  Projected Hartree-Fock theory (PHF) has a long history in quantum chemistry.
PHF is here understood as the variational determination of an N-electron broken
symmetry Slater determinant that minimizes the energy of a projected state with
the correct quantum numbers. The method was actively pursued for several
decades but seems to have been abandoned. We here derive and implement a
"variation after projection" PHF theory using techniques different from those
previously employed in quantum chemistry. Our PHF methodology has modest
mean-field computational cost, yields relatively simple expressions, can be
applied to both collinear and non-collinear spin cases, and can be used in
conjunction with deliberate symmetry breaking and restoration of other
molecular symmetries like complex conjugation and point group. We present
several benchmark applications to dissociation curves and singlet-triplet
energy splittings, showing that the resulting PHF wavefunctions are of high
quality multireference character. We also provide numerical evidence that in
the thermodynamic limit, the energy in PHF is not lower than that of
broken-symmetry HF, a simple consequence of the lack of size consistency and
extensivity of PHF.
