Research

2015
Neuhauser, D. ; Rabani, E. ; Cytter, Y. ; Baer, R. Stochastic Optimally Tuned Range-Separated Hybrid Density Functional Theory. J. Phys. Chem. A 2015, 120, 3071–3078.Abstract

We develop a stochastic formulation of the optimally tuned range-separated hybrid density functional theory that enables significant reduction of the computational effort and scaling of the nonlocal 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 a 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 that is central for additional postprocessing, for example, in the stochastic Bethe–Salpeter approach.

Rabani, E. ; Baer, R. ; Neuhauser, D. Time-dependent stochastic Bethe-Salpeter approach. Phys. Rev. B 2015, 91, 235302.Abstract

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 the configuration-interaction with single substitution (CIS) method. This results in a time-dependent Schrödinger-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.

Jacobi, S. ; Baer, R. Smoothing and extrapolating shifted-contour auxiliary-field Monte Carlo signals using discrete Laguerre functions. arXiv preprint arXiv:1504.05452 2015.Abstract

We develop a new smoothing or extrapolating method, based on discrete Laguerre functions, for systematically analyzing the stochastic signal of shifted-contour auxiliary-field Monte Carlo. We study the statistical errors and extrapolation errors using full configuration-interaction energies for the doubly stretched water molecule. The only free parameter is the order N of the fit. We show that low N emphasizes stability while higher N enable improved extrapolation, at the cost of increased statistical errors. Typically, one should use low order for signals based on a small number of iterations while higher order is efficacious for signals based on large number of iterations. We provide a heuristic algorithm for determining the order to be used and show its utility.

Gao, Y. ; Neuhauser, D. ; Baer, R. ; Rabani, E. Sublinear scaling for time-dependent stochastic density functional theory. J. Chem. Phys. 2015, 142, 034106.Abstract

A stochastic approach to time-dependent density functional theory 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 using real-space grids and find that the overall scaling of the algorithm is sublinear with computational time and memory.

2014
Neuhauser, D. ; Gao, Y. ; Arntsen, C. ; Karshenas, C. ; Rabani, E. ; Baer, R. Breaking the Theoretical Scaling Limit for Predicting Quasiparticle Energies: The Stochastic GW Approach. Phys. Rev. Lett. 2014, 113, 076402.Abstract

We develop a formalism to calculate the quasiparticle energy within the GW many-body perturbation correction to the density functional theory. The occupied and virtual orbitals of the Kohn-Sham Hamiltonian are replaced by stochastic orbitals used to evaluate the Green function G, the polarization potential W, and, thereby, the GW self-energy. The stochastic GW (sGW) formalism relies on novel theoretical concepts such as stochastic time-dependent Hartree propagation, stochastic matrix compression, and spatial or temporal stochastic decoupling techniques. Beyond the theoretical interest, the formalism enables linear scaling GW calculations breaking the theoretical scaling limit for GW as well as circumventing the need for energy cutoff approximations. We illustrate the method for silicon nanocrystals of varying sizes with Ne > 3000 electrons.

Neuhauser, D. ; Baer, R. ; Rabani, E. Communication: Embedded fragment stochastic density functional theory. J. Chem. Phys. 2014, 141, 041102.Abstract

We develop a method in which the electronic densities of small fragments determined by Kohn-Sham density functional theory (DFT) are embedded using stochastic DFT to form the exact density of the full system. The new method preserves the scaling and the simplicity of the stochastic DFT but cures the slow convergence that occurs when weakly coupled subsystems are treated. It overcomes the spurious charge fluctuations that impair the applications of the original stochastic DFT approach. We demonstrate the new approach on a fullerene dimer and on clusters of water molecules and show that the density of states and the total energy can be accurately described with a relatively small number of stochastic orbitals.

