Unrestricted Hartree-Fock

In unrestricted Hartree-Fock theory, the wave function model is a single Slater determinant that occupies \(N_\alpha\) \(\alpha\)-spinors and \(N_\beta\) \(\beta\)-spinors. Of course, this means that the underlying spinor basis is spin-separated. The wave function model is then expressed as: \[\begin{equation} \require{physics} \ket{\text{core}} = \qty( \prod_i^{N_\alpha} \hat{a}^\dagger_{i \alpha} ) \qty( \prod_i^{N_\beta} \hat{a}^\dagger_{i \beta} ) \ket{\text{vac}} \thinspace . \end{equation}\]

Using this wave function model, the same-spin \(1\)-DMs are given by: \[\begin{equation} D^\sigma_{ij} = \delta_{ij} \end{equation}\] and the \(2\)-DMs are given by: \[\begin{equation} d_{ijkl}^{\sigma \sigma \tau \tau} = \delta_{ij} \delta_{kl} - \delta_{\sigma \tau} \delta_{il} \delta_{kj} \thinspace . \end{equation}\]

Furthermore, the different-spin \(1\)-DMs vanish in this simple wave function model \[\begin{equation} D^{\alpha \beta}_{pq} = \ev{\hat{a}^\dagger_{p\alpha} \hat{a}_{p \beta}}{\text{core}} = 0 \thinspace . \end{equation}\]

The UHF energy can then be calculated as the expectation value of the Hamiltonian over the UHF wave function model, yielding: \[\begin{equation} E = \sum_{\sigma} \sum_i^{N_\sigma} h_{i\sigma, i\sigma} + \frac{1}{2} \sum_{\sigma \tau} \sum_i^{N_\sigma} \sum_{j}^{N_\tau} (i\sigma i\sigma | j\tau j\tau) - \frac{1}{2} \sum_{\sigma} \sum_{ij}^{N_\sigma} (i\sigma j\sigma | j\sigma i\sigma) \thinspace , \end{equation}\] which can be rewritten in the underlying AO basis as: \[\begin{equation} E = \frac{1}{2} \sum_\sigma \sum_{\mu \nu}^{K_\sigma} P^{\sigma \sigma}_{\mu \nu} \qty( h^{\sigma \sigma}_{\mu \nu} + F^{\sigma \sigma}_{\mu \nu} ) \thinspace , \end{equation}\] by introducing the UHF AO density matrices \(\vb{P}^{\sigma \sigma}\) and the UHF Fock matrices \(\vb{F}^{\sigma \sigma}\).

The UHF wave function model may be optimized through the use of Lagrange multipliers.

Due to the form of the UHF single Slater determinant, it is not an eigenfunction of \(\hat{S}^2\).