eQE: An open-source density functional embedding theory code for the condensed phase

In this work, we present the main features and algorithmic details of a novel imple- mentation of the Frozen Density Embedding (FDE) formulation of subsystem Density Functional Theory (DFT) that is specifically designed to enable ab initio molecu- lar dynamics (AIMD) simulations of large-scale condensed-phase systems containing 1000s of atoms. This code (available at http://eqe.rutgers.edu) has been given the moniker of embedded Quantum ESPRESSO (eQE) as it is a generalization of the open-source Quantum ESPRESSO (QE) suite of programs. The strengths of eQE reside in a hierarchical parallelization scheme that allows for an efficient and fully self- consistent treatment of the electronic structure (via the addition of an additional DIIS extrapolation layer) while simultaneously exploiting the inherent symmetries and pe- riodicities in the system (via sampling of subsystem-specific first Brillouin zones and utilization of subsystem-specific basis sets). While bulk liquids and molecular crystals are two classes of systems that exemplify the utility of the FDE approach (as these systems can be partitioned into weakly interacting subunits), we show that eQE has significantly extended this regime of applicability by outperforming standard semilocal Kohn-Sham DFT (KS-DFT) for large-scale heterogeneous catalysts with quite different layer-specific electronic structure and intrinsic periodicities. eQE features very favor- able strong parallel scaling for a model system of bulk liquid water composed of 256 water molecules, which allows for a significant decrease in the overall time to solution when compared to KS-DFT. We show that eQE achieves speedups greater than one order of magnitude (> 10×) when performing AIMD simulations of such large-scale condensed-phase systems as: (1) molecular liquids via bulk liquid water represented by 1024 independent water molecules (3072 atoms with a 25.3× speedup over KS-DFT), (2) polypeptide/biomolecule solvation via (gly) 6 solvated in (H 2 O) 395 (1230 atoms with a 38.6× speedup over KS-DFT), and (3) molecular crystals via a 3×3×3 peri- odic supercell of pentacene (1940 atoms with a 12.0× speedup over KS-DFT). These results represent a significant improvement over the current state-of-the-art and now enable subsystem DFT based AIMD simulations of realistically-sized condensed-phase systems of interest throughout chemistry, physics, and materials science.