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We review a suite of stochastic vector computational approaches for studying the electronic structure of extended condensed matter systems. These techniques help reduce algorithmic complexity, facilitate efficient parallelization, simplify computational tasks, accelerate calculations, and diminish memory requirements. While their scope is vast, we limit our study to ground-state and finite temperature density functional theory (DFT) and second-order many-body perturbation theory. More advanced topics, such as quasiparticle (charge) and optical (neutral) excitations and higher-order processes, are covered elsewhere. We start by explaining how to use stochastic vectors in computations, characterizing the associated statistical errors. Next, we show how to estimate the electron density in DFT and discuss effective techniques to reduce statistical errors. Finally, we review the use of stochastic vectors for calculating correlation energies within the second-order Møller-Plesset perturbation theory and its finite temperature variational form. Example calculation results are presented and used to demonstrate the efficacy of the methods.

We develop an improved stochastic formalism for the Bethe-Salpeter equation, based on an exact separation of the effective-interaction $W$ to two parts, $W=(W-v_W)+v_W$ where the latter is formally any translationally-invariant interaction $v_W(r-r')$. When optimizing the fit of $v_W$ exchange kernel to $W$, by using a stochastic sampling of $W$, the difference $W-v_W$ becomes quite small. Then, in the main BSE routine, this small difference is stochastically sampled. The number of stochastic samples needed for an accurate spectrum is then largely independent of system size. While the method is formally cubic in scaling, the scaling prefactor is small due to the constant number of stochastic orbitals needed for sampling $W$., 9 pages, 5 figures, 2 tables

Plasmonic nano-objects have shown great potential in enhancing applications like biological/chemical sensing, light harvesting and energy transfer, and optical/quantum computing. Therefore, an extensive effort has been vested in optimizing plasmonic systems and exploiting their field enhancement properties. Super-resolution imaging with quantum dots (QDs) is a promising method to probe plasmonic near-fields but is hindered by the distortion of the QD radiation pattern. Here, we investigate the interaction between QDs and "L-shaped" gold nanoantennas and demonstrate both theoretically and experimentally that this strong interaction can induce polarization-dependent modifications to the apparent QD emission intensity, polarization, and localization. Based on FDTD simulations and polarization-modulated single-molecule microscopy, we show that the displacement of the emitter's localization is due to the position-dependent interference between the emitter and the induced dipole, and can be up to 100 nm. Our results help pave a pathway for higher precision plasmonic near-field mapping and its underlying applications.

We develop the gapped-filtering method, whereby a short Chebyshev expansion accurately represents the density-matrix operator. The method optimizes the Chebyshev coefficients to give the correct density matrix at all energies except within the gapped region where there are no eigenstates. Gapped filtering reduces the number of required terms in the Chebyshev expansion compared to traditional expansion methods, as long as one knows or can determine efficiently the HOMO and LUMO positions. The reduction is especially noticeable (factors of 2-3) when high accuracy is sought. To exemplify the method, we use gapped-filtering to increase the efficiency of stochastic-GW calculations., Comment: 8 pages, 6 figures

We develop a stochastic resolution of identity approach to the real-time second-order Green's function (real-time sRI-GF2) theory, extending our recent work for imaginary-time Matsubara Green's function [ Takeshita et al. J. Chem. Phys. 2019 , 151 , 044114 ]. The approach provides a framework to obtain the quasi-particle spectra across a wide range of frequencies and predicts ionization potentials and electron affinities. To assess the accuracy of the real-time sRI-GF2, we study a series of molecules and compare our results to experiments as well as to a many-body perturbation approach based on the GW approximation, where we find that the real-time sRI-GF2 is as accurate as self-consistent GW. The stochastic formulation reduces the formal computatinal scaling from O(Ne5) down to O(Ne3) where Ne is the number of electrons. This is illustrated for a chain of hydrogen dimers, where we observe a slightly lower than cubic scaling for systems containing up to Ne ≈ 1000 electrons.

Molecular aggregates with long-range excitonic couplings have drastically different photophysical properties compared to their monomer counterparts. From Kasha’s model for 1-dimensional systems, positive or negative excitonic couplings lead to blue or red shifted optical spectra with respect to the monomers, labelled H-and J-aggregates respectively. The overall excitonic couplings in higher dimensional systems are much more complicated and cannot be simply classified from their spectral shifts alone. Here, we provide a unified classification for extended 2D aggregates using temperature dependent peak shifts, thermal broadening and quantum yields. We discuss the examples of six 2D aggregates with J-like absorption spectra but quite drastic changes quantum yields and superradiance. We find the origin of the differences is, in fact, a different excitonic band structure where the bright state is lower energy than the monomer but still away from the band edge. We call this an ‘I-aggregate’. Our results provide a description of the complex excitonic behaviors that cannot be explained solely on Kasha’s model. Further, such properties can be tuned with the packing geometries within the aggregates providing supramolecular pathways for controlling them. This will allow for precise optimizations of aggregate properties in their applications across the areas of optoelectronics, photonics, excitonic energy transfer, and shortwave infrared technologies.

