On the energy invariance of open-shell perturbation theory with respect to unitary transformations of molecular orbitals

A number of recently proposed single-reference open-shell perturbation theories based on a spin-restricted open-shell Hartree-Fock reference function are examined, with an emphasis on a consistent formalism within which the theories may be compared. In particular, the effect of unitary transformations among the molecular orbitals on the energy is discussed. Of the seven different perturbation theories examined here, the restricted Mo&slas;;ller–Plesset theory, open-shell perturbation theory method 1, the method of Hubacˇ and Cˇa´rsky, Z-averaged perturbation theory, and invariant open-shell perturbation theory methods are found to be invariant to all types of rotations for which the reference wave function is unaffected, though all are invariant to transformations of a more limited nature. Explicit equations for the generalized invariant forms of each perturbation theory are presented, in order to provide working equations for extension of the theories to local correlation schemes or coupled-cluster perturbational corrections, among others.