# Research

Submitted
Nazarov, V. U. ; Baer, R. The high frequency limit of spectroscopy. arXiv:2101.09467 [cond-mat] Submitted. Publisher's VersionAbstract

We consider a quantum-mechanical system, finite or extended, initially in its ground-state, exposed to a time-dependent potential pulse, with a slowly varying envelope and a carrier frequency \$\textbackslashomega\_0\$. By working out a rigorous solution of the time-dependent Schr\textbackslash"odinger equation in the high-\$\textbackslashomega\_0\$ limit, we show that the linear response is completely suppressed after the switch-off of the pulse. We show, at the same time, that to the lowest order in \$\textbackslashomega\_0ˆ\-1\\$, observables are given in terms of the linear density response function \$\textbackslashchi(\textbackslashrv,\textbackslashrv',\textbackslashomega)\$, despite the problem's nonlinearity. We propose a new spectroscopic technique based on these findings, which we name the Nonlinear High-Frequency Pulsed Spectroscopy (NLHFPS). An analysis of the jellium slab and sphere models reveals very high surface sensitivity of NLHFPS, which produces a richer excitation spectrum than accessible within the linear-response regime. Combining the advantages of the extraordinary surface sensitivity, the absence of constraints by the conventional dipole selection rules, and the ease of theoretical interpretation by means of the linear response time-dependent density functional theory, NLHFPS has the potential to evolve into a powerful characterization method in nanoscience and nanotechnology.

2022
Fabian, M. D. ; Shpiro, B. ; Baer, R. Linear Weak Scalability of Density Functional Theory Calculations without Imposing Electron Localization. J. Chem. Theory Comput. 2022, acs.jctc.1c00829. Publisher's VersionAbstract

Linear scaling density functional theory (DFT) approaches to the electronic structure of materials are often based on the tendency of electrons to localize in large atomic and molecular systems. However, in many cases of actual interest, such as semiconductor nanocrystals, system sizes can reach a substantial extension before signiﬁcant electron localization sets in, causing a considerable deviation from linear scaling. Herein, we address this class of systems by developing a massively parallel DFT approach which does not rely on electron localization and is formally quadratic scaling yet enables highly eﬃcient linear wall-time complexity in the weak scalability regime. The method extends from the stochastic DFT approach described in Fabian et al. (WIRES: Comp. Mol. Sci. 2019, e1412) but is entirely deterministic. It uses standard quantum chemical atomcentered Gaussian basis sets to represent the electronic wave functions combined with Cartesian real-space grids for some operators and enables 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.

Baer, R. ; Neuhauser, D. ; Rabani, E. Stochastic Vector Techniques in Ground-State Electronic Structure. Annu. Rev. Phys. Chem. 2022, 73, annurev–physchem–090519–045916. Publisher's VersionAbstract

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 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 highly effective techniques to reduce statistical errors. Finally, we review the use of stochastic vector techniques for calculating correlation energies within the secondorder Møller-Plesset perturbation theory and its finite temperature variational form. Example calculation results are presented and used to demonstrate the efficacy of the methods.

Shpiro, B. ; Fabian, M. D. ; Rabani, E. ; Baer, R. Forces from Stochastic Density Functional Theory under Nonorthogonal Atom-Centered Basis Sets. J. Chem. Theory Comput. 2022, 18, 1458–1466. Publisher's VersionAbstract

We develop a formalism for calculating forces on the nuclei within the linear-scaling stochastic density functional theory (sDFT) in a nonorthogonal atomcentered basis set representation (Fabian et al. Wiley Interdiscip. Rev.: Comput. Mol. Sci. 2019, 9, e1412, 10.1002/wcms.1412) and apply it to the Tryptophan Zipper 2 (Trpzip2) 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/Å (∼1.2 × 10−3Eh/a0) when 120 stochastic orbitals are used, independently of system 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.

2021
Nguyen, M. ; Li, W. ; Li, B. (Y. ); Baer, R. ; Rabani, E. ; Neuhauser, D. Tempering stochastic density functional theory. J. Chem. Phys. 2021, 5.0063266. Publisher's VersionAbstract

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.