Molecular aggregates with long-range excitonic couplings have drastically different photophysical properties compared to their monomer counterparts. From Kasha’s model for 1-dimensional systems, positive or negative excitonic couplings lead to blue or red shifted optical spectra with respect to the monomers, labelled H-and J-aggregates respectively. The overall excitonic couplings in higher dimensional systems are much more complicated and cannot be simply classified from their spectral shifts alone. Here, we provide a unified classification for extended 2D aggregates using temperature dependent peak shifts, thermal broadening and quantum yields. We discuss the examples of six 2D aggregates with J-like absorption spectra but quite drastic changes quantum yields and superradiance. We find the origin of the differences is, in fact, a different excitonic band structure where the bright state is lower energy than the monomer but still away from the band edge. We call this an ‘I-aggregate’. Our results provide a description of the complex excitonic behaviors that cannot be explained solely on Kasha’s model. Further, such properties can be tuned with the packing geometries within the aggregates providing supramolecular pathways for controlling them. This will allow for precise optimizations of aggregate properties in their applications across the areas of optoelectronics, photonics, excitonic energy transfer, and shortwave infrared technologies.

We present a highly efficient method for the extraction of optical properties of very large molecules via the Bethe-Salpeter equation. The crutch of this approach is the calculation of the action of the effective Coulombic interaction, $W$, through a stochastic TD Hartree propagation, which uses only 10 stochastic orbitals rather than propagating the full sea of occupied states. This leads to a scaling that is at most cubic in system size, with trivial MPI parallelization. We apply this new method to calculate the spectra and electronic density of the dominant excitons of a carbon-nanohoop bound fullerene system with 520 electrons, using less than 4000 core hours., 6 pages, 3 figures, communication

We introduce a tempering approach with stochastic density functional theory (sDFT), labeled t-sDFT, which reduces the statistical errors in the estimates of observable expectation values. This is achieved by rewriting the electronic density as a sum of a "warm" component complemented by "colder" correction(s). Since the "warm" component is larger in magnitude but faster to evaluate, we use many more stochastic orbitals for its evaluation than for the smaller-sized colder correction(s). This results in a significant reduction of the statistical fluctuations and the bias compared to sDFT for the same computational effort. We the method's performance on large hydrogen-passivated silicon nanocrystals (NCs), finding a reduction in the systematic error in the energy by more than an order of magnitude, while the systematic errors in the forces are also quenched. Similarly, the statistical fluctuations are reduced by factors of around 4-5 for the total energy and around 1.5-2 for the forces on the atoms. Since the embedding in t-sDFT is fully stochastic, it is possible to combine t-sDFT with other variants of sDFT such as energy-window sDFT and embedded-fragmented sDFT., Comment: 10 pages, 3 figures

We show that a stochastic approach enables calculations of the optical properties of large 2-dimensional and nanotubular excitonic molecular aggregates. Previous studies of such systems relied on numerically diagonalizing the dense and disordered Frenkel Hamiltonian, which scales approximately as $\mathcal{O}(N^3)$ for $N$ dye molecules. Our approach scales much more efficiently as $\mathcal{O}(N\log(N))$, enabling quick study of systems with a million of coupled molecules on the micron size scale. We calculate several important experimental observable including the optical absorption spectrum and density of states, and develop a stochastic formalism for the participation ratio. Quantitative agreement with traditional matrix diagonalization methods is demonstrated for both small- and intermediate-size systems. The stochastic methodology enables the study of the effects of spatial-correlation in site energies on the optical signatures of large 2D aggregates. Our results demonstrate that stochastic methods present a path forward for screening structural parameters and validating experiments and theoretical predictions in large excitonic aggregates., Comment: 11 pages, 7 figures, as submitted to JPC

We develop a range-separated stochastic resolution of identity (RS-SRI) approach for the four-index electron repulsion integrals, where the larger terms (above a predefined threshold) are treated using a deterministic RI and the remaining terms are treated using a SRI. The approach is implemented within a second-order Green's function formalism with an improved O(N3) scaling with the size of the basis set, N. Moreover, the RS approach greatly reduces the statistical error compared to the full stochastic version [T. Y. Takeshita et al., J. Chem. Phys. 151, 044114 (2019)], resulting in computational speedups of ground and excited state energies of nearly two orders of magnitude, as demonstrated for hydrogen dimer chains and water clusters.

Efficient Boltzmann-sampling using first-principles methods is challenging for extended systems due to the steep scaling of electronic structure methods with the system size. Stochastic approaches provide a gentler system-size dependency at the cost of introducing "noisy" forces, which could limit the efficiency of the sampling. When the forces are deterministic, the first-order Langevin dynamics (FOLD) offers efficient sampling by combining a well-chosen preconditioning matrix S with a time-step-bias-mitigating propagator [G. Mazzola and S. Sorella, Phys. Rev. Lett. 118, 015703 (2017)]. However, when forces are noisy, S is set equal to the force-covariance matrix, a procedure that severely limits the efficiency and the stability of the sampling. Here, we develop a new, general, optimal, and stable sampling approach for FOLD under noisy forces. We apply it for silicon nanocrystals treated with stochastic density functional theory and show efficiency improvements by an order-of-magnitude.

