Unrestricted Hartree-Fock (UHF) Equations

For open-shell systems, UHF uses separate spatial orbitals for alpha ($\alpha$) and beta ($\beta$) electrons. This breaks spin symmetry but allows different spatial distributions for each spin.

Pople-Nesbet Equations

The UHF Fock equations are:

$$\begin{array}{rl}{\mathbf{F}}^{\alpha }{\mathbf{C}}^{\alpha }& =\mathbf{S}{\mathbf{C}}^{\alpha }{\mathbf{\epsilon }}^{\alpha }\\ {\mathbf{F}}^{\beta }{\mathbf{C}}^{\beta }& =\mathbf{S}{\mathbf{C}}^{\beta }{\mathbf{\epsilon }}^{\beta }\end{array}$$

where ${\mathbf{F}}^{\alpha }$, ${\mathbf{F}}^{\beta }$ are the alpha and beta Fock matrices, ${\mathbf{C}}^{\alpha }$, ${\mathbf{C}}^{\beta }$ are the molecular orbital coefficient matrices, $\mathbf{S}$ is the overlap matrix, and ${\mathbf{\epsilon }}^{\alpha }$, ${\mathbf{\epsilon }}^{\beta }$ are the orbital energy vectors.

Fock Matrix Construction

The Fock matrices are:

$$\begin{array}{rl}{\mathbf{F}}^{\alpha }& ={\mathbf{H}}_{\text{core}}+\mathbf{J}-{\mathbf{K}}^{\alpha }\\ {\mathbf{F}}^{\beta }& ={\mathbf{H}}_{\text{core}}+\mathbf{J}-{\mathbf{K}}^{\beta }\end{array}$$

Components

The key difference from RHF:

• RHF: $\mathbf{F}={\mathbf{H}}_{\text{core}}+2\mathbf{J}-\mathbf{K}$ (factor 2 from spin summation)
• UHF: $\mathbf{J}$ uses total density (no factor 2); exchange uses respective spin density

Density Matrices

The alpha and beta density matrices are:

$$\begin{array}{rl}{P}_{\mu \nu }^{\alpha }& =\sum _i^{N_{\alpha }}{C}_{\mu i}^{\alpha }{C}_{\nu i}^{\alpha }\\ P_{\mu \nu }^{\beta }& =\sum _i^{N_{\beta }}{C}_{\mu i}^{\beta }{C}_{\nu i}^{\beta }\end{array}$$

where $N_{\alpha }$ and $N_{\beta }$ are the number of alpha and beta electrons (occupied orbitals).

Electronic Energy

The total electronic energy is the sum of alpha and beta contributions:

$$E_{\text{elec}}=\frac{1}{2}\text{Tr}\left[{\mathbf{P}}^{\alpha }({\mathbf{H}}_{\text{core}}+{\mathbf{F}}^{\alpha })\right]+\frac{1}{2}\text{Tr}\left[{\mathbf{P}}^{\beta }({\mathbf{H}}_{\text{core}}+{\mathbf{F}}^{\beta })\right]$$

Expanding:

$$E_{\text{elec}}=\frac{1}{2}\sum _{\mu \nu }\left[P_{\mu \nu }^{\alpha }(H_{\mu \nu }+F_{\mu \nu }^{\alpha })+P_{\mu \nu }^{\beta }(H_{\mu \nu }+F_{\mu \nu }^{\beta })\right]$$

The total energy includes nuclear repulsion:

