Second-Order Perturbation Theory with Spin-Symmetry-Projected Hartree-Fock

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 multireference 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.