Accelerating Density Matrix Embedding with Stochastic Density Fitting Theory – An Application to Hydrogen Bonded Clusters

In this work, we demonstrate how using semi-stochastic Density Fitting (ss-DF) can accelerate self-consistent Density Matrix Embedding Theory (DMET) calculations by reducing the number of auxiliary orbitals in the three-indexed DF integrals. This reduction results in significant time savings when building the Hartree Fock (HF) Coulomb and Exchange Matrices and in transforming integrals from the Atomic Orbital (AO) basis to the Embedding Orbital (EO) basis. We apply ss-DF to a range of hydrogen-bonded clusters to showcase its effectiveness. First, we examine how the amount of deterministic space impacts the quality of the calculation in a $\left(\mathrm{H}{2} \mathrm{O}\right)_{10}$ cluster. Next, we test the computational efficiency of ss-DF compared to deterministic DF (d-DF) in water clusters containing 6 to 30 water molecules using a triple-$\zeta$ basis set. Finally, we perform numerical structural optimizations on water and hydrogen fluoride clusters, revealing that DMET can recover weak interactions using a back-transformed energy formula. This work demonstrates the potential of using stochastic resolution of identity in quantum embedding theories and highlights its capability to recover weak interactions effectively.