The Pauli principle revisited (Q2517934)

Recall that the Pauli exclusion principle states that an arbitrary quantum state must be occupied by a single electron at most. For many-electron systems, the Pauli principle is equivalent to imposing the condition of antisymmetry on the total many-electron wavefunction that in turn implies some constraints that have to be imposed on the corresponding one-electron reduced density matrices (or 1-RDMs). This actually constitutes the so called famous \(N\)-representability problem for 1-RDMs that was first raised by \textit{Y. Mizuno} and \textit{T. Izuyama} [Prog. Theor. Phys. 18, 33--38 (1957; Zbl 0077.44904)], \textit{R. H. Tredgold} [Phys. Rev. 105, 1421 (1957)], \textit{F. Bopp} [Z. Phys. 156, 348--359 (1959; Zbl 0088.44805)], and by \textit{C. A. Coulson} [Rev. Mod. Phys. 32, 175 (1960)] and formulated by \textit{A. J. Coleman} [Rev. Mod. Phys. 32, 668 (1960)] as to find the complete set of constraints on the 1-RDMs which are generated either by \(N\)-electron wavefunctions (so called pure \(N\)-representability problem) or \(N\)-electron density matrices (so called ensemble \(N\)-representability problem). In the latter work, A. John Coleman ``solves the \(N\)-representability problem for the one-particle reduced density matrix of an ensemble'' [see also e. g. \textit{E. S. Kryachko} and \textit{E. V. Ludeña}, Energy Density Functional Theory of Many-Electron Systems, Kluwer, Dordrecht (1990)] by proving that the Pauli principle constraints are the necessary and sufficient conditions for the set of 1-RDMs to be ensemble \(N\)-representable. The authors of the present work extend the \(N\)-representability problem for 1-RDMs on any allowed type of permutational symmetry. It is worth finally to make the following couple of notes. First: the present work does not consider the precise content of the Pauli principle rather deals with its impact on the structure of \(N\)-representable 1-RDMs. On the other hand, the current discussion of the Pauli principle can be traced in the following recent papers by \textit{I. G. Kaplan} [in Fundamental World of Quantum Chemistry. A Tribute to the Memory of Per-Olov Löwdin, Ed. by E. J. Brändas and E. S. Kryachko, Kluwer, Dordrecht (2003), pp. 183--220; Int. J. Quantum Chem. 89, 268 (2002)]. Second: the independence of the diagonal element of 1-RDM on the permutational symmetry of the state and hence upon the total spin S of the state was proved by \textit{I. G. Kaplan} [Int. J. Quantum Chem. 107, 2595 (2007)].