Abstract
We present a metric-space approach to quantify the performance of approximations in lattice density-functional theory for interacting many-body systems and to explore the regimes where the Hohenberg-Kohn-type theorem on fermionic lattices is applicable. This theorem demonstrates the existence of one-to-one mappings between particle densities, wave functions and external potentials. We then focus on these quantities, and quantify how far apart in metric space the approximated and exact ones are. We apply our method to the one-dimensional Hubbard model for different types of external potentials, and assess the regimes where it is applicable to one of the most used approximations in density-functional theory, the local density approximation (LDA). We find that the potential distance may have a very different behaviour from the density and wave function distances, in some cases even providing the wrong assessments of the LDA performance trends. We attribute this to the systems reaching behaviours which are borderline for the applicability of the one-to-one correspondence between density and external potential. On the contrary the wave function and density distances behave similarly and are always sensitive to system variations. Our metric-based method correctly predicts the regimes where the LDA performs fairly well and the regimes where it fails. This suggests that our method could be a practical tool for testing the efficiency of density-functional approximations.
Highlights
The description and understanding of materials, nano-structures, atoms, molecules, and of their properties is clearly non trivial, as these are interacting and inhomogeneous many-body systems, and their main variable in usual quantum approaches, the wave function, is a 3N-dimensional function, N being the total number of particles
We evaluate the use of designed metrics for density, wave function and potential to appraise lattice-density-functional theory (DFT) approximations beyond total energy arguments, and use the same tools to explore the applicability of the Hohenberg-Kohn-type theorem on a lattice
To illustrate our method, we focus on one of the most used density-functional approximations, the local density approximation (LDA); we will introduce below all the elements of the method outlined above
Summary
DFT demonstrates that, for time-independent systems with a spin-independent external potential, ground-state particle density, the corresponding wave function, or the external potential are sufficient to describe a quantum system. We follow the route proposed in[22,23]: we will consider, together with the exact, that unique interacting system constructed to have the same density as the non-interacting Kohn-Sham system obtained by using the DFT approximation, and with the same particle-particle interaction operator as the exact interacting system We will construct it via reverse engineering, by using the iterative scheme in ref.[21] to find the potential of the interacting lattice system given the approximated density. At this point, the distance between all quantities of interest can be assessed by using appropriate metrics.
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