Excited-state theorem in both case of a nuclear potential and a more general external potential in correspondence to the Hohenberg-Kohn theorem

This paper is mainly aimed at presenting an excited-state theorem in both case of a nuclear potential and a more general external potential, in correspondence to the Hohenberg-Kohn (HK) theorem for the static ground state. The author uses the second-quantized scheme employed by HK. The electron field operator is expanded in terms of complete functions. State vectors are represented by electron occupation numbers of each basis state described by the complete function. The main idea is that for the electronic density as a function of spatial coordinates, the author found a one-to-one correspondence between the electronic density and a state vector via occupation numbers. It is found that in an atom and a more general system, there exists a one-to-one correspondence between the electronic density and external potential when the eigenenergy is specified. Thus, the theorem corresponding to the Hohenberg-Kohn theorem is presented for all states at the same level. The advantages and secondary works are as follows: (1) the Kohn-Sham formalism is extended to excited states with the electronic relaxation; (2) in the Schrödinger picture all time-dependent state vectors are derived within DFT; and (3) for simple cases numerical calculations were performed.