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arXiv:2207.02616

Reply to Comment on "Self-Consistent-Field Method for Correlated Many-Electron Systems with an Entropic Cumulant Energy", arXiv2202.05532v1

Jian Wang, Evert Jan Baerends

Published Jul 6, 2022• 2 citations
arXivHTMLPDF

Abstract

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 $i$-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$^+$.

DOI
10.48550/arxiv.2207.02616

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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 [Phys. Rev. Lett. 128, 013001 (2022)] a novel ground state method was proposed. It has been suggested that this iii-DMFT would be a method within one-particle reduced density matrix functional theory (DMFT), capable of describing accurately molecules at various geometries with an information-theoretical nature. We reassess this work and its suggestions from a conceptual and practical point of view, leading to the following conclusions: i) A method which assigns to each molecule M\mathcal{M}M its own functional F ⁣M\mathcal{F}_{\!\mathcal{M}}FM​ is not a functional theory (striking violation of "universality") ii) even for the simplest systems iii-DMFT yields incorrect one-particle reduced density matrices and iii) the use of an information-theoretical concept to describe molecular dissociation limits was not essential. The latter insight may help to fix the deficiency of iii-DMFT to not reproduce correctly the smaller occupation numbers and thus to not recover the important dynamic correlations.
The self-consistent-field method proposed recently [Wang et al., Phys. Rev. Lett. 128, 013001 (2022)] is discussed in more detail. The method leads to self-consistent eigenvalue equations for the natural spin orbitals and Fermi-Dirac distribution for the orbital occupation numbers. The entropic functional contains two parameters ($\ensuremath{\kappa}$ and bbb), which can be fitted to various experimental or theoretical data such as the dissociation energy, entropy, and total energy at a given geometry, etc. Calculations are demonstrated on the square H4{\mathrm{H}}_{4}H4​ and hydrogen chains of H50{\mathrm{H}}_{50}H50​, which are representative cases of degenerate or nondynamically correlated systems.
Partition density-functional theory is a formally exact method for solving the ground-state electronic structure problem. It operates by dividing the system into noninteracting fragments along with a global effective potential, which, when solved to self-consistency, yields the exact energy and density (as the sum of the fragment densities) of the interacting system. A recent paper [Phys. Rev. A 86, 012504 (2012)] questions the uniqueness of the fragment occupations and fragment densities. We demonstrate that their analysis is incorrect. We describe some of the technical details of partition density-functional calculations and discuss the relationship to other embedding schemes.
The question of which non-interacting Green's function 'best' describes an interacting many-body electronic system is both of fundamental interest as well as of practical importance in describing electronic properties of materials in a realistic manner. Here, we study this question within the framework of Baym-Kadanoff theory, an approach where one locates the stationary point of a total energy functional of the one-particle Green's function in order to find the total ground-state energy as well as all one-particle properties such as the density matrix, chemical potential, or the quasiparticle energy spectrum and quasiparticle wave functions. For the case of the Klein functional, our basic finding is that minimizing the length of the gradient of the total energy functional over non-interacting Green's functions yields a set of self-consistent equations for quasiparticles that is identical to those of the quasiparticle self-consistent GW (QSGW) (van Schilfgaarde et al 2006 Phys. Rev. Lett. 96 226402-4) approach, thereby providing an a priori justification for such an approach to electronic structure calculations. In fact, this result is general, applies to any self-energy operator, and is not restricted to any particular approximation, e.g., the GW approximation for the self-energy. The approach also shows that, when working in the basis of quasiparticle states, solving the diagonal part of the self-consistent Dyson equation is of primary importance while the off-diagonals are of secondary importance, a common observation in the electronic structure literature of self-energy calculations. Finally, numerical tests and analytical arguments show that when the Dyson equation produces multiple quasiparticle solutions corresponding to a single non-interacting state, minimizing the length of the gradient translates into choosing the solution with largest quasiparticle weight.