# Stochastic embedding DFT: theory and application to p-nitroaniline

## TL;DR

This paper introduces a stochastic embedding DFT method that reduces stochastic errors in specific sub-systems, demonstrated on a p-nitroaniline molecule in water, improving force accuracy significantly.

## Contribution

Developed a stochastic embedding DFT method that selectively reduces errors in targeted sub-systems, combining deterministic and stochastic orbitals for improved accuracy.

## Key findings

- Order of magnitude reduction in force errors for the embedded molecule
- Method is exact with infinite stochastic samples
- Applied successfully to p-nitroaniline in water

## Abstract

Over this past decade, we combined the idea of stochastic resolution of identity with a variety of electronic structure methods. In our stochastic Kohn-Sham DFT method, the density is an average over multiple stochastic samples, with stochastic errors that decrease as the inverse square root of the number of sampling orbitals. Here we develop a stochastic embedding density functional theory method (se-DFT) that selectively reduces the stochastic error (specifically on the forces) for a selected sub-system(s). The motivation, similar to that of other quantum embedding methods, is that for many systems of practical interest the properties are often determined by only a small sub-system. In stochastic embedding DFT two sets of orbitals are used: a deterministic one associated with the embedded subspace, and the rest which is described by a stochastic set. The method is exact in the limit of large number of stochastic samples. We apply se-DFT to study a p-nitroaniline molecule in water, where the statistical errors in the forces on the system (the p-nitroaniline molecule) are reduced by an order of magnitude compared with non-embedding stochastic DFT.