# Research

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.