Unrestricted Hartree–Fock

Unrestricted Hartree–Fock (UHF) theory is the most common molecular orbital method for open shell molecules where the number of electrons of each spin are not equal. While restricted Hartree–Fock theory uses a single molecular orbital twice, one multiplied by the α spin function and the other multiplied by the β spin function in the Slater determinant, unrestricted Hartree–Fock theory uses different molecular orbitals for the α and β electrons. This has been called a different orbitals for different spins (DODS) method. The result is a pair of coupled Roothaan equations, known as the Pople–Nesbet–Berthier equations.

Where ${\displaystyle \mathbf {F} ^{\alpha }\ }$ and ${\displaystyle \mathbf {F} ^{\beta }\ }$ are the Fock matrices for the ${\displaystyle \alpha \ }$ and ${\displaystyle \beta \ }$ orbitals, ${\displaystyle \mathbf {C} ^{\alpha }\ }$ and ${\displaystyle \mathbf {C} ^{\beta }\ }$ are the matrices of coefficients for the ${\displaystyle \alpha \ }$ and ${\displaystyle \beta \ }$ orbitals, ${\displaystyle \mathbf {S} }$ is the overlap matrix of the basis functions, and ${\displaystyle \mathbf {\epsilon } ^{\alpha }\ }$ and ${\displaystyle \mathbf {\epsilon } ^{\beta }\ }$ are the (diagonal, by convention) matrices of orbital energies for the ${\displaystyle \alpha \ }$ and ${\displaystyle \beta \ }$ orbitals. The pair of equations are coupled because the Fock matrix elements of one spin contains coefficients of both spin as the orbital has to be optimized in the average field of all other electrons. The final result is a set of molecular orbitals and orbital energies for the α spin electrons and a set of molecular orbitals and orbital energies for the β electrons.

...
Wikipedia