Gope, K. ; Livshits, E. ; Bittner, D. M. ; Baer, R. ; Strasser, D. Two pathways and an isotope effect in H3+ formation following double ionization of methanol. Natural Sciences 2021, ntls.10022. Publisher's VersionAbstract

The trihydrogen ion has a central role in creating complex molecules in the interstellar medium. Therefore, its formation and destruction mechanisms in high photon energy environments involving organic molecules are drawing significant experimental and theoretical attention. Here, we employ a combination of time-resolved ultrafast extreme-ultraviolet pump and near-infrared probe spectroscopy applied to the deuterated CH3OD methanol molecule. Similar to other double-ionization studies, the isotopic labeling reveals two competing pathways for forming trihydrogen: A) H+3 + COH+ and B) H+3 + HCO+. We validate our high-level ab initio nonadiabatic molecular dynamic simulations by showing that it closely reproduces the essential features of the measured kinetic energy release distribution and branching ratios of the two pathways of the deuterated system. The success of ab initio simulation in describing single photon double-ionization allows for an unprecedented peek into the formation pathways for the undeuterated species, beyond present experimental reach. For this case, we find that the kinetic energy release of pathway B shifts to lower energies by more than 0.6 eV due to a dynamical isotope effect. We also determine the mechanism for trihydrogen formation from excited states of the dication and elucidate the isotope effect’s role in the observed dynamics.

Chen, M. ; Baer, R. ; Neuhauser, D. ; Rabani, E. Stochastic density functional theory: Real- and energy-space fragmentation for noise reduction. J. Chem. Phys. 2021, 154, 204108. Publisher's VersionAbstract

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 observables 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 in 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/ Nχ according to the central limit theorem, where Nχ is the number of random orbitals. In this paper, we combined the embedding schemes mentioned above 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.

2020
Livshits, E. ; Luzon, I. ; Gope, K. ; Baer, R. ; Strasser, D. Time-resolving the ultrafast H2 roaming chemistry and H3+ formation using extreme-ultraviolet pulses. Communications Chemistry 2020, 3 49. Publisher's Version
Gope, K. ; Livshits, E. ; Bittner, D. M. ; Baer, R. ; Strasser, D. Absence of Triplets in Single-Photon Double Ionization of Methanol. The Journal of Physical Chemistry Letters 2020, 11, 8108–8113. Publisher's VersionAbstract

Despite the abundance of data concerning single-photon double ionization of methanol, the spin state of the emitted electron pair has never been determined. Here we present the ﬁrst evidence that identiﬁes the emitted electron pair spin as overwhelmingly singlet when the dication forms in low-energy conﬁgurations. The experimental data show that while the yield of the CH2O+ + H3+ Coulomb explosion channel is abundant, the metastable methanol dication is largely absent. According to high-level ab initio simulations, these facts indicate that photoionization promptly forms singlet dication states, where they quickly decompose through various channels, with signiﬁcant H3+ yields on the low-lying states. In contrast, if we assume that the initial dication is formed in one of the low-lying triplet states, the ab initio simulations exhibit a metastable dication, contradicting the experimental ﬁndings. Comparing the average simulated branching ratios with the experimental data suggests a \textgreater3 order of magnitude enhancement of the singlet:triplet ratio compared with their respective 1:3 multiplicities.

Lee, A. J. ; Chen, M. ; Li, W. ; Neuhauser, D. ; Baer, R. ; Rabani, E. Dopant levels in large nanocrystals using stochastic optimally tuned range-separated hybrid density functional theory. Physical Review B 2020, 102, 035112. Publisher's Version
Zhang, X. ; Lu, G. ; Baer, R. ; Rabani, E. ; Neuhauser, D. Linear-Response Time-Dependent Density Functional Theory with Stochastic Range-Separated Hybrids. Journal of Chemical Theory and Computation 2020, 16, 1064–1072. Publisher's VersionAbstract

Generalized Kohn−Sham density functional theory is a popular computational tool for the ground state of extended systems, particularly within range-separated hybrid (RSH) functionals that capture the long-range electronic interaction. Unfortunately, the heavy computational cost of the nonlocal exchange operator in RSH-DFT usually conﬁnes the approach to systems with at most a few hundred electrons. A signiﬁcant reduction in the computational cost is achieved by representing the density matrix with stochastic orbitals and a stochastic decomposition of the Coulomb convolution (J. Phys. Chem. A 2016, 120, 3071). Here, we extend the stochastic RSH approach to excited states within the framework of linear-response generalized Kohn−Sham time-dependent density functional theory (GKS-TDDFT) based on the plane-wave basis. As a validation of the stochastic GKS-TDDFT method, the excitation energies of small molecules N2 and CO are calculated and compared to the deterministic results. The computational eﬃciency of the stochastic method is demonstrated with a two-dimensional MoS2 sheet (∼1500 electrons), whose excitation energy, exciton charge density, and (excited state) geometric relaxation are determined in the absence and presence of a point defect.

