Stochastic density functional theory Marcel David Fabian∗ Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel Ben Shpiro∗ Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel Eran Rabaniy Department of Chemistry, University of California, Berkeley, California 94720, USA, and 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 Daniel Neuhauserz Department of Chemistry and Biochemistry, University of California, Los Angeles California 90095, USA Roi Baerx Fritz Haber Center for Molecular Dynamics and Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel Linear-scaling implementations of density functional theory (DFT) reach their intended efficiency regime only when applied to systems having a physical size larger than the range of their Kohn- Sham density matrix (DM). This causes a problem since many types of large systems of interest have a rather broad DM range and are therefore not amenable to analysis using DFT methods. For this reason, the recently proposed stochastic DFT (sDFT), avoiding exhaustive DM evaluations, is emerging as an attractive alternative linear-scaling approach. This review develops a general formulation of sDFT in terms of a (non)orthogonal basis representation and offers an analysis of the statistical errors (SEs) involved in the calculation. Using a new Gaussian-type basis-set implementation of sDFT, applied to water clusters and silicon nanocrystals, it demonstrates and explains how the standard deviation and the bias depend on the sampling rate and the system size in various types of calculations. We also develop basis-set embedded-fragments theory, demonstrating its utility for reducing the SEs for energy, density of states and nuclear force calculations. Finally, we discuss the algorithmic complexity of sDFT, showing it has CPU wall-time linear-scaling. The method parallelizes well over distributed processors with good scalability and therefore may find use in the upcoming exascale computing architectures. I. INTRODUCTION free DFT [36, 37] approaches using density-dependent kinetic energy functionals. The first two types of ap- proaches mentioned above are designed to answer ques- Density functional theory (DFT) is emerging as a tions typically asked about molecules, while for materials usefully-accurate general-purpose computational plat- and other large scale systems, we are more interested in form for predicting from first principles the ground-state coarse-grained properties. For example, with molecules, structure and properties of systems spanning a wide one is interested in bond orders, bond lengths spectral range of length scales, from single atoms and gas-phase lines; while for large systems we are more interested molecules, through macromolecules, proteins, nanocrys- in atomic densities, pair-correlation distributions (mea- tals, nanosheets, nanoribbons, surfaces, interfaces up to sured using neutron scattering) as well as charge/spin periodic or amorphous homogeneous or heterogeneous densities, polarizabilities and optical and electrical con- materials [1{5]. Significant efforts have been diverted to- ductivity. In molecules, we strive to understand each oc- wards the development of numerical and computational cupied/unoccupied Kohn-Sham eigenstate while in large methods enabling the use of DFT for studying exten- systems we are concerned with the density of hole and sive molecular systems. Several routes have been sug- electron states. gested: linear-scaling approaches [6{32], relying on the sparsity of the density matrix [33], DFT-based tight- Of course, detailed \molecular type" questions can also binding (DFTB) methods [34{36] which reduce the nu- arise in large systems, primarily when the processes of in- merical scaling using model Hamiltonians. Moreover, terest occur in small pockets or localized regions | for significant efforts have gone towards developing orbital- example, biochemical processes in proteins, localized cat- 2 alytic events on a surface, impurities in solids, etc. Here, A. Traditional basis-set formulation of Kohn-Sham a combination of methods, where the small subsystem equations with cubic scaling can be embedded in the larger environment is required. The Kohn-Sham (KS) density functional theory (KS- In this advanced review, we will focus on the stochas- DFT) is a molecular orbitals (MOs) approach which can tic DFT (sDFT) approach, developed using grids and be applied to a molecular system of N electrons using a plane-waves in recent years [38{42] but also based on e basis-set of atom-centered orbitals φ (r), α = 1; : : : ; K: ideas taken from works starting in the early 1990's, α The basis functions were developed to describe the elec- mainly within the tight-binding electronic structure tronic structure of the parent atom, and for molecules framework [43{50]. We