We apply a stochastic version of an optimally tuned range-separated hybrid functional to provide insight on the electronic properties of P- and B- doped Si nanocrystals of experimentally relevant sizes. We show that we can use the range-separation parameter for undoped systems to calculate accurate results for dopant activation energies. We apply this strategy for tuning functionals to study doped nanocrystals up to 2.5 nm in diameter at the hybrid functional level. In this confinement regime, the P- and B- dopants have large activation energies and have strongly localized states that lie deep within the energy gaps. Structural relaxation plays a greater role for B-substituted dopants and contributes to the increase in activation energy when the B dopant is near the nanocrystal surface.

The projector augmented wave (PAW) method of Bl\"ochl makes smooth but non-orthogonal orbitals. Here we show how to make PAW orthogonal, using a cheap transformation of the wave-functions. We show that the resulting Orthogonal PAW (OPAW), applied for DFT, reproduces (for a large variety of solids) band gaps from the ABINIT package. OPAW combines the underlying orthogonality of norm-conserving potentials with the large grid spacings and small spectral range in PAW. The OPAW framework can also be combined with other electronic structure theory methods., Comment: 8 pages, 2 figures, 2 tables

Summary: Neutral zinc alkoxide complexes show high activity toward the ring-opening polymerization of cyclic esters and carbonates, to generate biodegradable plastics applicable in several areas. Herein, we use a ferrocene-chelating heteroscorpionate complex in redox-switchable polymerization reactions, and we show that it is a moderately active catalyst for the ring-opening polymerization of L-lactide, ɛ-caprolactone, trimethylene carbonate, and δ-valerolactone. Uniquely for this type of catalyst, the oxidized complex has a similar polymerization activity as the corresponding reduced compound, but displays significantly different rates of reaction in the case of trimethylene carbonate and δ-valerolactone. Investigations of the oxidized compound suggest the presence of an organic radical rather than an Fe(III) complex. Electronic structure and density functional theory (DFT) calculations were performed to support the proposed electronic states of the catalytic complex and to help explain the observed reactivity differences. The catalyst was also compared with a monomeric phenoxide complex to show the influence of the phosphine-zinc interaction on catalytic properties. : Chemistry; Inorganic Chemistry; Catalysis; Polymer Chemistry Subject Areas: Chemistry, Inorganic Chemistry, Catalysis, Polymer Chemistry

Author(s): Cytter, Y; Rabani, E; Neuhauser, D; Preising, M; Redmer, R; Baer, R | Abstract: The Kubo-Greenwood (KG) formula is often used in conjunction with Kohn-Sham (KS) density functional theory (DFT) to compute the optical conductivity, particularly for warm dense matter. For applying the KG formula, all KS eigenstates and eigenvalues up to an energy cutoff are required and thus the approach becomes expensive, especially for high temperatures and large systems, scaling cubically with both system size and temperature. Here, we develop an approach to calculate the KS conductivity within the stochastic DFT framework, which requires knowledge only of the KS Hamiltonian but not its eigenstates and values. We show that the computational effort associated with the method scales linearly with system size and reduces in proportion to the temperature, unlike the cubic increase with traditional deterministic approaches. In addition, we find that the method allows an accurate description of the entire spectrum, including the high-frequency range, unlike the deterministic method which is compelled to introduce a high-frequency cutoff due to memory and computational time constraints. We apply the method to helium-hydrogen mixtures in the warm dense matter regime at temperatures of ∼60kK and find that the system displays two conductivity phases, where a transition from nonmetal to metal occurs when hydrogen atoms constitute ∼0.3 of the total atoms in the system.

Quasiparticle (QP) excitations are extremely important for understanding and predicting charge transfer and transport in molecules, nanostructures, and extended systems. Since density functional theory (DFT) within the Kohn-Sham (KS) formulation does not provide reliable QP energies, many-body perturbation techniques such as the GW approximation are essential. The main practical drawback of GW implementations is the high computational scaling with system size, prohibiting its use in extended, open boundary systems with many dozens of electrons or more. Recently, a stochastic formulation of GW (sGW) was presented (Phys. Rev. Lett. 2014, 113, 076402) with a near-linear-scaling complexity, illustrated for a series of silicon nanocrystals reaching systems of more than 3000 electrons. This advance provides a route for many-body calculations on very large systems that were impossible with previous approaches. While earlier we have shown the gentle scaling of sGW, its accuracy was not extensively demonstrated. Therefore, we show that this new sGW approach is very accurate by calculating the ionization energies of a group of sufficiently small molecules where a comparison to other GW codes is still possible. Using a set of 10 such molecules, we demonstrate that sGW provides reliable vertical ionization energies in close agreement with benchmark deterministic GW results (J. Chem. Theory Comput, 2015, 11, 5665), with mean (absolute) deviation of 0.05 and 0.09 eV. For completeness, we also provide a detailed review of the sGW theory and numerical implementation.

