Restricted HartreefockEdit
Restricted Hartree-Fock (RHF) is a foundational method in quantum chemistry for determining the ground-state electronic structure of atoms and closed-shell molecules. In RHF, each spatial orbital is doubly occupied by a pair of electrons with opposite spins, so the same spatial orbital is used for both α and β electrons. The overall wavefunction is a single Slater determinant, which provides a simple and transparent description of the electronic state while leveraging the mean-field idea that each electron moves in the average field created by all others. This spin-restricted construction makes RHF particularly well suited to closed-shell systems and serves as a standard reference point for more sophisticated correlation methods. See for example Hartree-Fock method and Slater determinant for foundational concepts, as well as Roothaan equations for the matrix formulation that underpins practical implementations.
As a building block in the hierarchy of electronic-structure methods, RHF delivers a clear, tractable picture of bonding and molecular structure while offering insight into trends across families of compounds. It is routinely used as a benchmark, a teaching tool, and a starting point for post-Hartree-Fock approaches such as Møller-Plesset perturbation theory and Coupled-cluster theory, as well as for comparative studies with Density functional theory results. The Roothaan framework, which recasts the self-consistent-field (SCF) problem into a matrix eigenvalue problem, enables calculations with modern Basis sets and large, flexible orbital representations.
Formalism and scope
RHF assumes a closed-shell singlet state in which all electrons are paired. The wavefunction is a single Slater determinant composed of doubly occupied spatial orbitals φi. Each φ_i is expanded in a finite set of basis functions χμ, so φi = Σμ C_μi χ_μ. The spin restriction means that the spatial part of the orbital is the same for both spins, and the spin density vanishes in the sense that there is no net spin polarization. See Molecular orbital and Basis set for related ideas and Spin for how spin appears in quantum descriptions.
The energy expression in RHF is built from one-electron and two-electron contributions. The one-electron part involves integrals h_μν over the basis, while the two-electron part comprises Coulomb (J) and exchange (K) interactions generated by the occupied orbitals. In compact terms, the energy is written using sums over occupied orbitals of h_ii plus half of the sum over occupied pairs of (J_ij − K_ij). The Coulomb and exchange pieces are commonly denoted by Coulomb integrals and Exchange integrals, and their accurate evaluation is central to the method’s performance.
The practical workhorse is the Fock operator F, which includes the one-electron part h and the mean-field two-electron terms built from the occupied orbitals. The core equation in the RHF formalism is the Roothaan equation FC = S C ε, where F is the Fock matrix, S is the overlap matrix between basis functions, C contains the orbital coefficients, and ε are the orbital energies. Solving these equations self-consistently yields the optimized orbitals and the RHF energy. See Fock matrix for the operator and Roothaan equations for the matrix form.
Self-consistent-field (SCF) iterations drive the process: start with a guess for the occupied orbitals, build F from those orbitals, solve the eigenvalue problem to obtain a new set of orbitals, and repeat until convergence. The spin-restricted construction ensures that the wavefunction remains an eigenfunction of the total spin operator for closed-shell systems, avoiding spin-contamination that can arise in some alternative formalisms. See Self-consistent field.
Mathematical formulation (high level)
- Basis expansion: each occupied orbital φi is expanded in a basis set χμ, with coefficients given by the matrix C. See Basis set.
- Energy expression: E = Σ_i h_ii + (1/2) Σ_ij (J_ij − K_ij), where h_ii are one-electron integrals, J_ij are Coulomb integrals, and K_ij are exchange integrals. See Coulomb integral and Exchange integral.
- Fock operator: F = h + Σj (2 J_j − K_j) over all occupied orbitals j, with the resulting matrix Fμν built in the same basis.
- Roothaan equations: FC = S C ε, solved for C and ε given the basis and the occupancy pattern.
- Spin restriction and purity: the method preserves spin symmetry for closed-shell species, which helps avoid certain artifacts that can appear in unrestricted approaches; see Spin contamination for related concerns in other formalisms.
Practical use and limitations
RHF is most appropriate for molecules in their ground-state, closed-shell configurations, where all electrons are paired and there is no net unpaired spin. It provides a straightforward and interpretable picture of bonding and geometry, and it serves as a robust, inexpensive baseline for qualitative trends across families of compounds.
However, RHF omits electron correlation beyond the mean-field, notably dynamic correlation, and therefore often falls short for quantitative energetics and for properties sensitive to correlation. It also struggles with bond dissociation and with systems that exhibit near-degeneracy or diradical character, where a single determinant cannot capture the essential physics. In such cases, post-Hartree-Fock methods or alternative approaches that include correlation more explicitly (for example Electron correlation-aware methods or Density functional theory) are typically preferred. See discussions under Restricted Open-Shell Hartree-Fock and Unrestricted Hartree-Fock for how open-shell cases are handled by related approaches.
Compared to ROHF (restricted open-shell Hartree-Fock) and UHF (unrestricted Hartree-Fock), RHF is spin-pure for closed-shell species but not flexible enough to describe all open-shell situations. ROHF maintains spin-purity for certain open-shell states while preserving a restricted orbital set, whereas UHF allows separate spatial orbitals for α and β electrons at the cost of potential spin contamination. These trade-offs are central to ongoing practical decisions in computational workflows, and they interplay with other methods such as Møller–Plesset perturbation theory and Coupled-cluster theory when moving beyond the HF level. See Unrestricted Hartree-Fock and Restricted Open-Shell Hartree-Fock for deeper contrasts.
In modern practice, RHF remains a staple in teaching and as a reference point, while larger systems or more demanding properties typically call for either spin-adapted open-shell variants or correlation-inclusive frameworks. It also provides a clear starting point for constructing and interpreting post-HF methods and for cross-checking results against density functional theory calculations, which often offer a practical balance between accuracy and computational effort. See Hartree-Fock method and Density functional theory for broader context.