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Stochastic orbital techniques offer reduced computational scaling and memory requirements to describe ground and excited states at the cost of introducing controlled statistical errors. Such techniques often rely on two basic operations, stochastic trace estimation and stochastic resolution of identity, both of which lead to statistical errors that scale with the number of stochastic realizations ($N_{\xi}$) as $\sqrt{N_{\xi}^{-1}}$. Reducing the statistical errors without significantly increasing $N_{\xi}$ has been challenging and is central to the development of efficient and accurate stochastic algorithms. In this work, we build upon recent progress made to improve stochastic trace estimation based on the ubiquitous Hutchinson's algorithm and propose a two-step approach for the stochastic resolution of identity, in the spirit of the Hutch++ method. Our approach is based on employing a randomized low-rank approximation followed by a residual calculation, resulting in statistical errors that scale much better than $\sqrt{N_{\xi}^{-1}}$. We implement the approach within the second-order Born approximation for the self-energy in the computation of neutral excitations and discuss three different low-rank approximations for the two-body Coulomb integrals. Tests on a series of hydrogen dimer chains with varying lengths demonstrate that the Hutch++-like approximations are computationally more efficient than both deterministic and purely stochastic (Hutchinson) approaches for low error thresholds and intermediate system sizes. Notably, for arbitrarily large systems, the Hutchinson-like approximation outperforms both deterministic and Hutch++-like methods.

Abstract Simulating mixed-state evolution in open quantum systems is crucial for various chemical physics, quantum optics, and computer science applications. These simulations typically follow the Lindblad master equation dynamics. An alternative approach known as quantum state diffusion unraveling is based on the trajectories of pure states generated by random wave functions, which evolve according to a nonlinear It\^o-Schr\"odinger equation (ISE). This study introduces weak first- and second-order solvers for the ISE based on directly applying the It\^o-Taylor expansion with exact derivatives in the interaction picture. We tested the method on free and driven Morse oscillators coupled to a thermal environment and found that both orders allowed practical estimation with a few dozen iterations. The variance was relatively small compared to the linear unraveling and did not grow with time. The second-order solver delivers much higher accuracy and stability with bigger time steps than the first-order scheme, with a small additional workload. However, the second-order algorithm has quadratic complexity with the number of Lindblad operators as opposed to the linear complexity of the first-order algorithm.

This study overviews and extends a recently developed stochastic finite-temperature Kohn-Sham density functional theory to study warm dense matter using Langevin dynamics, specifically under periodic boundary conditions. The method's algorithmic complexity exhibits nearly linear scaling with system size and is inversely proportional to the temperature. Additionally, a novel linear-scaling stochastic approach is introduced to assess the Kubo-Greenwood conductivity, demonstrating exceptional stability for DC conductivity. Utilizing the developed tools, we investigate the equation of state, radial distribution, and electronic conductivity of Hydrogen at a temperature of 30,000K. As for the radial distribution functions, we reveal a transition of Hydrogen from gas-like to liquid-like behavior as its density exceeds $4 g/cm^3$. As for the electronic conductivity as a function of the density, we identified a remarkable isosbestic point at frequencies around 7eV, which may be an additional signature of a gas-liquid transition in Hydrogen at 30,000K.

Linear-scaling techniques for Kohn-Sham density functional theory (KS-DFT) are essential to describe the ground state properties of extended systems. Still, these techniques often rely on the locality of the density matrix or on accurate embedding approaches, limiting their applicability. In contrast, stochastic density functional theory (sDFT) achieves linear- and sub-linear-scaling by statistically sampling the ground state density without relying on embedding or imposing localization. In return, ground state observables, such as the forces on the nuclei, fluctuate in sDFT, making the optimization of the nuclear structure a highly non-trivial problem. In this work, we combine the most recent noise-reduction schemes for sDFT with stochastic optimization algorithms to perform structure optimization within sDFT. We compare the performance of the stochastic gradient descent (sGD) approach and its variations (stochastic gradient descent with momentum (sGDM)) to stochastic optimization techniques that rely on the Hessian, such as the stochastic Broyden-Fletcher-Goldfarb-Shanno (sBFGS) algorithm. We further provide a detailed assessment of the computational efficiency and its dependence on the optimization parameters for each methods for determining the ground state structure of bulk silicon with varying supercell dimensions.

