What is the best expression of the second-order sum-over-state perturbation energy based on the Hartree--Fock wavefunction

We have improved in a stepwise manner the second-order sum-over-state perturbation energy which is written in the infinite sum over singly excited configurations based on the Hartree-Fock (HF) zeroth-order wavefunction. Firstly, the conventional equation [Eq. (10)] is shown to be dependent on the unitary transformations among singly excited configurations. The best unitary transformation gives the Tamm-Dancoff (or singly excited configuration interaction) approximation of the excited states. The resultant equation [Eq. (25)] includes in a simple form all of the coupling terms between different singly excited configurations. As a restrictive special case, this unitary transformation leads to the modified HF orbitals of Silverstone-Huzinaga and Morokuma-Iwata. Secondly, it is shown that the second-order energy of the coupled HF theory can also be written in a simple sum-over-state perturbation formula. The resultant equation [Eq. (50)] does not require an iterative solution. Moreover, it is shown that this is the best possible expression of the second-order energy based on the HF zeroth-order wavefunction. As a restrictive special case, this treatment produces new improved modified HF operators which are thought to be superior to those hitherto given.