Baratz, A. ; White, A. J. ; Galperin, M. ; Baer, R. Effects of Electromagnetic Coupling on Conductance Switching of a Gated Tunnel Junction. The Journal of Physical Chemistry Letters 2014, 5 3545–3550.Abstract

Using a combination of density functional theory and quantum master equations approach, we study the effect of electromagnetic (EM) coupling on the nonequilibrium steady-state behavior of a recently introduced gated molecular junction. This junction was demonstrated in a previous publication to exhibit sharp current switching near a certain critical DC field Ez*, which induces intramolecular charge transfer, and here, we analyze the steady-state population and current when an AC EM field (EMF) is present. The AC EMF at frequency $ømega_0$ produces pronounced population and current features at gate fields Ez = Ez* ± $\hbar ømega_0/ez$ (where $e_z$ is the dipole of the charge-transfer state) and thus allows additional sharp switching capability at lower gate fields. We found that even when EMF is absent, the EM coupling itself changes the overall steady-state population and current distributions because it allows for relaxation via spontaneous emission

Eisenberg, H. R. ; Baer, R. Exothermic Mechanism for the Abstraction of Hydrogen from Methane on Li-Doped MgO. The Journal of Physical Chemistry C 2014, 119, 196–215.Abstract

Despite its importance, it is still not known by which mechanism Li-doped MgO catalyzes the oxidative coupling of methane to ethane. Nevertheless, it is commonly assumed that the mechanism goes through catalytic H abstraction from methane via a Li+O– surface defect. In this paper we use first-principles density functional theory calculations to show that the reaction is significantly more exothermic when the Li+O– defect is situated on a step edge instead of on the flat surface. We find that the reaction on the step is exothermic by 0.25 eV, whereas it is endothermic by 0.3 eV on the flat surface. The presence of the Li dopant in the step edge is crucial for the exothermicity of the reaction. These findings suggest that surface steps which include lithium defects could be responsible for the catalytic behavior of Li/MgO. Following the binding of hydrogen to the Li+O– defect on the step edge the methyl radical can either depart to the gas phase or bind to an adjacent step-edge oxygen atom, increasing the exothermicity of the overall process to 0.8 eV. Activation energies of 0.2 eV for the first pathway and 0.5–0.8 eV for the second were calculated.

Cytter, Y. ; Neuhauser, D. ; Baer, R. Metropolis Evaluation of the Hartree–Fock Exchange Energy. J. Chem. Theory Comput. 2014, 10, 4317–4323.Abstract

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 $\rho(x,x')^2$, where $\rho(x,x')$ is the density matrix. The exchange energy per particle is then simply the average of the Coulomb repulsion energy U_C(|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.

Egger, D. A. ; Weissman, S. ; Refaely-Abramson, S. ; Sharifzadeh, S. ; Dauth, M. ; Baer, R. ; Kümmel, S. ; Neaton, J. B. ; Zojer, E. ; Kronik, L. Outer-valence Electron Spectra of Prototypical Aromatic Heterocycles from an Optimally Tuned Range-Separated Hybrid Functional. J. Chem. Theory Comput. 2014, 10, 1934–1952.Abstract

Density functional theory with optimally tuned range-separated hybrid (OT-RSH) functionals has been recently suggested [Refaely-Abramson et al. Phys. Rev. Lett. 2012, 109, 226405] as a nonempirical approach to predict the outer-valence electronic structure of molecules with the same accuracy as many-body perturbation theory. Here, we provide a quantitative evaluation of the OT-RSH approach by examining its performance in predicting the outer-valence electron spectra of several prototypical gas-phase molecules, from aromatic rings (benzene, pyridine, and pyrimidine) to more complex organic systems (terpyrimidinethiol and copper phthalocyanine). For a range up to several electronvolts away from the frontier orbital energies, we find that the outer-valence electronic structure obtained from the OT-RSH method agrees very well (typically within  0.1–0.2 eV) with both experimental photoemission and theoretical many-body perturbation theory data in the GW approximation. In particular, we find that with new strategies for an optimal choice of the short-range fraction of Fock exchange, the OT-RSH approach offers a balanced description of localized and delocalized states. We discuss in detail the sole exception found—a high-symmetry orbital, particular to small aromatic rings, which is relatively deep inside the valence state manifold. Overall, the OT-RSH method is an accurate DFT-based method for outer-valence electronic structure prediction for such systems and is of essentially the same level of accuracy as contemporary GW approaches, at a reduced computational cost.

