# Research

2019
Chen, M. ; Baer, R. ; Neuhauser, D. ; Rabani, E. Overlapped embedded fragment stochastic density functional theory for covalently-bonded materials. J. Chem. Phys. 2019, 150, 034106. Publisher's VersionAbstract

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

Luzon, I. ; Livshits, E. ; Gope, K. ; Baer, R. ; Strasser, D. Making Sense of Coulomb Explosion Imaging. J. Phys. Chem. Lett. 2019, 10, 1361–1367. Publisher's VersionAbstract

A multifaceted agreement between ab initio theoretical predictions and experimental measurements, including branching ratios, channel-speciﬁc kinetic energy release, and three-body momentum correlation spectra, leads to the identiﬁcation of new mechanisms in Coulomb-explosion (CE) induced two- and three-body breakup processes in methanol. These identiﬁed mechanisms include direct nonadiabatic Coulomb explosion responsible for CO bond-breaking, a long-range “ inverse harpooning” dominating the production of H2+ + HCOH+, a transient proton migration leading to surprising energy partitioning in three-body fragmentation and other complex dynamics forming products such as H2O+ and H3+. These mechanisms provide general concepts that should be useful for analyzing future time-resolved Coulomb explosion imaging of methanol as well as other molecular systems. These advances are enabled by a combination of recently developed experimental and computational techniques, using weak ultrafast EUV pulses to initiate the CE and a high-level quantum chemistry approach to follow the resulting ﬁeld-free nonadiabatic molecular dynamics.

2018
Vlček, V. ; Baer, R. ; Rabani, E. ; Neuhauser, D. Simple eigenvalue-self-consistent ΔGW0. J. Chem. Phys. 2018, 149, 174107. Publisher's Version
Vlček, V. ; Li, W. ; Baer, R. ; Rabani, E. ; Neuhauser, D. Swift G W beyond 10,000 electrons using sparse stochastic compression. Phys. Rev. B 2018, 98, 075107. Publisher's Version
Hernandez, S. ; Xia, Y. ; Vlček, V. ; Boutelle, R. ; Baer, R. ; Rabani, E. ; Neuhauser, D. First-principles spectra of Au nanoparticles: from quantum to classical absorption. Molecular Physics 2018, 116, 2506–2511. Publisher's VersionAbstract

Absorption cross-section spectra for gold nanoparticles were calculated using fully quantum Stochastic Density Functional Theory and a classical Finite-Difference Time Domain Maxwell solver. Spectral shifts were monitored as a function of size (1.3–) and shape (octahedron, cubeoctahedron and truncated cube). Even though the classical approach is forced to fit the quantum time-dependent density functional theory at , at smaller sizes there is a significant deviation as the classical theory is unable to account for peak splitting and spectral blueshifts 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 modelled 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.

Baer, R. ; Kronik, L. Time-dependent generalized Kohn–Sham theory. The European Physical Journal B 2018, 91, 170. Publisher's VersionAbstract

Generalized Kohn–Sham (GKS) theory extends the realm of density functional theory (DFT) by providing a rigorous basis for non-multiplicative potentials, the use of which is outside original Kohn–Sham theory. GKS theory is of increasing importance as it underlies commonly used approximations, notably (conventional or range-separated) hybrid functionals and meta-generalized-gradient-approximation (meta-GGA) functionals. While this approach is often extended in practice to time-dependent DFT (TDDFT), the theoretical foundation for this extension has been lacking, because the Runge–Gross theorem and the van Leeuwen theorem that serve as the basis of TDDFT have not been generalized to non-multiplicative potentials. Here, we provide the necessary generalization. Specifically, we show that with one simple but non-trivial additional caveat – upholding the continuity equation in the GKS electron gas – the Runge–Gross and van Leeuwen theorems apply to time-dependent GKS theory. We also discuss how this is manifested in common GKS-based approximations.

Ruan, Z. ; Baer, R. Unravelling open-system quantum dynamics of non-interacting Fermions. Mol. Phys. 2018, 116, 2490-2496. Publisher's VersionAbstract

ABSTRACTThe 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 HubbardStratonovich 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.