We develop a time dependent second order Green's function theory (GF2) for calculating neutral excited states in molecules. The equation of motion for the lesser Green's function (GF) is derived within the adiabatic approximation to the Kadanoff-Baym (KB) equation using the second order Born approximation for the self-energy. In the linear response regime, we recast the time dependent KB equation into a Bethe-Salpeter-like equation (GF2-BSE), with a kernel approximated by the second order Coulomb self-energy. We then apply our GF2-BSE to a set of molecules and atoms and find that GF2-BSE is superior to configuration interaction with singles (CIS) and/or time dependent Hartree-Fock (TDHF), particularly for charge transfer excitations, and is comparable to CIS with perturbative doubles (CIS(D)) in most cases.

Linear scaling density functional theory approaches to electronic structure are often based on the tendency of electrons to localize even in large atomic and molecular systems. However, in many cases of actual interest, for example in semiconductor nanocrystals, system sizes can reach very large extension before significant electron localization sets in and the scaling of the numerical methods may deviate strongly from linear. Here, we address this class of systems, by developing a massively parallel density functional theory (DFT) approach which doesn't rely on electron localizationa and is formally quadratic scaling, yet enables highly efficient linear wall-time complexity in the weak scalability regime. The approach extends from the stochastic DFT method described in Fabian et. al. WIRES: Comp. Mol. Science, e1412 2019 but is fully deterministic. It uses standard quantum chemical atom-centered Gaussian basis sets for representing the electronic wave functions combined with Cartesian real space grids for some of the operators and for enabling a fast solver for the Poisson equation. Our main conclusion is, that when a processor-abundant high performance computing (HPC) infrastructure is available, this type of approach has the potential to allow the study of large systems in regimes where quantum confinement or electron delocalization prevents linear-scaling.

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 $0.065\,eV/\r{A}$ ($\sim1.2\times10^{-3}E_{h}/a_{0}$) 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., Comment: 18 pages, 5 figures

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

Kohn-Sham DFT with optimally tuned range-separated hybrid (RSH) functionals provides accurate and nonempirical fundamental gaps for a wide variety of finite-size systems. The standard tuning procedure relies on calculation of total energies of charged systems and thus cannot be applied to periodic solids. Here, we develop a framework for tuning the range separation parameter that can be used for periodic and open boundary conditions. The basic idea is to choose the range parameter that results in a stationary point where the fundamental gap obtained by RSH matches the gap obtained from a G0W0 over RSH calculation. The proposed framework is therefore analogous to eigenvalue self-consistent GW (scGW). We assess the method for various solids and obtain very good agreement with scGW results.

We consider an arbitrary quantum mechanical system, initially in its ground-state, exposed to a time-dependent electromagnetic pulse with a carrier frequency $\omega_0$ and a slowly varying envelope of finite duration. By working out a solution to the time-dependent Schr\"odinger equation in the high-$\omega_0$ limit, we find that, to the leading order in $\omega_0^{-1}$, a perfect self-cancellation of the system's linear response occurs as the pulse switches off. Surprisingly, the system's observables are, nonetheless, describable in terms of a combination of its linear density response function and nonlinear functions of the electric field. An analysis of jellium slab and jellium sphere models reveals a very high surface sensitivity of the considered setup, producing a richer excitation spectrum than accessible within the conventional linear response regime. On this basis, we propose a new spectroscopic technique, which we provisionally name the Nonlinear High-Frequency Pulsed Spectroscopy (NLHFPS). Combining the advantages of the extraordinary surface sensitivity, the absence of constraints by the traditional dipole selection rules, and the clarity of theoretical interpretation utilizing the linear response time-dependent density functional theory, NLHFPS has a potential to evolve into a powerful characterization method for nanoscience and nanotechnology., Comment: 17 pages, 10 figures

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

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 approach for the $4$-index electron repulsion integrals, where the larger terms (above a predefined threshold) are treated using a deterministic resolution of identity and the remaining terms are treated using a stochastic resolution of identity. The approach is implemented within a second-order Greens function formalism with an improved $O(N^3)$ scaling with the size of the basis set, $N$. Moreover, the range-separated approach greatly reduces the statistical error compared to the full stochastic version ({\it J. Chem. Phys.} {\bf 151}, 044144 (2019)), resulting in computational speedups of ground and excited state energies of nearly two orders of magnitude, as demonstrated for hydrogen dimer chains.

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.

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 serve to limit the efficiency of the sampling. In the first-order Langevin dynamics (FOLD), efficient sampling is achievable by combining a well-chosen preconditioning matrix S with a time-step-bias-mitigating propagator (Mazzola et al., Phys. Rev. Lett., 118, 015703 (2017)). However, when forces are noisy, S is set equal to the force-covariance matrix, a procedure which 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 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 {\em J. Chem. Phys.} {\bf 151}, 044114 (2019)). The approach provides a framework to obtain the quasi-particle spectra across a wide range of frequencies as well as predict 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 and 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 scaling to $O(N_e^3)$, where $N_e$ 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 $N_e \approx 1000$.

