Forces from stochastic density functional theory under nonorthogonal atom-centered basis sets

Ben Shpiro Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel

Marcel David Fabian Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel

Eran Rabani∗ Department of Chemistry, University of California, Berkeley, California 94720, USA Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA and The Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 69978, Israel

Roi Baer† Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel

We develop a formalism for calculating forces on the nuclei within the linear-scaling stochastic density functional theory (sDFT) in a nonorthogonal atom-centered basis-set representation (Fabian et al. WIREs Comput Mol Sci. 2019;e1412. https://doi.org/10.1002/wcms.1412) and apply it to Tryptophan Zipper 2 (Trp-zip2) peptide solvated in water. We use an embedded-fragment approach to reduce the statistical errors (fluctuation and systematic bias), where the entire peptide is the main fragment and the remaining 425 water molecules are grouped into small fragments. We analyze the magnitude of the statistical errors in the forces and find that the systematic bias is of the order of 3 0.065 eV/Å( 1.2 10− Eh/a0) when 120 stochastic orbitals are used, independently of systems ∼ × size. This magnitude of bias is sufficiently small to ensure that the bond lengths estimated by stochastic DFT (within a Langevin molecular dynamics simulation) will deviate by less than 1% from those predicted by a deterministic calculation.

I. INTRODUCTION an expected value and a fluctuation. In sDFT the mag- nitude of fluctuations can be controlled by increasing the Kohn-Sham density functional theory (KS-DFT) is of- sampling and/or by variance-reducing techniques, such ten used for estimating the forces on the nuclei in ab- as the embedded-fragment method [40–43] or the energy initio molecular dynamics simulations, with which reli- windowing approach [44, 45]. Due to the nonlinear na- able predictions concerning structure and properties of ture of the Kohn-Sham theory, the fluctuations create molecules can be made. Despite the fact that it can be bias errors, i.e. the random variable’s expected value de- used to study extended molecular systems relevant to viates from the precise quantum mechanical expectation biomolecular chemistry and materials science [1–4], the value [43]. The magnitude of the bias can be controlled conventional implementations are slow due to cubic al- by using the above-mentioned variance-reducing meth- gorithmic complexity. Therefore, several approaches to ods. KS-DFT have been developed and are routinely used for Early implementations of sDFT were based on real- treating such extended systems. These include linear- space grid representations of the electron density [39, scaling approaches which rely on electron localization 41, 42, 46, 47] and were applied to relatively homoge- within the system’s interior volume [5–33], or the tight- neous systems, either to bulk silicon or H-He mixtures binding DFT approach, which uses a very small basis set with periodic boundary conditions [42, 45, 47, 48] or to finite-sized hydrogen-passivated silicon nanocrystals arXiv:2108.06770v1 [physics.chem-ph] 15 Aug 2021 complemented by approximations calibrated with empir- ical data [34–36], and the orbital-free DFT, which is ap- (with impurities) and water cluster [41, 42, 49, 50]. We plicable to relatively homogeneous systems [37, 38]. also developed a Langevin dynamics approach to sam- Here, we focus on the stochastic DFT (sDFT) [39], a ple the Boltzmann-weighted configurations [41, 51]. The linear-scaling a