Second quantization for systems with a constant number of particles in the Dirac notation

Abstract

AbstractThe relation between the completeness condition for an appropriate one‐particle basis set and the occupation number representation (second quantization) is shown for the time‐independent case. The explicit expressions for the basic symmetric operators are derived in the Dirac bra–ket notation. The physical meaning of these operators, the algebra as well as the connections with the one‐electron density matrix and with the projector on the Fermi sea in the one‐electron approximation, follow directly from these expressions. The generalization for a nonorthogonal basis and the algebra for corresponding basic operators are formulated. The connection with the notion of the molecular diagrams of different kinds for the nonorthogonal atomic orbitals is shown. The Mulliken populations and the Chirgwin–Coulson bond orders are equal to the diagonal and offdiagonal elements of the molecular diagram 1, respectively. The matrix elements of the projector on the Fermi sea in the one‐electron approximation in the representation of nonorthogonal atomic orbitals are elements of the molecular diagram 2.