$$E_{\text{total}}=E_{\text{elec}}+E_{\text{nuc}}$$

SCF Iteration Procedure

1. Initial guess: Build initial ${\mathbf{P}}^{\alpha }$, ${\mathbf{P}}^{\beta }$ from core Hamiltonian or Hückel/SAD
2. Build Fock matrices: ${\mathbf{F}}^{\alpha }=\mathbf{H}+\mathbf{J}-{\mathbf{K}}^{\alpha }$, ${\mathbf{F}}^{\beta }=\mathbf{H}+\mathbf{J}-{\mathbf{K}}^{\beta }$
3. Solve eigenproblems: Diagonalize ${\mathbf{F}}^{\alpha }{\mathbf{C}}^{\alpha }=\mathbf{S}{\mathbf{C}}^{\alpha }{\mathbf{\epsilon }}^{\alpha }$ and similarly for $\beta$
4. Form new densities: ${\mathbf{P}}_{\text{new}}^{\alpha }$, ${\mathbf{P}}_{\text{new}}^{\beta }$ from occupied orbitals
5. Check convergence: RMS density change $\Delta P=\sqrt{\frac{1}{N}{\sum }_{\mu \nu }(\Delta P_{\mu \nu }^{\alpha }{)}^2+(\Delta P_{\mu \nu }^{\beta }{)}^2}$
6. Mix/damp/DIIS: Apply acceleration if enabled
7. Repeat until convergence

Convergence Criteria

OpenQuantum uses the combined RMS density change:

$$\Delta P_{\text{RMS}}=\sqrt{\frac{{\sum }_{\mu \nu }(P_{\mu \nu }^{\alpha ,\text{new}}-P_{\mu \nu }^{\alpha ,\text{old}}{)}^2+(P_{\mu \nu }^{\beta ,\text{new}}-P_{\mu \nu }^{\beta ,\text{old}}{)}^2}{N_{\text{basis}}^2}}$$

Default threshold: $10^{-8}$.

Acceleration Methods

Implementation in OpenQuantum

Fock Build (crates/scf/src/fock.rs::build_uhf_fock_matrices)

#![allow(unused)]
fn main() {
pub fn build_uhf_fock_matrices(
    h_core: &DMatrix<f64>,
    density_alpha: &DMatrix<f64>,
    density_beta: &DMatrix<f64>,
    eri: &(impl IntegralProvider + Sync),
) -> (DMatrix<f64>, DMatrix<f64>) {
    // Total density P^T = P^α + P^β
    let total_density = density_alpha + density_beta;

    // Coulomb matrix from total density (same for both spins)
    let j = build_coulomb_matrix(&total_density, eri);

    // Exchange matrices (different for each spin)
    let k_alpha = build_exchange_matrix(density_alpha, eri);
    let k_beta  = build_exchange_matrix(density_beta, eri);

    // F^α = H + J - K^α
    let fock_alpha = h_core + &j - &k_alpha;

    // F^β = H + J - K^β
    let fock_beta = h_core + &j - &k_beta;

    (fock_alpha, fock_beta)
}
}

SCF Driver (crates/scf/src/uhf.rs::uhf_scf)

The main loop handles:

• Separate alpha/beta diagonalization with generalized_eigensolve
• DIIS with combined error vectors
• Level shifting on both Fock matrices
• Spin contamination calculation via compute_s_squared
• Convergence checking on combined RMS density change

Initial Guess

OpenQuantum supports three initial guesses for UHF:

1. Core Hamiltonian (default): Diagonalize $H_{\text{core}}$ in orthogonal basis; same MOs for both spins, different occupations
2. Extended Hückel: Approximate MO coefficients from atomic orbital parameters
3. SAD (Superposition of Atomic Densities): Build initial density from atomic fragments

Comparison with RHF

ROHF Connection

Restricted open-shell HF (ROHF) uses a single set of orbitals for both spins but with different occupation numbers. The Fock matrix is a linear combination:

$${\mathbf{F}}^{\text{ROHF}}={\mathbf{F}}^{\text{closed}}+\frac{1}{2}({\mathbf{F}}^{\alpha }+{\mathbf{F}}^{\beta })-\frac{1}{2}({\mathbf{K}}^{\alpha }+{\mathbf{K}}^{\beta })$$

ROHF maintains spin purity ($⟨S^2⟩=S(S+1)$) but often gives higher energies than UHF for open-shell systems.

Further Reading

• Pople & Nesbet, "Self-consistent orbitals for radicals" (1954)
• Szabo & Ostlund, Modern Quantum Chemistry, Ch. 3
• Helgaker, Jørgensen & Olsen, Molecular Electronic-Structure Theory, Ch. 10