2013
Zohar, G. ; Baer, R. ; Rabani, E. Multiexciton generation in IV–VI nanocrystals: the role of carrier effective mass, band mixing, and phonon emission. The journal of physical chemistry letters 2013, 4 317–322.Abstract

We study the role of the effective mass, band mixing, and phonon emission on multiexciton generation in IV–VI nanocrystals. A four-band k · p effective mass model, which allows for an independent variation of these parameters, is adopted to describe the electronic structure of the nanocrystals. Multiexciton generation efficiencies are calculated using a Green’s function formalism, providing results that are numerically similar to impact excitation. We find that multiexciton generation efficiencies are maximized when the effective mass of the electron and hole are small and similar. Contact with recent experimental results for multiexciton generation in PbS and PbSe is made.

Neuhauser, D. ; Rabani, E. ; Baer, R. Expeditious Stochastic Approach for MP2 Energies in Large Electronic Systems. J. Chem. Theory Comput. 2013, 9 24–27.Abstract

A fast stochastic method for calculating the second order Møller-Plesset (MP2) correction to the correlation energy of large systems of electrons is presented. The approach is based on reducing the exact summation over occupied and unoccupied states to a time-dependent trace formula amenable to stochastic sampling. We demonstrate the abilities of the method to treat systems with thousands of electrons using hydrogen passivated silicon spherical nanocrystals represented on a real space grid, much beyond the capabilities of present day MP2 implementations.

Neuhauser, D. ; Rabani, E. ; Baer, R. Expeditious Stochastic Calculation of Random-Phase Approximation Energies for Thousands of Electrons in Three Dimensions. J. Phys. Chem. Lett. 2013, 4 1172–1176.Abstract

A fast method is developed for calculating the random phase approximation (RPA) correlation energy for density functional theory. The correlation energy is given by a trace over a projected RPA response matrix, and the trace is taken by a stochastic approach using random perturbation vectors. For a fixed statistical error in the total energy per electron, the method scales, at most, quadratically with the system size; however, in practice, due to self-averaging, it requires less statistical sampling as the system grows, and the performance is close to linear scaling. We demonstrate the method by calculating the RPA correlation energy for cadmium selenide and silicon nanocrystals with over 1500 electrons. We find that the RPA correlation energies per electron are largely independent of the nanocrystal size. In addition, we show that a correlated sampling technique enables calculation of the energy difference between two slightly distorted configurations with scaling and a statistical error similar to that of the total energy per electron.

Refaely-Abramson, S. ; Sharifzadeh, S. ; Jain, M. ; Baer, R. ; Neaton, J. B. ; Kronik, L. Gap renormalization of molecular crystals from density-functional theory. Phys. Rev. B 2013, 88, 081204.Abstract

Fundamental gap renormalization due to electronic polarization is a basic phenomenon in molecular crystals. Despite its ubiquity and importance, all conventional approaches within density-functional theory completely fail to capture it, even qualitatively. Here, we present a new screened range-separated hybrid functional, which, through judicious introduction of the scalar dielectric constant, quantitatively captures polarization-induced gap renormalization, as demonstrated on the prototypical organic molecular crystals of benzene, pentacene, and C60. This functional is predictive, as it contains system-specific adjustable parameters that are determined from first principles, rather than from empirical considerations.

Baratz, A. ; Galperin, M. ; Baer, R. Gate-Induced Intramolecular Charge Transfer in a Tunnel Junction: A Nonequilibrium Analysis. J. Phys. Chem. C 2013, 117, 10257–10263.Abstract

A recently introduced molecular junction, for which the gate acts as an on/off switch for intrajunction electron transfer between localized donor and acceptor sites is studied. We demonstrate that a Landauer + density functional (DFT) approach is fundamentally flawed for describing the electronic conductance in this system. By comparing the Landauer + DFT conductance to that predicted by the Redfield quantum master equations, we point out several effects that cannot be explained by the former approach. The molecular junction is unique in the small number of conductance channels and their sharp response to the gate.

