Generalized Pauli conditions on the spectra of one-electron reduced density matrices of atoms and molecules

Romit Chakraborty and David A. Mazziotti

Phys. Rev. A 89, 042505 – Published 16 April 2014

Abstract

The Pauli exclusion principle requires the spectrum of the occupation numbers of the one-electron reduced density matrix (1-RDM) to be bounded by one and zero. However, for a 1-RDM from a wave function, there exist additional conditions on the spectrum of occupation numbers, known as pure $N$ -representability conditions or generalized Pauli conditions. For atoms and molecules, we measure through a Euclidean-distance metric the proximity of the 1-RDM spectrum to the facets of the convex set (polytope) generated by the generalized Pauli conditions. For the ground state of any spin symmetry, as long as time-reversal symmetry is considered in the definition of the polytope, we find that the 1-RDM's spectrum is pinned to the boundary of the polytope. In contrast, for excited states, we find that the 1-RDM spectrum is not pinned. Proximity of the 1-RDM to the boundary of the polytope provides a measurement and classification of electron correlation and entanglement within the quantum system. For comparison, this distance to the boundary of the generalized Pauli conditions is also compared to the distance to the polytope of the traditional Pauli conditions, and the distance to the nearest 1-RDM spectrum from a Slater determinant. We explain the difference in pinning in the ground- and excited-state 1-RDMs through a connection to the $N$ -representability conditions of the two-electron reduced density matrix.

Received 23 January 2014

DOI:https://doi.org/10.1103/PhysRevA.89.042505

Authors & Affiliations

Romit Chakraborty and David A. Mazziotti^*

Department of Chemistry and The James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA

^*damazz@uchicago.edu

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Issue

Vol. 89, Iss. 4 — April 2014

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Images

Figure 1
The sets of (a) ensemble and (b) pure $N$ -representable 1-RDMs are shown for a general three-electron ( $N = 3$ ) and six-orbital ( $r = 6$ ) quantum system. The plane defined by the Borland-Dennis equalities in Eq. (6) causes the pure $N$ -representable set of 1-RDMs in (b) to be significantly smaller than the ensemble $N$ -representable set of 1-RDMs in (a). The sets are shown in terms of the first three natural occupation numbers, $λ_{1}$ , $λ_{2}$ , and $λ_{3}$ , ordered from highest to lowest, relative to a fixed set of natural orbitals. These three occupation numbers provide a complete three-dimensional description of the pure 1-RDM spectra because the other occupation numbers are determined from the Borland-Dennis equalities in Eq. (5); they provide a partial description of the full five-dimensional ensemble 1-RDM spectra.
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Figure 2
For the ground state of the lithium atom, the logarithm of the pure distance decreases linearly with the precision of the floating-point calculations. The plot demonstrates that the ground-state 1-RDM spectrum of lithium is pinned to a facet to at least 35 digits of floating-point precision.
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Figure 3
As functions of the bond angle in H $_{3}$ , (a) the Hartree-Fock and correlation energies as well as (b) the minimum Euclidean distances to the boundaries of the pure and ensemble sets and the nearest Slater point are shown. Both the ensemble and Slater distances in (b) are greater than 0.01 for all bond angles, while the pure distance is zero for all angles. The large distance to the nearest Slater point shows that H $_{3}$ is significantly correlated. The vanishing pure distance shows that the generalized Pauli conditions can be saturated even when the traditional Pauli conditions are far from being saturated.
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Figure 4
As functions of the distance between H $_{2}$ dimers, the figure shows (a) the potential-energy surfaces from FCI and the variational 2-RDM method as well as (b) the minimum Euclidean distances from the 1-RDM spectra to the ensemble set and a Slater point from FCI and the variational 2-RDM method. The peak in the Slater distance at a dimer distance of 1 Å shows that the maximum electron correlation occurs when the two H $_{2}$ dimers form a square H $_{4}$ molecule. While the ensemble distance is about 0.01 or larger for all dimer distances, the 1-RDM spectra are pinned to Smith's set of pure $N$ -representable 1-RDMs with time-reversal symmetry. The figures also show that the FCI and variational 2-RDM methods give similar results for both energies and Euclidean distances.
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