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Physical Review A

Reply to “Comment on “Nonuniqueness of algebraic first-order density-matrix functionals” ”

Jian Wang, Peter J. Knowles

Published Feb 14, 2018• 5 citations
Publisher

Abstract

It is shown that symmetry considerations do not alter the conclusions of our original work [Phys. Rev. A 92, 012520 (2015)], and that there exists an example of an electronic system for which at several geometries the one-matrix eigenvalues are identical, but the two-matrix spectrum is not. It is still therefore the case that JK and related functionals that depend on the one-matrix eigenvalues to model the two-matrix cannot be made arbitrarily accurate.

DOI
10.1103/physreva.97.026502

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Nonuniqueness of algebraic first-order density-matrix functionals

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Comment on “Nonuniqueness of algebraic first-order density-matrix functionals”

O. V. Gritsenko

Wang and Knowles (WK) [Phys. Rev. A 92, 012520 (2015)] have given a counterexample to the conventional in reduced density-matrix functional theory representation of the second-order reduced density matrix (2RDM) ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ in the basis of the natural orbitals as a function ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$ of the orbital occupation numbers (ONs) ${n}_{i}$. The observed nonuniqueness of ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ for prototype systems of different symmetry has been interpreted as the inherent inability of ON functions to reproduce the 2RDM, due to the insufficient information contained in the 1RDM spectrum. In this Comment, it is argued that, rather than totally invalidating ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$, the WK example exposes its symmetry dependence which, as well as the previously established analogous dependence in density functional theory, is demonstrated with a general formulation based on the Levy constrained search.

Physical Review A•Feb 14, 2018•7 citations
It is shown that symmetry considerations do not alter the conclusions of our original work [Phys. Rev. A 92, 012520 (2015)], and that there exists an example of an electronic system for which at several geometries the one-matrix eigenvalues are identical, but the two-matrix spectrum is not. It is still therefore the case that JK and related functionals that depend on the one-matrix eigenvalues to model the two-matrix cannot be made arbitrarily accurate.
Recently we proposed an information entropy based method for electronic structure calculations within the density-matrix functional theory(DMFT) (Phys. Rev. Lett. 128, 013001), dubbed as iii-DMFT. Comments have been raised regarding the accuracy of the one-particle density-matrix compared to that from the wave function, and the universality of the functional. We address these questions and the problematic case of HeH+^++.
In a recent paper [Phys. Rev. A 77, 062511 (2008)], it was advocated to modify the variational principle for the action-integral functional in the Runge-Gross foundation of time-dependent density-functional theory. This was criticised in a subsequent paper [Phys. Rev. A 62, 052510 (2010)] by the present author. In a Comment [Phys. Rev. A 83, 046501 (2011)], it is argued that the criticism is unfounded. This is a response as given to the four specific points raised in the Comment, clarifying and confirming the essence of the original criticism.
Wang and Knowles (WK) [Phys. Rev. A 92, 012520 (2015)] have given a counterexample to the conventional in reduced density-matrix functional theory representation of the second-order reduced density matrix (2RDM) ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ in the basis of the natural orbitals as a function ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$ of the orbital occupation numbers (ONs) ni{n}_{i}ni​. The observed nonuniqueness of ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}$ for prototype systems of different symmetry has been interpreted as the inherent inability of ON functions to reproduce the 2RDM, due to the insufficient information contained in the 1RDM spectrum. In this Comment, it is argued that, rather than totally invalidating ${\mathrm{\ensuremath{\Gamma}}}_{ij,kl}(n)$, the WK example exposes its symmetry dependence which, as well as the previously established analogous dependence in density functional theory, is demonstrated with a general formulation based on the Levy constrained search.