  Spin-projected Hartree-Fock is introduced as a particle-hole excitation
ansatz over a symmetry-adapted reference determinant. Remarkably, this
expansion has an analytic expression that we were able to decipher. While the
form of the polynomial expansion is universal, the excitation amplitudes need
to be optimized. This is equivalent to the optimization of orbitals in the
conventional projected Hartree-Fock framework of non-orthogonal determinants.
Using the inverse of the particle-hole expansion, we similarity transform the
Hamiltonian in a coupled-cluster style theory. The left eigenvector of the
non-hermitian Hamiltonian is constructed in a similar particle-hole expansion
fashion, and we show that to numerically reproduce variational projected
Hartree-Fock results, one needs as many pair excitations in the bra as the
number of strongly correlated entangled pairs in the system. This
single-excitation polynomial similarity transformation theory is an alternative
to our recently presented double excitation theory, but supports projected
Hartree-Fock and coupled cluster simultaneously rather than interpolating
between them.
