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 approach based on the paradigm that the real-space implementation of sDFT is useful as a start- expectation values of the system observables can be re- ing point for post-processing DFT-based methods, such garded as random variables in a stochastic process with as the stochastic GW for charge excitations [52, 53], the stochastic time-dependent DFT and Bethe-Salpeter equations for neutral excitations [54–56], and for conduc- ∗ [email protected] tance calculations in warm dense matter [48]. † [email protected] Recently, we developed an sDFT approach based on a 2 non-orthogonal atom-centered basis-set representation in pending on the electron density, n (r): combination with norm-conserving pseudopotentials [43], δEHxc [n] n (r0) 3 attempting to exploit the small energy range and the v [n](r) = = d r0 + v [n](rˆ) Hxc δn (r) r r xc compact basis set for DFT calculations of extended sys- Z | − 0| tems. Our original work focused on describing the total (2) energy per electron and the density of states, but lacked a where EHxc [n] is the Hartree and exchange-correlation description of the forces on the nuclei, which is the main energy functional. subject of the current work. Here, we present a method We use a nonorthogonal atom-centered basis set, to calculate these forces maintaining the linear-scaling of φα (r), α = 1,...,K, with an overlap matrix Sαγ = φ φ , α, γ = 1,...,K. Within such a basis set ap- sDFT and discuss the statistical fluctuations and the re- h α | γ i proach, the K K density matrix (DM) is given as sulting biases for a heterogeneous system of Tryptophan × Zipper 2 (Trp-zip2) peptide solvated in water. Unlike 1 1 P = S− f HS− ; β, µ (3) the QM/MM [57, 58] approach which uses quantum me- chanics (QM) to describe the forces on the active site and ˆ molecular mechanics (MM) to the remaining degrees of where Hαγ = φα hKS φγ and freedom (DOF), we aim to develop a fully quantum me- D E 1 chanical approach, combining deterministic DFT applied f (ε; β, µ) β(ε µ) . (4) to the active site and sDFT to couple it to the remaining ≡ 1 + e − DOF. is the Fermi-Dirac distribution function. The DM is used The manuscript is organized as follows: In Section II, to calculate expected values of single-electron observables we introduce the new formalism for the stochastic forces oˆ as: calculations. Then, in Section III, we present the bench- mark calculations on the Tryptophan Zipper 2 (Trp-zip2) oˆ = 2 Tr [OP ] , (5) h i × peptide in solution. Finally, we summarize and discuss the results in Section IV. where O is the matrix representing oˆ in the basis, with elements: Oαγ = φα oˆ φγ , (6) II. FORCE CALCULATIONS IN STOCHASTIC h | | i DENSITY FUNCTIONAL THEORY and the factor of 2 accounts for the electron’s spin in a closed shell representation. For example, the expectation In this section we describe the theory of the electronic value of the density operator nˆ (r) is the electron density, forces on nuclei within the finite temperature KS-DFT given by: formalism. We set the notations and describe the basis- set representation we use for Kohn-Sham DFT in subsec- n [P ](r) = δ (r rˆ) = 2 P φ (r) φ (r) , (7) tion II A with the combined implementation using real h − i × αγ α γ αγ space grids briefly described in subsection II B. Expres- X sions for the forces are given in subsection II C with a de- The DM in Eq. (3) minimizes the total electronic free- tailed derivation given in Appendix A. Finally, in subsec- energy: tions II D-II E we provide the detail behind the stochastic evaluation of the electronic density and any other ob- 1 Ω[P ] = E [P ] µN [P ] (k β)− [P ] . servables in sDFT (including the forces), and present the − − B Sent statistical errors involved. Here E [P ] is the electronic internal energy, nl loc E [P ] = 2 Tr Ts + VPP + VPP P A. Setting the stage × + EHxc [n [P ]] The KS Hamiltonian is given by: and the number of electrons is given by ˆ ˆ nl loc N [P ] = 2 Tr [SP ] . hKS = ts +v ˆpp +v ˆpp + vHxc [n](r) , (1) × The actual value we use for the chemical potential µ is ˆ 1 2 where ts = 2 (we use atomic units throughout the tuned to enforce N [P ] to be equal to the actual number − ∇ nl paper) is the electron kinetic energy operator, vˆ = of electrons in the system. Finally ent [P ] is the entropy pp S vˆnl , and vˆloc = vloc (rˆ R ) of the non-interacting electrons of the KS system, given C nuclei pp(C) pp C nuclei pp(C) − C are∈ the non-local and local norm-conserving∈ pseudopo- by: P P tential terms in the Kleinman-Bylander form [59, 60] for ent [P ] = 2 kBTr [SP ln SP + nucleus C, at position RC . The last potential term, vˆHxc, S − × (1 SP ) ln (1 SP )] is the Hartree and exchange correlation potential, de- − − 3 Equations (1)-(7) must be solved together, and the re- is given solely in terms of the variations in the Hamilto- sulting solution for the density n (r) and the DM P nian, is called the self-consistent field (SCF) solution to the (δ H) = φ δ vˆnl +v ˆloc φ KS equations. The procedure for reaching SCF solution C αβ α C pp pp β is iterative: in each iteration, called an SCF cycle, P + δC φ α hˆKS φβ + φα hˆKS δC φβ (11) is calculated from H using Eq. (3), n (r) from P from D E D E which vHxc [n](r) is calculated and a new KS Hamilto- and the overlap nian matrix H is built.