Stochastic density functional theory (sDFT) is becoming a valuable tool for studying ground state properties of extended materials. The computational complexity of describing the Kohn-Sham orbitals is replaced by introducing a set of random (stochastic) orbitals leading to linear and often sub-linear scaling of certain ground-state observable at the account of introducing a statistical error. Schemes to reduce the noise are essential, for example, for determining the structure using the forces obtained from sDFT. Recently we have introduced two embedding schemes to mitigate the statistical fluctuations in the electron density and resultant forces on the nuclei. Both techniques were based on fragmenting the system either in real-space or slicing the occupied space into energy windows, allowing for a significant reduction of the statistical fluctuations. For chemical accuracy further reduction of the noise is required, which could be achieved by increasing the number of stochastic orbitals. However, the convergence is relatively slow as the statistical error scales as $1/\sqrt{N_\chi}$ according to the central limit theorem, where $N_\chi$ is the number of random orbitals. In this paper we combined the aforementioned embedding schemes and introduced a new approach that builds on overlapped fragments and energy windows. The new approach significantly lowers the noise for ground state properties such as the electron density, total energy, and forces on the nuclei, as demonstrated for a G-center in bulk silicon., Comment: The following article has been submitted to the Journal of Chemical Physics

A stochastic orbital approach to the resolution of identity (RI) approximation for 4-index electron repulsion integrals (ERIs) is presented. The stochastic RI-ERIs are then applied to second order Moller–Plesset perturbation theory (MP2) utilizing a multiple stochastic orbital approach. The introduction of multiple stochastic orbitals results in an O(NAO3) scaling for both the stochastic RI-ERIs and stochastic RI-MP2, NAO being the number of basis functions. For a range of water clusters we demonstrate that this method exhibits a small prefactor and observed scalings of O(Ne2.4) for total energies and O(Ne3.1) for forces (Ne being the number of correlated electrons), outperforming MP2 for clusters with as few as 21 water molecules.

Linear scaling density functional theory is important for understanding electronic structure properties of nanometer scale systems. Recently developed stochastic density functional theory can achieve linear or even sublinear scaling for various electronic properties without relying on the sparsity of the density matrix. The basic idea relies on projecting stochastic orbitals onto the occupied space by expanding the Fermi-Dirac operator and repeating this for Nχ stochastic orbitals. Often, a large number of stochastic orbitals are required to reduce the statistical fluctuations (which scale as Nχ -1/2) below a tolerable threshold. In this work, we introduce a new stochastic density functional theory that can efficiently reduce the statistical fluctuations for certain observable and can also be integrated with an embedded fragmentation scheme. The approach is based on dividing the occupied space into energy windows and projecting the stochastic orbitals with a single expansion onto all windows simultaneously. This allows for a significant reduction of the noise as illustrated for bulk silicon with a large supercell. We also provide theoretical analysis to rationalize why the noise can be reduced only for a certain class of ground state properties, such as the forces and electron density.

Nonmonotonic bending-induced changes of fundamental band gaps and quasiparticle energies are observed for realistic nanoscale phosphorene nanosheets. Calculations using stochastic many-body perturbation theory (sGW) show that even slight curvature causes significant changes in the electronic properties. For small bending radii (< 4 nm) the band-gap changes from direct to indirect. The response of phosphorene to deformation is strongly anisotropic (different for zig-zag vs. armchair bending) due to an interplay of exchange and correlation effects. Overall, our results show that fundamental band gaps of phosphorene sheets can be manipulated by as much as 0.7 eV depending on the bending direction., 23 pages, 5 figures, 3 tables

We develop a stochastic approach to time-dependent density functional theory with optimally tuned range-separated hybrids containing nonlocal exchange, for calculating optical spectra. The attractive electron-hole interaction, which leads to the formation of excitons, is included through a time-dependent linear-response technique with a nonlocal exchange interaction which is computed very efficiently through a stochastic scheme. The method is inexpensive and scales quadratically with the number of electrons, at almost the same (low) cost of time dependent Kohn-Sham with local functionals. Our results are in excellent agreement with experimental data, and the efficiency of the approach is demonstrated on large finite phosphorene sheets containing up to 1958 valence electrons.

The stochastic density functional theory (DFT) [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 to 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 $N_{e}=16384$ electrons) and show that overlapped fragments reduce significantly the statistical noise even for systems with a delocalized density matrix., 9 pages, 6 figures

Absorption cross-section spectra for gold nanoparticles were calculated using fully quantum Stochastic Density Functional Theory and a classical Finite-Difference Time Domain (FDTD) Maxwell solver. Spectral shifts were monitored as a function of size (1.3-3.1 nm) and shape (octahedron, cubeoctahedron, and truncated cube). Even though the classical approach is forced to fit the quantum TDDFT at 3.1nm, at smaller sizes there is a significant deviation as the classical theory is unable to account for peak splitting and spectral blue shifts even after quantum spectral corrections. We attribute the failure of classical methods at predicting these features to quantum effects and low density of states in small nanoparticles. Classically, plasmon resonances are modeled as collective conduction electron excitations, but at small nanoparticle size these excitations transition to few or even individual conductive electron excitations, as indicated by our results.

