ABSTRACT
Note: The atomic units are used throughout the paper; in the adopted notation P denotes the square or rectangular matrix, P stands for the row vector, and P represents the scalar quantity.
The internal degrees-of-freedom of molecular systems are of either electronic or nuclear (geometric) origins. In the Born-Oppenheimer (BO) approximation, the equilibrium (ground) state of the externally closed molecule is specified by the overall number of electrons N (integer) in the system and the external potential v(r; Q) due to the nuclei located in the parametrically specified locations corresponding to the internal geometric coordinates Q. Alternatively, the state-parameters N and Q uniquely identify the system (Coulombic) Hamiltonian H^(N, v)¼ H^(N,Q), its ground state [N, v]¼(N, Q), the electronic energy E[N, v]¼h[N, v]jH^(N, v)j[N, v]i¼ E(N,Q), and the BO potential W(N, Q)¼E(N, Q)þVnn(Q), where Vnn(Q) stands for the nuclear repulsion energy. One similarly specifies the equilibrium state of an externally open-system characterized by the fractional average number of electrons,
which is coupled to an external electron reservoir controlling the system chemical potential m¼ @E(N, Q)=@N. In the geometrically rigid molecule this equilibrium state is identified by m and Q. The corresponding equilibrium (relaxed) geometries are identified by the vanishing forces F¼[@W(N, Q)=@Q]T¼ 0, giving rise to the alternative sets of state-parameters, (N, F) and (m, F), in the electronically closed and open (relaxed) systems, respectively (e.g., Chattaraj and Parr, 1993; Cohen, 1996; Geerlings et al., 2003; Nalewajski, 1993, 1995, 1997, 2002a, 2003, 2006a; Nalewajski and Korchowiec, 1997; Nalewajski et al., 1996; Parr and Yang, 1989).
