General form of the total one-electron Hamiltonian in the restricted open shell Hartree-Fock method

The structure of the effective one-electron Hamiltonian \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat R$$\end{document} in the Hartree-Fock equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat R \varphi _i = \varepsilon _i \varphi _i$$\end{document} is discussed in many works. The most general definitions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat R$$\end{document}, satisfying all necessary conditions imposed by the variational principle for the energy in open shell systems are derived by Dyadyusha and Kuprievich and by Hirao and Nakasutji. In this work it is shown that these definitions cannot be concordant with additional variational conditions imposed by Koopmans’ theorem. A more general form of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat R$$\end{document} is proposed which provides a combination of the variational conditions imposed on the Hartree-Fock orbitals by the variational principle and Koopmans’ theorem.