We develop a stochastic formulation of the optimally-tuned range-separated hybrid density functional theory which enables significant reduction of the computational effort and scaling of the non-local exchange operator at the price of introducing a controllable statistical error. Our method is based on stochastic representations of the Coulomb convolution integral and of the generalized Kohn-Sham density matrix. The computational cost of the approach is similar to that of usual Kohn-Sham density functional theory, yet it provides much more accurate description of the quasiparticle energies for the frontier orbitals. This is illustrated for a series of silicon nanocrystals up to sizes exceeding 3000 electrons. Comparison with the stochastic GW many-body perturbation technique indicates excellent agreement for the fundamental band gap energies, good agreement for the band-edge quasiparticle excitations, and very low statistical errors in the total energy for large systems. The present approach has a major advantage over one-shot GW by providing a self-consistent Hamiltonian which is central for additional post-processing, for example in the stochastic Bethe-Salpeter approach., 7 pages, 3 figures

We introduce the concept of sparse stochastic compression, an efficient stochastic sampling of any general function. The technique uses sparse stochastic orbitals (SSOs), short vectors that sample a small number of space points. As a first demonstration, SSOs are applied in conjunction with simple direct projection to accelerate our recent stochastic $GW$ technique; the new developments enable accurate prediction of ${G}_{0}{W}_{0}$ quasiparticle energies and gaps for systems with up to ${N}_{e}g10,000$ electrons, with small statistical errors of $\ifmmode\pm\else\textpm\fi{}0.05\phantom{\rule{0.28em}{0ex}}\mathrm{eV}$ and using less than 2000 core CPU hours. Overall, stochastic $GW$ scales now linearly (and often sublinearly) with ${N}_{e}.$

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 density functional theory (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 subsystem(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 subsystem. 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 agrees exactly with deterministic calculations in the limit of a 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 nonembedding stochastic DFT.

A stochastic cumulant GW method is presented, allowing us to map the evolution of photoemission spectra, quasiparticle energies, lifetimes and emergence of collective excitations from molecules to bulk-like systems with up to thousands of valence electrons, including Si nanocrystals and nanoplatelet. The quasiparticle energies rise due to their coupling with collective shake-up (plasmon) excitations, and this coupling leads to significant spectral weight loss (up to 50% for the low energy states), shortening the lifetimes and shifting the spectral features to lower energy by as much as 0.6 eV. Such features are common to all the systems studied irrespective of their size and shape. For small and low dimensional systems the surface plasmon resonances affect the frequency of the collective excitation and position of the satellites., Comment: 6 pages, 4 figures

Simulations in the warm dense matter regime using finite temperature Kohn-Sham density functional theory (FT-KS-DFT), while frequently used, are computationally expensive due to the partial occupation of a very large number of high-energy KS eigenstates which are obtained from subspace diagonalization. We have developed a stochastic method for applying FT-KS-DFT, that overcomes the bottleneck of calculating the occupied KS orbitals by directly obtaining the density from the KS Hamiltonian. The proposed algorithm scales as $O\left(N{T}^{\ensuremath{-}1}\right)$ and is compared with the high-temperature limit scaling $O\left({N}^{3}{T}^{3}\right)$ of the deterministic approach, where $N$ is the system size (number of electrons, volume, etc.) and $T$ is the temperature. The method has been implemented in a plane-waves code within the local density approximation (LDA); we demonstrate its efficiency, statistical errors, and bias in the estimation of the free energy per electron for a diamond structure silicon. The bias is small compared to the fluctuations and is independent of system size. In addition to calculating the free energy itself, one can also use the method to calculate its derivatives and obtain the equations of state.

Author(s): Fabian, MD; Shpiro, B; Rabani, E; Neuhauser, D; Baer, R | Abstract: 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 g Ab Initio Electronic Structure Methods Structure and Mechanism g Computational Materials Science Electronic Structure Theory g Density Functional Theory.

The second-order Matsubara Green's function method (GF2) is a robust temperature dependent quantum chemistry approach, extending beyond the random-phase approximation. However, till now the scope of GF2 applications was quite limited as they require computer resources which rise steeply with system size. In each step of the self-consistent GF2 calculation there are two parts: the estimation of the self-energy from the previous step's Green's function, and updating the Green's function from the self-energy. The first part formally scales as the fifth power of the system size while the second has a much gentler cubic scaling. Here, we develop a stochastic approach to GF2 (sGF2) which reduces the fifth power scaling of the first step to merely quadratic, leaving the overall sGF2 scaling as cubic. We apply the method to linear hydrogen chains containing up to 1000 electrons, showing that the approach is numerically stable, efficient and accurate. The stochastic errors are very small, of the order of 0.1% or less of the correlation energy for large systems, with only a moderate computational effort. The first iteration of GF2 is an MP2 calculation that is done in linear scaling, hence we obtain an extremely fast stochastic MP2 (sMP2) method as a by-product. While here we consider finite systems with large band gaps where at low temperatures effects are negligible, the sGF2 formalism is temperature dependent and general and can be applied to finite or periodic systems with small gaps at finite temperatures., 9 pages

