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

Abstract

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.