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 mater. 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 (sDFT) 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 cut-off due to memory and computational time constraints. We apply the method to helium-hydrogen mixtures in the warm dense matter regime at temperatures of \sim60\text{kK} and find that the system displays two conductivity phases, where a transition from non-metal to metal occurs when hydrogen atoms constitute \sim0.3 of the total atoms in the system.

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.

We develop a stochastic approach to time-dependent DFT with optimally-tuned range-separated hybrids containing non-local 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 non-local 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 (TDKS) 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.

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., Comment: 23 pages, 5 figures, 3 tables

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., Comment: 9 pages, 6 figures

The rates of elementary photophysical processes in nanocrystals, such as carrier cooling, multiexciton generation, Auger recombination etc., are determined by monitoring the transient occupation of the lowest exciton band. The underlying assumption is that hot carriers relax rapidly to their lowest quantum level. Using femtosecond transient absorption spectroscopy in CdSe/CdS nanodots we challenge this assumption. Results show, that in nanodots containing a preexisting cold exciton "spectator", \emph{only half of the photoexcited electrons}relax directly to the band-edge and the complementary half is blocked in an excited state level due to Pauli exclusion. Full relaxation occurs only after \textasciitilde 15 ps, as the blocked electrons flip spin. This novel spin-blockade effect may offer the key for the long-sought-for bottleneck mechanism for multiexciton energy dissipation.

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.

We introduce the concept of fractured stochastic orbitals (FSOs), short vectors that sample a small number of space points and enable an efficient stochastic sampling of any general function. As a first demonstration, FSOs 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 $N_{e}>10,000$ electrons, with small statistical errors of $\pm0.05\,{\rm eV}$ and using less than 2000 core CPU hours. Overall, stochastic $GW$ scales now linearly (and often sub-linearly) with $N_{e}.$, Comment: 9 pages, 2 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., Comment: 10 pages, 2 figures, 2 tables

The Lindblad equation is commonly used for studying quantum dynamics in open systems that cannot be completely isolated from an environment, relevant to a broad variety of research fields, such as atomic physics, materials science, quantum biology and quantum information and computing. For electrons in condensed matter systems, the Lindblad dynamics is intractable even if their mutual Coulomb repulsion could somehow be switched off. This is because they would still be able to affect each other by interacting with the bath. Here, we develop an approximate approach, based on the Hubbard-Stratonovich transformation, which allows to evolve non-interacting Fermions in open quantum systems. We discuss several applications for systems of trapped 1D Fermions showing promising results.

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 $O\left(NT^{-1}\right)$ 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.

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., Comment: 9 pages

The reduced density matrix (RDM) is a fundamental contraction of the Bose-Einstein condensate wave function, encapsulating its one-body properties. It serves as a major analysis tool with which the condensed component of the density can be identified. Despite its cardinal importance, calculating the ground-state RDM of trapped interacting bosons is challenging and has been fully achieved only for specific models or when the pairwise interaction is weak. In this paper we discuss a new approach to compute the RDM based on a double-walker diffusion Monte Carlo random walk coupled with a stochastic permanent calculation. We here describe the new method and study some of its statistical convergence and properties applying it to some model systems.

A stochastic orbital approach to the resolution of identity (RI) approximation for 4-index 2-electron electron repulsion integrals (ERIs) is presented. The stochastic RI-ERIs are then applied to M\o ller-Plesset perturbation theory (MP2) utilizing a \textit{multiple stochastic orbital approach}. The introduction of multiple stochastic orbitals results in an $N^3$ scaling for both the stochastic RI-ERIs and stochastic RI-MP2. We demonstrate that this method exhibits a small prefactor and an observed scaling of $N^{2.4}$ for a range of water clusters, already outperforming MP2 for clusters with as few as 21 water molecules., Comment: 6 pages, 2 figures

We develop a new scheme for determining molecular partial atomic charges (PACs) with external electrostatic potential (ESP) closely mimicking that of the molecule. The PACs are the "minimal corrections" to a reference-set of PACs necessary for reproducing exactly the tensor components of the Cartesian zero- first- and second- molecular electrostatic multipoles. We evaluate the quality of ESP reproduction when "minimally correcting" (MC) Mulliken, Hirshfeld or iterated-Hirshfeld reference PACs. In all these cases the MC-PACs significantly improve the ESP while preserving the reference PACs' invariance under the molecular symmetry operations. When iterative-Hirshfeld PACs are used as reference the MC-PACs yield ESPs of comparable quality to those of the ChElPG charge fitting method.

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.