## Recent Publications

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 \textgreater Ab Initio Electronic Structure Methods Structure and Mechanism \textgreater Computational Materials Science Electronic Structure Theory \textgreater Density Functional Theory

Over this past decade, we combined the idea of stochastic resolution of identity with a variety of electronic structure methods. In our stochastic Kohn-Sham DFT method, the density is an average over multiple stochastic samples, with stochastic errors that decrease as the inverse square root of the number of sampling orbitals. Here we develop a stochastic embedding density functional theory method (se-DFT) that selectively reduces the stochastic error (specifically on the forces) for a selected sub-system(s). The motivation, similar to that of other quantum embedding methods, is that for many systems of practical interest the properties are often determined by only a small sub-system. In stochastic embedding DFT two sets of orbitals are used: a deterministic one associated with the embedded subspace, and the rest which is described by a stochastic set. The method is exact in the limit of large number of stochastic samples. We apply se-DFT to study a p-nitroaniline molecule in water, where the statistical errors in the forces on the system (the p-nitroaniline molecule) are reduced by an order of magnitude compared with non-embedding stochastic DFT.

The conjecture that, as in bulk semiconductors, hot multiexcitons in nanocrystals cool rapidly to the lowest available energy levels is tested here by recording the eﬀects of a single cold “spectator” exciton on the relaxation dynamics of a subsequently deposited hot counterpart. Results in CdSe/CdS nanodots show that a preexisting cold “spectator exciton” allows only half of the photoexcited electrons to relax directly to the band-edge. The rest are blocked in an excited quantum state due to conﬂicts in spin orientation. The latter fully relax in this sample only after ∼25 ps as the blocked electrons spins ﬂip, prolonging the temporal window of opportunity for harvesting the retained energy more than 100 fold! Common to all quantum-conﬁned nanocrystals, this process will delay cooling and impact the spectroscopic signatures of hot multiexcitons in all envisioned generation scenarios. How the spin-ﬂipping rate scales with particle size and temperature remains to be determined.

The stochastic density functional theory (DFT) [R. Baer et al., Phys. Rev. Lett. 111, 106402 (2013)] is a valuable linear-scaling approach to Kohn-Sham DFT that does not rely on the sparsity of the density matrix. Linear (and often sub-linear) scaling is achieved by introducing a controlled statistical error in the density, energy, and forces. The statistical error (noise) is proportional to the inverse square root of the number of stochastic orbitals and thus decreases slowly; however, by dividing the system into fragments that are embedded stochastically, the statistical error can be reduced significantly. This has been shown to provide remarkable results for non-covalently-bonded systems; however, the application to covalently bonded systems had limited success, particularly for delocalized electrons. Here, we show that the statistical error in the density correlates with both the density and the density matrix of the system and propose a new fragmentation scheme that elegantly interpolates between overlapped fragments. We assess the performance of the approach for bulk silicon of varying supercell sizes (up to Ne = 16 384 electrons) and show that overlapped fragments reduce significantly the statistical noise even for systems with a delocalized density matrix.