Canonical form of the Hartree-Fock orbitals in open-shell systems

This work compares different approaches to deriving Hartree-Fock (HF) orbitals and orbital energies for open-shell systems. We compare the basic HF equations underlying both the classic open-shell HF methods, which are the restricted open-shell HF (ROHF) and unrestricted HF (UHF) methods, and a number of the novel (amended) versions of these methods. The main attention is paid to a treatment of the validity of Brillouin's and Koopmans' theorems in the amended versions. We show that these two theorems are fully obeyed only in the special (canonical) form of the ROHF method developed by Plakhutin et al. [J. Chem. Phys. 125, 204110 (2006)] and by Davidson and Plakhutin [J. Chem. Phys. 132, 184110 (2010)], while each of the amended UHF methods suffers from some deficiencies inherent to original UHF and ROHF methods. To compare the HF orbitals derived by different methods in two different forms - DODS (different orbitals for different spins) and SODS (the same orbitals for different spins), we develop the new ROHF-DODS method which combines the use of DODS underlying amended UHF methods and the main advantage of the canonical ROHF method which is a fulfillment of the rigorous Koopmans' conditions. The main result of this work is that the orbitals and orbital energies derived with the new ROHF-DODS method appear identical to those derived with the canonical ROHF method based on the use of SODS. A discussion is presented of some related problems arising in open-shell HF methods such as a violation of the Aufbau principle.