  We propose two different schemes for second-order perturbation theory with
spin-projected Hartree-Fock. Both schemes employ the same ansatz for the
first-order wave function, which is a linear combination of spin-projected
configurations. The first scheme is based on the normal-ordered projected
Hamiltonian, which is partitioned into the Fock-like component and the
remaining two-particle-like contribution. In the second scheme, the generalized
Fock operator is used to construct a spin-free zeroth-order Hamiltonian. To
avoid the intruder state problem, we adopt the level-shift techniques
frequently used in other multi-reference perturbation theories. We describe
both real and imaginary shift schemes and compare their performances on small
systems. Our results clearly demonstrate the superiority of the second
perturbation scheme with an imaginary shift over other proposed approaches in
various aspects, giving accurate potential energy curves, spectroscopic
constants, and singlet-triplet splitting energies. We also apply these methods
to the calculation of spin gaps of transition metal complexes as well as the
potential energy curve of the chromium dimer.