make the point that the effi- they are the building blocks from which the orthonormal ciency of sDFT results from its adherence to answering MOs are built as superpositions: the coarse-grained \large system questions" mentioned above, rather than those asked for molecules. K X The new viewpoint taken here is that of stochastic (r) = φ (r) C ; n = 1; : : : ; K: (1) DFT using non-orthogonal localized basis-sets. The pri- n α αn α=1 mary motivation behind choosing local basis-sets is that they are considerably more compact than plane-waves In the simplest \population" model, each MO can either and therefore may enable studying significantly larger \occupy" two electrons (of opposing spin) or be empty. systems. Deterministic calculations using local basis-sets The occupied MOs (indexed as the first Nocc = Ne=2 are more readily applicable to large systems, and thus MOs) are used to form the total electron density: can generate useful benchmarks with which the statisti- Nocc cal errors and other properties characterizing sDFT can X 2 be studied in detail. n (r) = 2 × j n (r)j : (2) n The review includes three additional sections, further divided into subsections, to be described later. Sec- The coefficient matrix C in Eq. (1) can be obtained tion II reviews the theory and techniques used for non- from the variational principle applied to the Schr¨odinger orthogonal sDFT and studies in detail the statistical er- equation, leading to the Roothaan-Hall generalized eigen- rors and their dependence on sampling and system size. value equations [51, 52] (we follow the notations in In section III we explain the use of embedded fragments refs. [53{55]): and show their efficacy in reducing the stochastic errors FC = SCE: (3) of sDFT. Section IV summarizes and discusses the find- ings. Here, F = T + V en + J [n] + V xc [n] is the K × K KS Fock matrix, Sαα0 = hφαjφα0 i is the overlap matrix of the AO's and E is a diagonal matrix containing the MO energies, "1;:::;"K . The Fock matrix Fαα0 includes 1 2 the kinetic energy integrals, Tαα0 = φα − 2 r φα0 , en II. THEORY AND METHODS the nuclear attraction integrals Vαα0 = hφα jv^enj φα0 i, wherev ^en is the electron-nuclear interaction opera- tor, the Coulomb integrals Jαα0 = hφα jvH [n](r^)j φα0 i, 0 In this section, we discuss three formulations of KS- R n(r ) 3 0 where vH [n](r) = jr−r0j d r is the Hartree potential, DFT represented in non-orthogonal basis-sets. Since the xc and finally, the exchange-correlation integrals, V 0 = issue of algorithmic scaling is at the heart of develop- αα hφα jvxc [n](r^)j φα0 i where vxc [n](r^) is the exchange cor- ing DFT methods for large systems, we emphasize for relation potential. each formulation the associated algorithmic complexity In KS theory, the Fock matrix F and the electron den- (so-called system-size scaling). We start with the tra- sity n (r) are mutually dependent on each other and must ditional basis-set formulation of the Kohn-Sham equa- be obtained self-consistently. This is usually achieved by tions leading to standard cubic-scaling (subsection II A). converging an iterative procedure, Then, showing how, by focusing on observables and ex- ploiting the sparsity of the matrices, a quadratic-scaling · · · −! n (r) −! fvH [n](r) ; vxc [n](r)g −! (4) approach can be developed with no essential loss of rigor O(K3) or accuracy (subsection II B). Most of the discussion will −! F −−−−!fC; Eg −! n (r) −! :::; revolve around the third and final approach, stochastic DFT, which estimates expectation values using stochas- where in each iteration, a previous density iterate n (r) tic sampling methods, as described in subsection II C. is used to generate the Hartree vH [n](r) and exchange- This latter approach leads, to linear-scaling complexity. correlation vxc [n](r) potentials from which we construct 3 the Fock matrix F . Then, by solving Eq. (3) the coef- Here, S−1F is \plugged" in place of " into the function ficient matrix C is obtained from which a new density f of Eq. (7) [58]. Just like P is an operator, our method iterate n (r) is generated via Eqs. (1)-(2). The iterations also views S−1 as an operator which is applied to any continue until convergence (density stops changing with vector u with linear-scaling cost using a preconditioned a predetermined threshold), and a self-consistent field so- conjugate gradient method [59,60]). The operator P , ap- lution is thus obtained. plied to an arbitrary vector u, uses a Chebyshev expan- PNC l This implementation of the basis-set based approach sion [9,17,44,61] of length NC : P u = l=0 al (T; µ) u 0 −1 becomes computationally expensive for very large sys- where al are the expansion coefficients and u = S u, tems due to the cubic scaling of solving the algebraic u1 = Hu0 and then ul+1 = 2Hul − ul−1, l = 2; 3; :::.