Cytter, Y. ; Rabani, E. ; Neuhauser, D. ; Baer, R. Stochastic Density Functional Theory at Finite Temperatures. Phys. Rev. B 2018, 97, 115207. Publisher's VersionAbstract

Simulations in the warm dense matter regime using finite temperature Kohn-Sham density functional theory (FT-KS-DFT), while frequently used, are extremely expensive computationally 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 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.

2017
Luzon, I. ; Jagtap, K. ; Livshits, E. ; Lioubashevski, O. ; Baer, R. ; Strasser, D. Single-photon Coulomb explosion of methanol using broad bandwidth ultrafast EUV pulses. Phys. Chem. Chem. Phys. 2017, 19, 13488–13495.Abstract

Single-photon Coulomb explosion of methanol is instigated using the broad bandwidth pulse achieved through high-order harmonics generation. Using 3D coincidence fragment imaging of one molecule at a time, the kinetic energy release (KER) and angular distributions of the products are measured in different Coulomb explosion (CE) channels. Two-body CE channels breaking either the C–O or the C–H bonds are described as well as a proton migration channel forming H2O+, which is shown to exhibit higher KER. The results are compared to intense-field Coulomb explosion measurements in the literature. The interpretation of broad bandwidth single-photon CE data is discussed and supported by ab initio calculations of the predominant C–O bond breaking CE channel. We discuss the importance of these findings for achieving time resolved imaging of ultrafast dynamics.

Takeshita, T. Y. ; de Jong, W. A. ; Neuhauser, D. ; Baer, R. ; Rabani, E. Stochastic Formulation of the Resolution of Identity: Application to Second Order Møller–Plesset Perturbation Theory. J. Chem. Theory Comput. 2017, 13, 4605. Publisher's VersionAbstract

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 Møller-Plesset perturbation theory (MP2) utilizing a multiple stochastic orbital approach. The introduction of multiple stochastic orbitals results in an O(N_AO^3 ) 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(Ne^2.4) for total energies and O(Ne^3.1) for forces (Ne being the number of correlated electrons), outperforming MP2 for clusters with as few as 21 water molecules.

Vlček, V. ; Rabani, E. ; Neuhauser, D. ; Baer, R. Stochastic GW calculations for molecules. J. Chem. Theory Comput. 2017, 13, 4997–5003.Abstract

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 Kohn-Sham (KS) formulation does not provide reliable QP energies, a many-body perturbation technique within the GW approximation are essential. The steep computational scaling of GW prohibits its use in extended, open boundary, systems with thousands of electrons and more. Recently, a stochastic formulation of GW has been proposed [Phys. Rev. Lett. 113, 076402 (2014)] which scales nearly linearly with the system size, as illustrated for a series of silicon nanocrystals exceeding 3000 electrons. Here, we implement the stochastic GW (sGW) approach to study the ionization potential (IP) of a subset of molecules taken from the "GW 100" benchmark. We show that sGW provides a reliable results in comparison to GW WEST code and to experimental results, numerically establishing its validity. For completeness, we also provide a detailed review of sGW and a summary of the numerical algorithm.

Neuhauser, D. ; Baer, R. ; Zgid, D. Stochastic self-consistent second-order Green’s function method for correlation energies of large electronic systems. J. Chem. Theory Comput. 2017, 13, 5396−5403.Abstract

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.

Hadad, R. E. ; Baer, R. Minimally corrected partial atomic charges for non-covalent electrostatic interactions. Mol. Phys. 2017, 115, 3155-3163. Publisher's VersionAbstract

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 necessaryfor 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 iterative-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.

Buchman, O. ; Baer, R. Stochastic method for calculating the ground-state one-body density matrix of trapped Bose particles in one dimension. Phys. Rev. A 2017, 96, 033626.
Arnon, E. ; Rabani, E. ; Neuhauser, D. ; Baer, R. Equilibrium configurations of large nanostructures using the embedded saturated-fragments stochastic density functional theory. J. Chem. Phys. 2017, 146, 224111.Abstract

An ab initio Langevin dynamics approach is developed based on stochastic density functional theory (sDFT) within a new embedded fragment formalism. 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–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 3nm corresponding to Ne = 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.

