Significant conditions for the two-electron reduced density matrix from the constructive solution ofNrepresentability

Abstract

We recently presented a constructive solution to the N -representability problem of the twoelectron reduced density matrix (2-RDM)-a systematic approach to constructing complete conditions to ensure that the 2-RDM represents a realistic N -electron quantum system [D. A. Mazziotti, Phys. Rev. Lett. 108, 263002 (2012)]. In this paper we provide additional details and derive further N -representability conditions on the 2-RDM that follow from the constructive solution. The resulting conditions can be classified into a hierarchy of constraints, known as the (2, q)-positivity conditions where the q indicates their derivation from the nonnegativity of q-body operators. In addition to the known T1 and T2 conditions, we derive a new class of (2,3)-positivity conditions. We also derive 3 classes of (2,4)-positivity conditions, 6 classes of (2,5)-positivity conditions, and 24 classes of (2,6)-positivity conditions. The constraints obtained can be divided into two general types: (i) lifting conditions, that is conditions which arise from lifting lower (2, q)-positivity conditions to higher (2, q + 1)-positivity conditions and (ii) pure conditions, that is conditions which cannot be derived from a simple lifting of the lower conditions. All of the lifting conditions and the pure (2, q)-positivity conditions for q > 3 require tensor decompositions of the coefficients in the model Hamiltonians. Subsets of the new N -representability conditions can be employed with the previously known conditions to achieve polynomially scaling calculations of ground-state energies and 2-RDMs of many-electron quantum systems even in the presence of strong electron correlation.

Quasipinning and selection rules for excitations in atoms and molecules

“…It is worth mentioning that the Nrepresentability constraints for the ensemble 2-RDM contain as a subset all of these "classical" N -representability conditions [26,27].…”

Functional <i>N</i>-Representability in 2-Matrix, 1-Matrix, and Density Functional Theories

“…Summing up, as in the variational upper bound ations based on wave functions, where the accuracy is improved when the variational space is made larger, in the 2-RDM theory, the accuracy is also improved by imposition of more N -representability conditions. Finally, we are f oming a fourth stage marke P bounds e t calcul ath d by im onstructive solution of N-representability establis portant mathematical developments such as Mazziotti's very recent discovery of complete N-representability conditions for the 2-RDM and higher order reduced density matrices [59,60]. No doubt, faster and more efficient computer codes will be developed to implement these new ideas.…”

“…Lower bounds on the ground-state energy can be computed and improved systematically within the theory. While not all of the 2-RDM conditions will be imposed in practical calculations, a complete knowledge of the conditions their form and functions can be invaluable in devising and testing approximate N-representability conditions for different types of quantum systems and interactions" [60]. …”

Reduced density-matrix functionals from many-particle theory

“…The approximate 2-RDM should be hermitian, antisymmetric and ensemble or purestate N-representable. The complete set of ensemble N -representability conditions of the 2-RDM D (2) were given by Mazziotti [36,37]. A set of necessary pure-state Nrepresentability conditions for the 2-RDM are also known [38].…”