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arXiv:1710.07221 · Physical Review A

Comment on “Generalization of the Kohn-Sham system that can represent arbitrary one-electron density matrices”

Mario Piris, Katarzyna Pernal

Published Oct 19, 2017• 2 citations
arXivHTMLPDF

Abstract

van Dam [Phys. Rev. A 93, 052512 (2016)] claims that the one-particle reduced density matrix (1RDM) of an interacting system can be represented by means of a single-determinant wave function of fictitious noninteracting particles. van Dam [Phys. Rev. A 93, 052512 (2016)] introduced orbitals within a mean-field framework that produce energy levels similar to Hartree-Fock orbital energies, therefore he also claims that conventional analyses based on Koopmans' theorem are possible in 1RDM functional theory. In this Comment, we demonstrate that both claims are unfounded.

DOI
10.1103/physreva.96.046501

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van Dam [Phys. Rev. A 93, 052512 (2016)] claims that the one-particle reduced density matrix (1RDM) of an interacting system can be represented by means of a single-determinant wave function of fictitious noninteracting particles. van Dam [Phys. Rev. A 93, 052512 (2016)] introduced orbitals within a mean-field framework that produce energy levels similar to Hartree-Fock orbital energies, therefore he also claims that conventional analyses based on Koopmans' theorem are possible in 1RDM functional theory. In this Comment, we demonstrate that both claims are unfounded.
Density functional theory is currently the most widely applied method in electronic structure theory. The Kohn-Sham method, based on a fictitious system of non-interacting particles, is the work horse of the theory. The particular form of the Kohn-Sham wavefunction admits only idem-potent one electron density matrices whereas wavefunctions of correlated electrons in post-Hartree-Fock methods invariably have fractional occupation numbers. Here we show that by generalizing the orbital concept, and introducing a suitable dot-product as well as a probability density a non-interacting system can be chosen that can represent the one-electron density matrix of any system, even one with fractional occupation numbers. This fictitious system ensures that the exact electron density is accessible within density functional theory. It can also serve as the basis for reduced density matrix functional theory. Moreover, to aid the analysis of the results the orbitals may be assigned energies from a mean-field Hamiltonian. This produces energy levels that are akin to Hartree-Fock orbital energies such that conventional analyses based on Koopmans theorem are available. Finally, this system is convenient in formalisms that depend on creation and annihilation operators as they are trivially applied to single determinant wavefunctions.
A system of electrons in a local or nonlocal external potential can be described with one-matrix functional theory (1MFT), which is similar to density-functional theory (DFT) but takes the one-particle reduced density matrix (one-matrix) instead of the density as its basic variable. Within 1MFT, Gilbert derived [Phys. Rev. B 12, 2111 (1975)] effective single-particle equations analogous to the Kohn-Sham (KS) equations in DFT. The self-consistent solution of these 1MFT-KS equations reproduces not only the density of the original electron system but also its one-matrix. While in DFT it is usually possible to reproduce the density using KS orbitals with integer (0 or 1) occupancy, in 1MFT reproducing the one-matrix requires in general fractional occupancies. The variational principle implies that the KS eigenvalues of all fractionally occupied orbitals must collapse at self-consistency to a single level. We show that as a consequence of the degeneracy, the iteration of the KS equations is intrinsically divergent. Fortunately, the level-shifting method, commonly introduced in Hartree-Fock calculations, is always able to force convergence. We introduce an alternative derivation of the 1MFT-KS equations that allows control of the eigenvalue collapse by constraining the occupancies. As an explicit example, we apply the 1MFT-KS scheme to calculate the ground state one-matrix of an exactly solvable two-site Hubbard model.
By introducing a set of auxiliary equations representing a many-body system, we have derived an extension of the Kohn-Sham scheme for the density functional theory. These equations consist of a Kohn-Sham-type equation determining single-particle orbitals and an eigen-value equation for an effective many-body problem. A variational method similar to the Kohn-Sham technique was utilized to derive effective interactions as well as effective potentials without artificial substitution of a Hubbard-type interaction and a mean-field correction in the energy functional. The second equation is described by an effective many-body Hamiltonian with both 2-body interactions and mean-field terms. Rigorous formulation of the extended Kohn-Sham equation is also given in accordance with the Hadjisavvas-Theophilou formulation. Our formulation can be interpreted as a way to define models of the strongly correlated electron systems, e.g. the Hubbard model
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.
Two possible routes are considered to arrive at a one-particle reduced density matrix formulation of electronic structure theory. In the first scheme, an extended Fock matrix H is defined that has twice the dimension of the one-particle basis set. The corresponding Green’s function, defined as the upper left block of (ω1−H)−1, yields the exact one-particle density matrix and energy. The poles of the Green’s function are precisely the ionization potentials and electron affinities of the extended Koopmans theorem. In the second scheme, a generalized Fock equation [F(ρ),ρ]=X is derived that is satisfied by the exact non-idempotent one-particle density matrix. The antisymmetric matrix X on the right-hand side is obtained from the irreducible part of the two-particle reduced density matrix, while F is the usual Fock matrix defined using the correlated one-matrix. The generalized Fock equation is a necessary condition but does not determine ρ uniquely. Alternatively, the one-matrix can be obtained from the irreducible part of the two-matrix directly, using a sum rule. The analysis leads to some additional desiderata and separability properties that may be imposed on traditional wave function based approaches. Possibilities for practical computational schemes are addressed briefly.
We develop a systematic approach to construct energy functionals of the one-particle reduced density matrix (1RDM) for equilibrium systems at finite temperature. The starting point of our formulation is the grand potential Ω [ G ] regarded as variational functional of the Green’s function G based on diagrammatic many-body perturbation theory and for which we consider either the Klein or Luttinger–Ward form. By restricting the input Green’s function to be one-to-one related to a set on one-particle reduced density matrices (1RDM) this functional becomes a functional of the 1RDM. To establish the one-to-one mapping we use that, at any finite temperature and for a given 1RDM γ in a finite basis, there exists a non-interacting system with a spatially non-local potential v [ γ ] which reproduces the given 1RDM. The corresponding set of non-interacting Green’s functions defines the variational domain of the functional Ω . In the zero temperature limit we obtain an energy functional E [ γ ] which by minimisation yields an approximate ground state 1RDM and energy. As an application of the formalism we use the Klein and Luttinger–Ward functionals in the GW-approximation to compute the binding curve of a model hydrogen molecule using an extended Hubbard Hamiltonian. We compare further to the case in which we evaluate the functionals on a Hartree–Fock and a Kohn–Sham Green’s function. We find that the Luttinger–Ward version of the functionals performs the best and is able to reproduce energies close to the GW energy which corresponds to the stationary point.