Restricted and Unrestricted Hartree–Fock

Introduction

A closed-shell molecule can pair α and β electrons in the same spatial orbitals, but a radical has unpaired electrons and needs a different treatment. Hartree–Fock variants differ mainly in constraints placed on α and β orbitals. Restricted Hartree–Fock is compact for closed shells, restricted open-shell Hartree–Fock keeps a spin-adapted open-shell framework, and unrestricted Hartree–Fock allows the two spin sets to relax independently. Flexibility can improve energies but can also break spin symmetry.

Core explanation

In restricted Hartree–Fock, RHF, each occupied spatial orbital is doubly occupied by one α and one β electron. The two spin functions share the same spatial shape. This ansatz is natural for a closed-shell singlet near an equilibrium geometry and leads to a simple density expression with factor two. It cannot directly represent a state with an odd electron count using only doubly occupied orbitals. Even for an even-electron system, a single paired-orbital determinant may become inadequate as bonds stretch and electron configurations become nearly degenerate.

Restricted open-shell Hartree–Fock, ROHF, allows singly occupied orbitals while keeping paired electrons in common spatial orbitals. It can maintain a cleaner total-spin description for many high-spin open-shell states, though canonical orbital energies and implementation details are less uniquely defined than the basic RHF picture. The method is still a single-determinant or closely related restricted mean-field treatment and does not capture all correlation.

Unrestricted Hartree–Fock, UHF, lets α and β electrons occupy different spatial orbitals. This freedom can lower the variational energy within its larger determinant space and describe spin polarisation. The α and β densities can differ spatially, useful for radicals and stretched bonds. UHF fixes Nα and Nβ, so S z=(Nα−Nβ)/2 is specified, but its determinant is not always an eigenfunction of total S². It can contain admixtures of states with higher total spin, called spin contamination.

For an intended doublet with S=1/2, a pure spin state has ⟨S²⟩=S(S+1)ħ²=3ħ²/4. A UHF output with substantially larger ⟨S²⟩/ħ² indicates higher-spin admixture. The precise significance depends on the question, and a small deviation need not invalidate every property. Reporting only a low energy without inspecting spin can, however, hide a state that is not a good approximation to the intended electronic symmetry.

Hydrogen molecule bond stretching illustrates the tradeoff. Near its equilibrium bond length, an RHF paired-orbital description is useful. Far apart, the correct singlet has roughly one electron on each atom with correlated opposite spins. RHF's single shared spatial orbital can retain unphysical ionic contributions. UHF can lower energy by localising α density toward one H and β density toward the other, but this broken-symmetry determinant is not a pure singlet. A multi-determinant description can represent the separated-atom singlet more faithfully.

Restricted and unrestricted labels describe an orbital ansatz, not whether the underlying physics itself is restricted. A different method or basis may change numerical results. Stability tests can reveal whether an RHF solution is unstable toward a lower-energy UHF solution. That lower energy does not by itself settle which approximate wavefunction best represents a spin-resolved observable or dissociation limit.

Electron count and target multiplicity should be specified before selecting a method. If Nα−Nβ=1, the determinant has S z=1/2 and may target a doublet. If the difference is two, it can target a triplet high-spin component. Total spin still needs checking in UHF. Chemical interpretation should consider both energy and the spin-state quality of the solution.

Step-by-step reasoning

Count electrons and choose a target spin multiplicity. For a closed-shell singlet, RHF is a natural reference. For open shells, compare ROHF's paired-orbital constraint with UHF's separate spin orbitals. After SCF convergence, inspect energy, Nα/Nβ, ⟨S²⟩ and orbital stability. For bond breaking or near degeneracy, consider whether a multi-configuration method is needed.

Visual explanation

Draw RHF as one spatial orbital shape containing an up and a down arrow. Draw UHF as two differently shaped α and β clouds. For stretched H₂, show α density on one atom and β density on the other in a broken-symmetry UHF sketch, then add a note that a pure singlet requires a correlated combination of both spin assignments.

Real-world analogy

Two people sharing one identical route resemble paired electrons constrained to one spatial shape; allowing separate routes resembles UHF flexibility. The extra freedom can make a numerical arrangement easier but may violate a desired collective symmetry. The analogy cannot replace spin algebra or the fact that electrons are indistinguishable.

Real-world example

A calculation on an organic radical often uses UHF or ROHF as a starting reference. UHF can show spin density distributed across a conjugated framework, which is chemically useful. The computed ⟨S²⟩ should be compared with the target value before interpreting spin densities or reaction energies, especially if nearby spin states may mix.

Why?

Why can UHF produce a lower energy than RHF for the same fixed basis and geometry? It optimises over a larger set of determinants because α and β spatial orbitals need not be identical. The variational search therefore cannot be higher merely due to this relaxed constraint. A lower energy may come with broken spin symmetry, so the energetic gain has an interpretive cost.

Common misconception

Unrestricted does not mean exact or free of approximations; UHF remains a one-determinant mean-field method. A fixed S z is not proof of a pure total-spin S state. Conversely, a restricted calculation is not automatically more physically accurate for every geometry, particularly when bond dissociation needs multiple configurations.

Worked example

An OH radical has nine total electrons. A simple doublet determinant uses Nα=5 and Nβ=4, giving S z=(5−4)/2=1/2. For a pure doublet, ⟨S²⟩/ħ² should be S(S+1)=1/2(3/2)=0.75. If an unrestricted calculation reports 1.10, its density may contain higher-spin contamination even though Nα−Nβ remains one and the SCF equations converged.

Quick check

1. Can RHF with only doubly occupied spatial orbitals describe an odd-electron radical directly? Answer: No. An odd electron requires an open-shell treatment such as ROHF or UHF. 2. Does specifying Nα−Nβ determine a UHF wavefunction's exact total S? Answer: No. It fixes S z, while a UHF determinant may mix total-spin eigenstates.

Exam focus

Define the orbital constraint of each method and distinguish S z from S². Use ⟨S²⟩=S(S+1)ħ² for a pure target spin. When comparing RHF and UHF energies, discuss symmetry breaking and correlation rather than assuming the lower number always gives the best physical state description.

Advanced insight

Broken-symmetry UHF can approximate parts of static correlation at stretched geometries through spin polarisation, but it is not a replacement for a spin-pure multi-reference wavefunction. Spin projection methods can extract approximate spin-pure components from broken-symmetry references, with their own assumptions and limitations.

Summary

RHF pairs opposite-spin electrons in common spatial orbitals, ROHF treats open shells with restricted pairing, and UHF allows separate α and β orbital shapes. UHF flexibility can lower energy and reveal spin polarisation but may contaminate total spin. Converged energies, spin diagnostics and bond-breaking behaviour should be interpreted together.

Practice questions

1. What is the target ⟨S²⟩/ħ² for a pure triplet state with S=1? Answer: S(S+1)=1(2)=2. A substantially different UHF value suggests spin contamination or state-assignment trouble. 2. Why may RHF fail qualitatively for widely separated H₂ atoms? Answer: Its doubly occupied shared spatial orbital cannot represent the correlated one-electron-per-atom singlet cleanly and retains unphysical ionic contributions at dissociation. 3. An unrestricted radical calculation has lower energy than a restricted-open-shell calculation. What else should be checked before calling it superior? Answer: Inspect total-spin expectation, orbital stability, relevant properties and whether the determinant represents the intended state; lower energy alone does not settle spin purity.