2016
Vlček, V. ; Eisenberg, H. R. ; Steinle-Neumann, G. ; Rabani, E. ; Neuhauser, D. ; Baer, R. Spontaneous charge carrier localization in extended one-dimensional systems. Phys. Rev. Lett. 2016, 116, 186401.Abstract

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 length. Moreover, we find that 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.

Eshet, H. ; Baer, R. ; Neuhauser, D. ; Rabani, E. Theory of highly efficient multiexciton generation in type-II nanorods. Nat. Commun. 2016, 7.Abstract

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.

2015
Vlček, V. ; Eisenberg, H. R. ; Steinle-Neumann, G. ; Kronik, L. ; Baer, R. Deviations from piecewise linearity in the solid-state limit with approximate density functionals. J. Chem. Phys. 2015, 142, 034107.Abstract

In exact density functional theory, the total ground-state energy is a series of linear segments between integer electron points, a condition known as “piecewise linearity.” Deviation from this condition is indicative of poor predictive capabilities for electronic structure, in particular of ionization energies, fundamental gaps, and charge transfer. In this article, we take a new look at the deviation from linearity (i.e., curvature) in the solid-state limit by considering two different ways of approaching it: a large finite system of increasing size and a crystal represented by an increasingly large reference cell with periodic boundary conditions. We show that the curvature approaches vanishing values in both limits, even for functionals which yield poor predictions of electronic structure, and therefore cannot be used as a diagnostic or constructive tool in solids. We find that the approach towards zero curvature is different in each of the two limits, owing to the presence of a compensating background charge in the periodic case. Based on these findings, we present a new criterion for functional construction and evaluation, derived from the size-dependence of the curvature, along with a practical method for evaluating this criterion. For large finite systems, we further show that the curvature is dominated by the self-interaction of the highest occupied eigenstate. These findings are illustrated by computational studies of various solids, semiconductor nanocrystals, and long alkane chains.

Shao, Y. ; Gan, Z. ; Epifanovsky, E. ; Gilbert, A. T. B. ; Wormit, M. ; Kussmann, J. ; Lange, A. W. ; Behn, A. ; Deng, J. ; Feng, X. ; et al. Advances in molecular quantum chemistry contained in the Q-Chem 4 program package. Mol. Phys. 2015, 113, 184-215. Publisher's VersionAbstract
A summary of the technical advances that are incorporated in the fourth major release of the Q-Chem quantum chemistry program is provided, covering approximately the last seven years. These include developments in density functional theory methods and algorithms, nuclear magnetic resonance (NMR) property evaluation, coupled cluster and perturbation theories, methods for electronically excited and open-shell species, tools for treating extended environments, algorithms for walking on potential surfaces, analysis tools, energy and electron transfer modelling, parallel computing capabilities, and graphical user interfaces. In addition, a selection of example case studies that illustrate these capabilities is given. These include extensive benchmarks of the comparative accuracy of modern density functionals for bonded and non-bonded interactions, tests of attenuated second order Møller–Plesset (MP2) methods for intermolecular interactions, a variety of parallel performance benchmarks, and tests of the accuracy of implicit solvation models. Some specific chemical examples include calculations on the strongly correlated Cr2 dimer, exploring zeolite-catalysed ethane dehydrogenation, energy decomposition analysis of a charged ter-molecular complex arising from glycerol photoionisation, and natural transition orbitals for a Frenkel exciton state in a nine-unit model of a self-assembling nanotube.
Petsalakis, I. D. ; Theodorakopoulos, G. ; Buchman, O. ; Baer, R. Applicability of Mulliken's formula for photoinduced and intramolecular charge-transfer energies. Chem. Phys. Lett. 2015, 625, 98–103.Abstract

The applicability of Mulliken’s theory for photoinduced as well as intramolecular charge-transfer states is examined for several systems of interest by comparing its predictions to TDDFT excitation energies, obtained using functionals appropriate for charge-transfer (CT) states. The results show that it is possible to estimate the energy of the CT state of a donor–acceptor pair on the basis of information on the separate donor and acceptor moieties, along with structural data, within 0.3 eV of TDDFT values. The novelty and usefulness of the proposed method lies mainly in PET applications where the TDDFT determination of the CT state is challenging.