Dou, W. ; Chen, M. ; Takeshita, T. Y. ; Baer, R. ; Neuhauser, D. ; Rabani, E. Range-separated stochastic resolution of identity: Formulation and application to second-order Green’s function theory. The Journal of Chemical Physics 2020, 153, 074113. Publisher's VersionAbstract

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.

Arnon, E. ; Rabani, E. ; Neuhauser, D. ; Baer, R. Efficient Langevin dynamics for "noisy" forces. J. Chem. Phys. 2020, 152, 161103. Publisher's VersionAbstract

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.

2019
Chen, M. ; Baer, R. ; Neuhauser, D. ; Rabani, E. Energy window stochastic density functional theory. The Journal of Chemical Physics 2019, 151, 114116. Publisher's Version
Dou, W. ; Takeshita, T. Y. ; Chen, M. ; Baer, R. ; Neuhauser, D. ; Rabani, E. Stochastic Resolution of Identity for Real-Time Second-Order Green’s Function: Ionization Potential and Quasi-Particle Spectrum. Journal of Chemical Theory and Computation 2019. Publisher's VersionAbstract

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.

Cytter, Y. ; Rabani, E. ; Neuhauser, D. ; Preising, M. ; Redmer, R. ; Baer, R. Transition to metallization in warm dense helium-hydrogen mixtures using stochastic density functional theory within the Kubo-Greenwood formalism. Physical Review B 2019, 100. Publisher's VersionAbstract

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 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 \textbackslashsim60\textbackslashtext\kK\ and find that the system displays two conductivity phases, where a transition from non-metal to metal occurs when hydrogen atoms constitute \textbackslashsim0.3 of the total atoms in the system.

Ospadov, E. ; Rothstein, S. M. ; Baer, R. Quantum Monte Carlo assessment of density functionals for π-electron molecules: ethylene and bifuran. Molecular Physics 2019, 117, 2241–2250. Publisher's VersionAbstract

We perform all-electron, pure-sampling quantum Monte Carlo (QMC) calculations on ethylene and bifuran molecules. The orbitals used for importance sampling with a single Slater determinant are generated from Hartree-Fock and density functional theory (DFT). Their fixed-node energy provides an upper bound to the exact energy. The best performing density functionals for ethylene are BP86 and M06, which account for 99% of the electron correlation energy. Sampling from the π-electron distribution with these orbitals yields a quadrupole moment comparable to coupled cluster CCSD(T) calculations. However, these, and all other density functionals, fail to agree with CCSD(T) while sampling from electron density in the plane of the molecule. For bifuran, as well as ethylene, a correlation is seen between the fixed-node energy and deviance of the QMC quadrupole moment estimates from those calculated by DFT. This suggests that proximity of DFT and QMC densities correlates with the quality of the exchange nodes of the DFT wave function for both systems.

Li, W. ; Chen, M. ; Rabani, E. ; Baer, R. ; Neuhauser, D. Stochastic embedding DFT: Theory and application to p-nitroaniline in water. 2019, 151, 174115. Publisher's VersionAbstract

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.

Vlček, V. ; Rabani, E. ; Baer, R. ; Neuhauser, D. Nonmonotonic band gap evolution in bent phosphorene nanosheets. Phys. Rev. Materials 2019, 3 064601. Publisher's VersionAbstract

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 show that even slight curvature causes significant changes in the electronic properties. For small bending radii (\textless4 nm) the band gap changes from direct to indirect. The response of phosphorene to deformation is strongly anisotropic (different for zigzag 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.

Fabian, M. D. ; Shpiro, B. ; Rabani, E. ; Neuhauser, D. ; Baer, R. Stochastic density functional theory. Wiley Interdisciplinary Reviews: Computational Molecular Science 2019, 10.1002/wcms.1412, e1412. Publisher's VersionAbstract

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