An \emph{ab initio} Langevin dynamics approach is developed based on stochastic density functional theory (sDFT) within a new \emph{embedded saturated } \emph{fragment }formalism, applicable to covalently bonded systems. The forces on the nuclei generated by sDFT contain a random component natural to Langevin dynamics and its standard deviation is used to estimate the friction term on each atom by satisfying the fluctuation\textendash dissipation relation. The overall approach scales linearly with system size even if the density matrix is not local and is thus applicable to ordered as well as disordered extended systems. We implement the approach for a series of silicon nanocrystals (NCs) of varying size with a diameter of up to $3$nm corresponding to $N_{e}=3000$ electrons and generate a set of configurations that are distributed canonically at a fixed temperature, ranging from cryogenic to room temperature. We also analyze the structure properties of the NCs and discuss the reconstruction of the surface geometry.

We examine the possibility of using a Metropolis algorithm for computing the exchange energy in a large molecular system. Following ideas set forth in a recent publication (Baer, Neuhauser, and Rabani, Phys. Rev. Lett. 111, 106402 (2013)) we focus on obtaining the exchange energy per particle (ExPE, as opposed to the total exchange energy) to a predefined statistical error and on determining the numerical scaling of the calculation achieving this. For this we assume that the occupied molecular orbitals (MOs) are known and given in terms of a standard Gaussian atomic basis set. The Metropolis random walk produces a sequence of pairs of three-dimensional points (x,x'), which are distributed in proportion to ρ(x,x')(2), where ρ(x,x') is the density matrix. The exchange energy per particle is then simply the average of the Coulomb repulsion energy υC(|x-x'|) over these pairs. To reduce the statistical error we separate the exchange energy into a short-range term that can be calculated deterministically in a linear scaling fashion and a long-range term that is treated by the Metropolis method. We demonstrate the method on water clusters and silicon nanocrystals showing the magnitude of the ExPE standard deviation is independent of system size. In the water clusters a longer random walk was necessary to obtain full ergodicity as Metropolis walkers tended to get stuck for a while in localized regions. We developed a diagnostic tool that can alert a user when such a situation occurs. The calculation effort scales linearly with system size if one uses an atom screening procedure that can be made numerically exact. In systems where the MOs can be localized efficiently the ExPE can even be computed with "sublinear scaling" as the MOs themselves can be screened.

The stochastic formulation of multiexciton generation (MEG) rates is extended to provide access to MEG efficiencies in nanostructures containing thousands of atoms. The formalism is applied to a series of CdSe/CdS seeded nanorod heterostructures with different core and shell dimensions. At energies above 3Eg (where Eg is the band gap), the MEG yield increases with decreasing core size, as expected for spherical nanocrystals. Surprisingly, this behavior is reversed for energies below this value, and is explained by the dependence of the density of states near the valence band edge, which increases with the core diameter. Our predictions indicate that the onset of MEG can be shifted to lower energies by manipulating the density of states in complex nanostructure geometries.

For metallic nanoparticles less than 10 nm in diameter, localized surface plasmon resonances (LSPRs) become sensitive to the quantum nature of conduction electrons. In this regime, experimental probes of size-dependent LSPRs are particularly challenging, and contradictory results are often reported. Unfortunately, quantum mechanical simulations based on time-dependent Kohn-Sham density functional theory (TD-KSDFT) are computationally too expensive to tackle metal particles larger than 2 nm. Herein, we present a time-dependent orbital-free density functional theory (TD-OFDFT) that accurately captures the dynamic response of electrons in the presence of realistic ionic potentials. The TD-OFDFT method offers a comparable accuracy as TD-KSDFT but with a much lower computational cost. Using TD-OFDFT, we study size-dependent LSPRs on Na nanoparticles with diameters from 0.7 to 12.3 nm. The optical absorption spectra exhibit a nonmonotonic behavior from blue shift to red shift and back to blue shift as the particle size decreases. Three principal plasmon modes are identified, and their physical origins are elucidated. Competing physical mechanisms responsible for the nonmonotonic size dependence are discussed. The TD-OFDFT provides a unified theoretical framework that bridges the gap between classical electromagnetic theory and quantum mechanical theory for plasmonics and nanophotonics.

Multiexciton generation, by which more than a single electron–hole pair is generated on optical excitation, is a promising paradigm for pushing the efficiency of solar cells beyond the Shockley–Queisser limit of 31%. Utilizing this paradigm, however, requires the onset energy of multiexciton generation to be close to twice the band gap energy and the efficiency to increase rapidly above this onset. This challenge remains unattainable even using confined nanocrystals, nanorods or nanowires. Here, we show how both goals can be achieved in a nanorod heterostructure with type-II band offsets. Using pseudopotential atomistic calculation on a model type-II semiconductor heterostructure we predict the optimal conditions for controlling multiexciton generation efficiencies at twice the band gap energy. For a finite band offset, this requires a sharp interface along with a reduction of the exciton cooling and may enable a route for breaking the Shockley–Queisser limit., Multiple exciton generation could help limit thermalization losses in solar cells, but the efficiency of the process is still limited. Here, the authors show by atomistic calculations that type-II interfaces in nanostructures along with a change in exciton cooling rate favour multiple exciton generation.

