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WIREs Comput Mol Sci
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Stochastic density functional theory

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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 a 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. This article is categorized under: Electronic Structure Theory > Ab Initio Electronic Structure Methods Structure and Mechanism > Computational Materials Science Electronic Structure Theory > Density Functional Theory
The timing of DFT calculations of (H2O)N water clusters using the 6‐31G basis‐set within the LDA. Left panel: The sDFT wall time as function of x = N log N normalized to one random orbital per thread for a full SCF calculation (blue symbols) and for a single SCF cycle (orange symbols). Dashed lines are functions t = Axn, where n is best‐fitted to the data and shown in the legend. Right panel: Wall time of a conventional SCF calculation (using Q‐CHEM), performed on a single node, as a function of x = NlogN for a full SCF calculation (blue symbols) and for a single SCF cycle (orange symbols). The calculations were run on an Intel Xeon CPU E3‐1230 v5 @ 3.40GHz 64 GB RAM (without Infiniband networking). Each processor supports eight threads. The sDFT results were calculated with 800 random vectors and fragments of a representative size of 128 water molecules (denoted /f128)
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The SD (σ, triangles) and errors (δE squares) of the stochastic estimate of the energy per electron as a function of the number of water molecules Nwater using no fragments (/f0, blue markers) and single water molecule fragments (/f1, yellow markers). The dotted lines are fits to the σ values. These results were calculated using the STO‐3G basis‐set within the LDA and employed I = 100 random vectors
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The SD (σ,circles) and errors (δE, squares) of the stochastic estimate of the energy per electron as a function of the number of random vectors (I) in (H2O)237 without fragments (/f0, blue) and with H2O fragments (/f1, yellow, discussed in Section 3) . The dashed lines are best fit functions αIn to the data, where n = 1/2 for fitting the SDs and n = 1 for the bias. These results were calculated using the STO‐3G basis‐set within the LDA
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Top panel: The estimate of energy per electron as a function of the inverse number of random vectors (1/I) for water molecule clusters of indicated sizes, without fragments (/f0) and with fragments (discussed in Section 3) of single H2O molecules (/f1). The dotted lines are linear fit to the data (weighted by the inverse error bar length). The deterministic results are represented at 1/I = 0 by star symbols. Bottom panel: A zoomed view of the /f1 results. All these results were calculated using the STO‐3G basis‐set within the LDA
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Scalability plot of the calculation, showing the speedup as a function of the number of threads used when calculating a SCF iteration of (H2O)1120 (at the 6‐31G basis‐set level within LDA) using a total of 2,400 random vectors. Calculations were performed on several 2.30GHz Intel Xeon E5‐2650 v3 with 252 GB and Infiniband networking
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The DOS as a function of energy for a hydrogen‐saturated silicon cluster (Si87H76) calculated using the all‐electron Q‐CHEM and the bs‐Inbar codes. Comparison is made for three standard Gaussian basis‐sets as indicated in the panels. We used the local density approximation (LDA) for the exchange‐correlation energy. Both calculations plot the DOS of Equation (13) using kBT = 0.01Eh
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The force on a marked water molecule in (H2O)237 (red dashed line is the deterministic DFT value) calculated as δE/δx where δE is the energy difference between two positions of the molecule displaced by a distance δx = 0.05a0. On the left (right) panel we present /f12 (/f32) results. The arrow points to the force F(I = 0) (the deterministic force on the molecule when in the parent fragment). The “error bars” are 95% confidence intervals for E{F(I)}. These results were calculated using the STO‐3G basis‐set within the LDA
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Left panels: The LDA DOS of a cluster of 1,120 water molecules using the 6‐31G basis‐set computed with I = 400 and 1,600 random vectors and using single‐molecule fragments (/f1 top panel) and 128 molecule fragments (/f128 bottom left). The insets zoom on the region of the band gap. Right panels: The LDA DOS of Si705H300, a hydrogen‐terminated silicon nanocrystal, using the STO‐3G basis‐set computed with I = 400 and 1,600 random vectors and using no fragments (/f0 top panel) and 16 atom fragments (/f16 bottom panel). In all panels the results are compared to deterministic calculations under the same conditions
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A schematic depiction of a bare fragment (blue region) as a localized set of atoms or molecules within the large system. The fragment is first saturated by coating it with capping atoms (red region), its saturated‐DM is calculated using a deterministic DFT calculation, from which a bare DM Pf is “carved” out by an algebraic procedure
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Electronic Structure Theory > Ab Initio Electronic Structure Methods
Electronic Structure Theory > Density Functional Theory
Structure and Mechanism > Computational Materials Science

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