Ge, Q. ; Gao, Y. ; Baer, R. ; Rabani, E. ; Neuhauser, D. A Guided Stochastic Energy-Domain Formulation of the Second Order Møller–Plesset Perturbation Theory. J. Phys. Chem. Lett. 2013, 5 185–189.Abstract

We develop an alternative formulation in the energy-domain to calculate the second order Møller–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.

Baer, R. ; Rabani, E. Communication: Biexciton generation rates in CdSe nanorods are length independent. J. Chem. Phys. 2013, 138, 051102–4.Abstract

We study how shape affects multiexciton generation rates in a semiconducting nanocrystal by considering CdSe nanorods with varying diameters and aspect ratios. The calculations employ an atomistic semiempirical pseudopotential model combined with an efficacious stochastic approach applied to systems containing up to 20 000 atoms. The effect of nanorod diameter and aspect ratio on multiexciton generation rates is analyzed in terms of the scaling of the density of trion states and the scaling of the Coulomb couplings. Both show distinct scaling from spherical nanocrystals leading to a surprising result where the multiexciton generation rates are roughly independent of the nanorod length.

Baer, R. ; Neuhauser, D. ; Rabani, E. Self-Averaging Stochastic Kohn-Sham Density-Functional Theory. Phys. Rev. Lett. 2013, 111, 106402. Publisher's VersionAbstract

We formulate the Kohn-Sham density functional theory (KS-DFT) as a statistical theory in which the electron density is determined from an average of correlated stochastic densities in a trace formula. The key idea is that it is sufficient to converge the total energy per electron to within a predefined statistical error in order to obtain reliable estimates of the electronic band structure, the forces on nuclei, the density and its moments, etc. The fluctuations in the total energy per electron are guaranteed to decay to zero as the system size increases. This facilitates “self-averaging” which leads to the first ever report of sublinear scaling KS-DFT electronic structure. The approach sidesteps calculation of the density matrix and thus, is insensitive to its evasive sparseness, as demonstrated here for silicon nanocrystals. The formalism is not only appealing in terms of its promise to far push the limits of application of KS-DFT, but also represents a cognitive change in the way we think of electronic structure calculations as this stochastic theory seamlessly converges to the thermodynamic limit.

2012
Stein, T. ; Autschbach, J. ; Govind, N. ; Kronik, L. ; Baer, R. Curvature and Frontier Orbital Energies in Density Functional Theory. J. Phys. Chem. Lett. 2012, 3 3740–3744.Abstract

Perdew et al. discovered two different properties of exact Kohn–Sham density functional theory (DFT): (i) The exact total energy versus particle number is a series of linear segments between integer electron points. (ii) Across an integer number of electrons, the exchange-correlation potential “jumps” by a constant, known as the derivative discontinuity (DD). Here we show analytically that in both the original and the generalized Kohn–Sham formulation of DFT the two properties are two sides of the same coin. The absence of a DD dictates deviation from piecewise linearity, but the latter, appearing as curvature, can be used to correct for the former, thereby restoring the physical meaning of orbital energies. A simple correction scheme for any semilocal and hybrid functional, even Hartree–Fock theory, is shown to be effective on a set of small molecules, suggesting a practical correction for the infamous DFT gap problem. We show that optimally tuned range-separated hybrid functionals can inherently minimize both DD and curvature, thus requiring no correction, and that this can be used as a sound theoretical basis for novel tuning strategies.

Salzner, U. ; Baer, R. Density functional theory orbital energies for predicting ionization energies. In AIP Conference Proceedings; AIP, 2012; Vol. 1504, pp. 1257–1260.Abstract

The range-separated Baer-Neuhauser-Livshits functional with optimized range-separation parameter \gamma was employed to predict ionization energies of alkanes and oligothiophenes. For all systems negative orbital energies of neutral species are consistent with explicitly calculated states of cations. For \sigma-systems excellent agreement with experiment is obtained while for conjugated π-systems IPs are underestimated.