# Self-Consistent Density-Functional Embedding: a Novel Approach for Density-Functional Approximations

## TL;DR

This paper introduces a self-consistent embedding method in density functional theory that directly targets the density-to-potential mapping, achieving accurate results with small fragments and reproducing key features of the exact potential.

## Contribution

It presents a novel self-consistent embedding approach that directly approximates the density-to-potential mapping, improving accuracy over traditional energy-functional methods.

## Key findings

- Accurately reproduces steps and peaks in the exchange-correlation potential.
- Achieves convergence to exact results with increasing fragment size.
- Performs well in molecular bond stretching in 1D and 2D systems.

## Abstract

In the present work, we introduce a Self-Consistent Density-Functional Embedding technique, which leaves the realm of standard energy-functional approaches in Density Functional Theory and targets directly the density-to-potential mapping that lies at its heart. Inspired by the Density Matrix Embedding Theory, we project the full system onto a set of small interacting fragments that can be solved accurately. Based on the rigorous relation of density and potential in Density Functional Theory, we then invert the fragment densities to local potentials. Combining these results in a continuous manner provides an update for the Kohn-Sham potential of the full system, which is then used to update the projection. The scheme proposed here converges to an accurate approximation for the density and the Kohn-Sham potential of the full system. Convergence to exact results can be achieved by increasing the fragment size. We find, however, that already for small embedded fragments accurate results are obtained. We benchmark our approach for molecular bond stretching in one and two dimensions and demonstrate that it reproduces the known steps and peaks that are present in the exact exchange-correlation potential with remarkable accuracy.