We develop a stochastic resolution of identity representation to the second-order Matsubara Green's function (sRI-GF2) theory. Using a stochastic resolution of the Coulomb integrals, the second order Born self-energy in GF2 is decoupled and reduced to matrix products/contractions, which reduces the computational cost from O(N5) to O(N3) (with N being the number of atomic orbitals). The current approach can be viewed as an extension to our previous work on stochastic resolution of identity second order Moller-Plesset perturbation theory [T. Y. Takeshita et al., J. Chem. Theory Comput. 13, 4605 (2017)] and offers an alternative to previous stochastic GF2 formulations [D. Neuhauser et al., J. Chem. Theory Comput. 13, 5396 (2017)]. We show that sRI-GF2 recovers the deterministic GF2 results for small systems, is computationally faster than deterministic GF2 for N > 80, and is a practical approach to describe weak correlations in systems with 103 electrons and more.

We show that a rigid scissors-like GW self-consistency approach, labeled here Δ¯GW0, can be trivially implemented at zero additional cost for large scale one-shot G0W0 calculations. The method significantly improves one-shot G0W0 and for large systems is very accurate. Δ¯GW0 is similar in spirit to evGW0 where the self-consistency is only applied on the eigenvalues entering Green’s function, while both W and the eigenvectors of Green’s function are held fixed. Δ¯GW0 further assumes that the shift of the eigenvalues is rigid scissors-like so that all occupied states are shifted by the same amount and analogously for all the unoccupied states. We show that this results in a trivial modification of the time-dependent G0W0 self-energy, enabling an a posteriori self-consistency cycle. The method is applicable for our recent stochastic-GW approach, thereby enabling self-consistent calculations for giant systems with thousands of electrons. The accuracy of Δ¯GW0 increases with the system size. For molecules, it is up to 0.4-0.5 eV away from coupled-cluster single double triple (CCSD(T)), but for tetracene and hexacene, it matches the ionization energies from both CCSD(T) and evGW0 to better than 0.05 eV. For solids, as exemplified here by periodic supercells of semiconductors and insulators with 6192 valence electrons, the method matches evGW0 quite well and both methods are in good agreement with the experiment.We show that a rigid scissors-like GW self-consistency approach, labeled here Δ¯GW0, can be trivially implemented at zero additional cost for large scale one-shot G0W0 calculations. The method significantly improves one-shot G0W0 and for large systems is very accurate. Δ¯GW0 is similar in spirit to evGW0 where the self-consistency is only applied on the eigenvalues entering Green’s function, while both W and the eigenvectors of Green’s function are held fixed. Δ¯GW0 further assumes that the shift of the eigenvalues is rigid scissors-like so that all occupied states are shifted by the same amount and analogously for all the unoccupied states. We show that this results in a trivial modification of the time-dependent G0W0 self-energy, enabling an a posteriori self-consistency cycle. The method is applicable for our recent stochastic-GW approach, thereby enabling self-consistent calculations for giant systems with thousands of electrons. The accuracy of Δ¯GW0 increases with the system size. For molecules, it ...

We propose a general theme, labeled mechanical electrodynamics, where the relative three-dimensional (3-D) orientation of particles with nontrivial geometries is tracked based on the details of the...

We demonstrate a far-field single molecule super-resolution method that maps plasmonic near-fields. The method is largely invariant to fluorescence quenching (arising from probe proximity to a metal), has reduced point-spread-function distortion compared to fluorescent dyes (arising from strong coupling to nanoscopic metallic features), and has a large dynamic range (of 2 orders of magnitude) allowing mapping of plasmonic field-enhancements regions. The method takes advantage of the sensitivity of quantum dot (QD) stochastic blinking to plasmonic near-fields. The modulation of the blinking characteristics thus provides an indirect measure of the local field strength. Since QD blinking can be monitored in the far-field, the method can measure localized plasmonic near-fields at high throughput using a simple far-field optical setup. Using this method, propagation lengths and penetration depths were mapped-out for silver nanowires of different diameters and for different dielectric environments, with a spatial accuracy of ∼15 nm. We initially use sparse sampling to ensure single molecule localization for accurate characterization of the plasmonic near-field with plans to increase density of emitters in further studies. The measured propagation lengths and penetration depths values agree well with Maxwell finite-difference time-domain calculations and with published literature values. This method offers advantages such as low cost, high throughput, and superresolved mapping of localized plasmonic fields at high sensitivity and fidelity.

