Unrestricted & Restricted Open-Shell HF

For open-shell systems, OpenQuantum supports both Unrestricted Hartree–Fock (UHF) and Restricted Open-Shell Hartree–Fock (ROHF).

UHF Equations

UHF uses separate spatial orbitals for α and β electrons:

$${\mathbf{F}}^{\alpha }=\mathbf{H}+\mathbf{J}-{\mathbf{K}}^{\alpha }$$ $${\mathbf{F}}^{\beta }=\mathbf{H}+\mathbf{J}-{\mathbf{K}}^{\beta }$$

where:

• $\mathbf{J}$ is the Coulomb matrix from total density ${\mathbf{P}}^T={\mathbf{P}}^{\alpha }+{\mathbf{P}}^{\beta }$
• ${\mathbf{K}}^{\alpha }$ and ${\mathbf{K}}^{\beta }$ are exchange matrices from respective spin densities

The density matrices are: $$P_{\mu \nu }^{\alpha }=\sum _{i(\text{occ}{)}^{\alpha }}{C}_{\mu i}^{\alpha }{C}_{\nu i}^{\alpha }$$ $$P_{\mu \nu }^{\beta }=\sum _{i(\text{occ}{)}^{\beta }}{C}_{\mu i}^{\beta }{C}_{\nu i}^{\beta }$$

The Fock matrices are coupled through the common Coulomb matrix $\mathbf{J}$.

UHF Energy

$$E=\sum _{\sigma =\alpha ,\beta }\mathrm{Tr}[{\mathbf{P}}^{\sigma }(\mathbf{H}+{\mathbf{F}}^{\sigma })]+E_{\text{nuc}}$$

Spin Contamination

The S² expectation value measures spin contamination:

$$⟨S^2⟩=S(S+1)+N_{\beta }-\sum _{i,j}|⟨{\varphi }_i^{\alpha }|{\varphi }_j^{\beta }⟩{|}^2$$

Ideal value: $S(S+1)$ where $S=(N_{\alpha }-N_{\beta })/2$

OpenQuantum reports $⟨S^2⟩$ in the output for every UHF calculation. Significant deviation from the ideal value indicates spin contamination (the determinant is not a pure spin eigenfunction).

ROHF

Restricted Open-Shell HF uses a single set of doubly-occupied orbitals plus singly-occupied orbitals. The Fock matrix is constructed to maintain orbital degeneracy within the open shell. ROHF avoids spin contamination but has a more complex Fock matrix construction.

At convergence, the occupied–occupied and virtual–virtual blocks of the Fock matrix are diagonalized separately (QCPsuC), producing canonical-like orbital energies required for subsequent post-HF calculations (MP2, CCSD(T)).