A quadratic form of the Coulomb operator and an optimization scheme for the extended Kohn-Sham models

To construct an optimization scheme for an extension of the Kohn-Sham approach, I introduce an operator form of the Coulomb interaction. This form is the sum of quadratic form pairs, which can be redefined in a self-consistent calculation of a multi-reference density functional theory. A detailed derivation of the form is given. A fluctuation term introduced in the extended Kohn-Sham scheme is expressed in this form for regularization. The present procedure also provides an exact derivation of effective negative interactions in charge fluctuation channels. Relevance to high-temperature superconductors is discussed.