We present density functional theory (DFT) and time-dependent DFT (TD-DFT) study of the structures and electronic spectra of small CdSe nanocluster-adenine complexes Cd(n)Se(n)-adenine (n = 3, 6, 10, 13). We examine the changes in the geometries and excitation spectra of the nanoclusters induced by DNA base-binding. By comparing the results calculated for the bare (Cd(n)Se(n)), hydrogen-passivated (Cd(n)Se(n)H(2n)), as well as the corresponding adenine (Ade)-bound clusters (Cd(n)Se(n)-Ade, Cd(n)Se(n)H(2n)-Ade, Cd(n)Se(n)H(2n-2)-Ade), we find that binding with Ade slightly blue-shifts (up to 0.18 eV) the electronic excitations of bare nanoclusters but strongly red-shifts (1.2 eV) those of hydrogen-passivated nanoclusters. Natural bond orbital analysis shows that the LUMO of Cd(n)Se(n)H(2n)-Ade is a pi* orbital located on the purine ring.

We develop an alternative formulation in the energy-domain to calculate the second order Moller−Plesset (MP2) perturbation energies. The approach is based on repeatedly choosing four random energies using a nonseparable guiding function, filtering four random orbitals at these energies, and averaging the resulting Coulomb matrix elements to obtain a statistical estimate of the MP2 correlation energy. In contrast to our time-domain formulation, the present approach is useful for both quantum chemistry and real-space/plane wave basis sets. The scaling of the MP2 calculation is roughly linear with system size, providing a useful tool to study dispersion energies in large systems. This is demonstrated on a structure of 64 fullerenes within the SZ basis as well as on silicon nanocrystals using real-space grids.

Charge carrier localization in extended atomic systems has been described previously as being driven by disorder, point defects or distortions of the ionic lattice. Here we show for the first time by means of first-principles computations that charge carriers can spontaneously localize due to a purely electronic effect in otherwise perfectly ordered structures. Optimally-tuned range-separated density functional theory and many-body perturbation calculations within the GW approximation reveal that in trans-polyacetylene and polythiophene the hole density localizes on a length scale of several nanometers. This is due to exchange-induced translational symmetry breaking of the charge density. Ionization potentials, optical absorption peaks, excitonic binding energies and the optimally-tuned range parameter itself all become independent of polymer length as it exceeds the critical localization scale. Moreover, lattice disorder and the formation of a polaron result from the charge localization in contrast to the traditional view that lattice distortions precede charge localization. Our results can explain experimental findings that polarons in conjugated polymers form instantaneously after exposure to ultrafast light pulses., Comment: 5 pages, 4 figures

A time-dependent formulation for electron-hole excitations in extended finite systems, based on the Bethe-Salpeter equation (BSE), is developed using a stochastic wave function approach. The time-dependent formulation builds on the connection between time-dependent Hartree-Fock (TDHF) theory and configuration-interaction with single substitution (CIS) method. This results in a time dependent Schr\"odinger-like equation for the quasiparticle orbital dynamics based on an effective Hamiltonian containing direct Hartree and screened exchange terms, where screening is described within the Random Phase Approximation (RPA). To solve for the optical absorption spectrum, we develop a stochastic formulation in which the quasiparticle orbitals are replaced by stochastic orbitals to evaluate the direct and exchange terms in the Hamiltonian as well as the RPA screening. This leads to an overall quadratic scaling, a significant improvement over the equivalent symplectic eigenvalue representation of the BSE. Application of the time-dependent stochastic BSE (TDsBSE) approach to silicon and CdSe nanocrystals up to size of ~3000 electrons is presented and discussed., Comment: 12 pages, 5 figures

We reformulate 1D boson-fermion duality in path-integral terms. The result is a 1D counterpart of the boson-fermion duality in the 2D Chern-Simons gauge theory. The theory is consistent and enables, using standard resummation techniques, to obtain the long-wavelength asymptotics of the collective mode in 1D boson systems at the Tonks-Girardeau regime. The collective mode has the dispersion of Bogoliubov phonons: ω(q) = q(U(q)/m)1/2, where is the bosons density and U(q) is a Fourier component of the two-body potential.

We demonstrate an unexpectedly strong surface-plasmonic absorption at the interface of silver and high-index dielectrics based on electron and photon spectroscopy. The measured bandwidth and intensity of absorption deviate significantly from the classical theory. Our density-functional calculation well predicts the occurrence of this phenomenon. It reveals that due to the low metal-to-dielectric work function at such interfaces, conduction electrons can display a drastic quantum spillover, causing the interfacial electron-hole pair production to dominate the decay of surface plasmons. This finding can be of fundamental importance in understanding and designing quantum nano-plasmonic devices that utilize noble metals and high-index dielectrics., Comment: 5 pages, 5 figures

A stochastic approach to time-dependent density functional theory (TDDFT) is developed for computing the absorption cross section and the random phase approximation (RPA) correlation energy. The core idea of the approach involves time-propagation of a small set of stochastic orbitals which are first projected on the occupied space and then propagated in time according to the time-dependent Kohn-Sham equations. The evolving electron density is exactly represented when the number of random orbitals is infinite, but even a small number (? 16) of such orbitals is enough to obtain meaningful results for absorption spectrum and the RPA correlation energy per electron. We implement the approach for silicon nanocrystals (NCs) using real-space grids and find that the overall scaling of the algorithm is sublinear with computational time and memory., 7 pages, 4 figures