Quasipinning and entanglement in the lithium isoelectronic series

Abstract

The Pauli exclusion principle gives an upper bound of 1 on the natural occupation numbers. Recently there has been an intriguing amount of theoretical evidence that there is a plethora of additional generalized Pauli restrictions or (in)equalities, of kinematic nature, satisfied by these numbers [1]. Here for the first time a numerical analysis of the nature of such constraints is effected in real atoms. The inequalities are nearly saturated, or quasi-pinned. For rank-six and rank-seven approximations for lithium, the deviation from saturation is smaller than the lowest occupancy number. For a rank-eight approximation we find well-defined families of saturation conditions.

Optimal Slater-determinant approximation of fermionic wave functions

“…Only recently has the problem in the simplest case of m = 1 been solved [10], generalizing the equalities and inequalities in (1) systematically. This has led to a burst of studies of the relevance and implications of the so-called generalized Pauli constraints like (1) in atoms, molecules, and model systems [11][12][13][14][15][16][17][18][19][20]. As for the next case of m = 2 (which is possibly of more interest from the point of view of calculating the * wdlang06@163.com † mauser@courant.nyu.edu ground state energy of a multi-electron system), a systematic procedure for generating the N -representability conditions on the two-particle reduced density matrix has been derived by Mazziotti [21].…”

Quasipinning and selection rules for excitations in atoms and molecules

“…Such a configuration is in general favorable for the energy in comparison with other types of arrangements [6]. The 1-RDM (a 6 × 6 matrix) is the direct sum of two (3 × 3) matrices, one related to the spin up and the other related to the spin down:…”

“…In a seminal work, Borland and Dennis [2] observed that for the rank 6 approximation of a pure-state N = 3 system, belonging to the Hilbert space ∧ 3 H 6 , the occupation numbers satisfy the following additional conditions: n 1 + n 6 = n 2 + n 5 = n 3 + n 4 = 1 and n 4 n 5 + n 6 . The set of equalities allows exactly one electron in the natural orbitals r and 7 − r. The analysis by Klyachko and coworkers [3,4] of the pure N -representability problem for the 1-RDM establishes a systematical approach, generalizing this type of constraint.…”

“…The nature of those conditions has been explored till now in a few systems: a model of three spinless fermions confined to a one-dimensional harmonic potential [5], the lithium isoelectronic series [6], and ground and excited states of some three-and four-electron molecules with the rank being equal to twice the number of electrons [7]. For reasons that remain mysterious, for all these systems some inequalities are (quite often) nearly saturated; that is, in equations like (1), equality almost holds [8].…”

Conditions for Describing Triplet States in Reduced Density Matrix Functional Theory

“…The class of spin-compensated systems with time-reversal symmetry is a notable exception, since, in that case, the conditions for pure-state N -representability collapse to the ensemble conditions [32]. The necessary and sufficient conditions for pure-state N -representability, also called generalized Pauli constraints, have only recently been discussed and explicitly expressed for systems with a small number of particles and specific finite sizes of the Hilbert space [33][34][35][36][37][38][39]. Recently, it has been demonstrated that with enforcing only the ensemble conditions in a RDMFT calculation for open-shell systems, the pure-state conditions will be violated for many functionals of the 1RDM [40].…”