.org/abs/0805.1724v1) 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.