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This erratum corrects a factor 2 error in previous work on current-dependent TDDFT.
For a large class of orthogonal basis functions, there has been a recent identification of expansion methods for computing accurate, stable approximations of a quantity of interest. Here this paper presents, within the context of uncertainty quantification, a practical implementation using basis adaptation, and coherence motivated sampling, which under assumptions has satisfying guarantees. This implementation is referred to as Basis Adaptive Sample Efficient Polynomial Chaos (BASE-PC). A key component of this is the use of anisotropic polynomial order which admits evolving global bases for approximation in an efficient manner, leading to consistently stable approximation for a practical class of smooth functionals. This fully adaptive, non-intrusive method, requires no a priori information of the solution, and has satisfying theoretical guarantees of recovery. A key contribution to stability is the use of a presented correction sampling for coherence-optimal sampling in order to improve stability and accuracy within the adaptive basis scheme. Theoretically, the method may dramatically reduce the impact of dimensionality in function approximation, and numerically the method is demonstrated to perform well on problems with dimension up to 1000.
Here, the damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter $\kappa$ is analyzed, when the external force is periodic in space and given by $\mathcal{f(x) = r}$ cos(Kx), both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our variational approximation satisfies exactly the low-order moment equations. Because of competition between the spatial period of the external force λ = 2π/K , and the soliton width $l_s$, which is a function of the nonlinearity $\kappa$ as well as the initial frequency ω 0 of the solitary wave, there is a transition (at fixed ω 0 ) from trapped to unbound behavior of the soliton, which depends on the parameters $\mathcal{r}$ and K of the external force and the nonlinearity parameter $\kappa$. We previously studied this phenomena when $\kappa$ = 1 where we showed that for λ $\gg l_s$ the soliton oscillates in an effective potential, while forλ $\ll l_s$ it moves uniformly as a free particle. In this paper we focus on the $\kappa$ dependence of the transition from oscillatory to particle behavior and explicitly compare the curves of the transition regime found in the collective coordinate approximation as a function of $\mathcal{r}$ and K when $\kappa$ = 1/2,1,2 at fixed value of the frequency ω 0 . Since the solitary wave gets narrower for fixed ω 0 as a function of $\kappa$, we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase $\kappa$.
We develop a novel method for the calculation and the analysis of the one-electron reduced densities in open-shell molecular systems using the natural orbitals and approximate spin projected occupation numbers obtained from broken symmetry (BS), i.e., spin-unrestricted (U), density functional theory (DFT) calculations. The performance of this approximate spin projection (ASP) scheme is examined for the diradical character dependence of the second hyperpolarizability (γ) using several exchange-correlation functionals, i.e., hybrid and long-range corrected UDFT schemes. It is found that the ASP-LC-UBLYP method with a range separating parameter μ = 0.47 reproduces semi-quantitatively the strongly-correlated [UCCSD(T)] result for p-quinodimethane, i.e., the γ variation as a function of the diradical character.
The electron cyclotron current drive (ECCD) is described by the solution of a drift-kinetic equation (DKE) with the quasi-linear heating term which is highly localised in phase space. Within the adjoint approach, an equivalent DKE must be solved without this highly localised source, and the ECCD is described by a straightforward convolution in phase space. The parallel momentum correction technique [Maassberg et al., Phys. Plasmas 16, 072504 (2009)] reduces the problem to the solution of a simplified mono-energetic DKE. The mono-energetic distribution functions are calculated by the DKES code [W. I. van Rij and S. P. Hirshman, Phys. Fluids B 1, 563 (1989)]. Parallel momentum correction, requiring only an energy-dependent weighting factor, is applied to these distribution functions allowing for the direct estimation of the EC-driven current. For small yet finite collisionalities, a rather simple model for ECCD is presented. In this approach, the symmetric portion of the electron distribution function with respect to v{sub Parallel-To} (the Ware pinch contribution) is neglected and an 'off-set' only in the passing particle domain adds to the collisionless (anti-symmetric) distribution function. For this approximation, only the mono-energetic parallel conductivity coefficient as function of the collisionality is needed. The impact of small yet finite collisionalities on ECCD is shown for X2- and O2-scenarios at W7-X.
This paper is devoted to the theory of approximation of coefficient inverse problems for differential equations of parabolic, elliptic, and hyperbolic types in functional spaces. We present general statements of problems and their approximations and review results obtained earlier in the literature.
Predicting phonon relaxation time from molecular dynamics (MD) requires a long simulation time to compute the mode energy auto-correlation function. Here, we present an alternative approach to infer the phonon life-time from an approximate form of the energy auto-correlation function. The method requires as an input a set of sampled equilibrium configurations. A stochastic sampling method is used to generate the equilibrium configurations. We consider a truncated Taylor series expansion of the phonon energy auto-correlation function. The different terms in the truncated correlation function are obtained using the stochastic sampling approach. The expansion terms, thus, obtained are in good agreement with the corresponding values obtained using MD. We then use the approximate function to compute the phonon relaxation time. The relaxation time computed using this method is compared with that obtained from the exact correlation function. The two values are in agreement with each other.
We propose an ensemble score filter (EnSF) for solving high-dimensional nonlinear filtering problems with superior accuracy. A major drawback of existing filtering methods, e.g., particle filters or ensemble Kalman filters, is the low accuracy in handling high-dimensional and highly nonlinear problems. EnSF attacks this challenge by exploiting the score-based diffusion model, defined in a pseudo-temporal domain, to characterizing the evolution of the filtering density. EnSF stores the information of the recursively updated filtering density function in the score function, in stead of storing the information in a set of finite Monte Carlo samples (used in particle filters and ensemble Kalman filters). Unlike existing diffusion models that train neural networks to approximate the score function, we develop a training-free score estimation that uses mini-batch-based Monte Carlo estimator to directly approximate the score function at any pseudo-spatial-temporal location, which provides sufficient accuracy in solving high-dimensional nonlinear problems as well as saves tremendous amount of time spent on training neural networks. Another essential aspect of EnSF is its analytical update step, gradually incorporating data information into the score function, which is crucial in mitigating the degeneracy issue faced when dealing with very high-dimensional nonlinear filtering problems. High-dimensional Lorenz systems are used to demonstrate the performance of our method. EnSF provides surprisingly impressive performance in reliably tracking extremely high-dimensional Lorenz systems (up to 1,000,000 dimension) with highly nonlinear observation processes, which is a well-known challenging problem for existing filtering methods.
Here Tthis work describes a newly developed, arbitrarily high-order Cartesian-grid method for reconstructing material interfaces from a volume fraction field. The method begins by identifying all of the grid cells in the volume fraction field that are intersected by the interface and need to be approximated by the reconstruction scheme. Finite-differences are used to calculate the gradient of the volume fraction field and provide an estimate of the surface normal in all of the interfacial grid cells. Groups of connected grid cells are then identified which all have the same dominant component of the normal vector. This grouping by orientation determines the proper dependent variable to use in the surface reconstruction (e.g. for a 2D curve, this step determines if the surface will be approximated by a function of x or y). A cumulative integral over the surface is constructed and fit using b-splines for two-dimensional problems or tensor-product b-splines for three-dimensional problems. This construction allows for the interface to be recovered through application of the second fundamental theorem of calculus. Fitting the cumulative integral with $\mathscr{N}$ th-order b-splines (or tensor-product b-splines) yields an ($\mathscr{N}$-1) th-order convergence rate of the interface shape. Differentiation of the b-spline interface function(s) allows for the high-order approximation of the normal vector and curvature to be obtained directly anywhere along b-spline. Together, the proposed reconstruction technique can achieve arbitrarily high mesh convergence rates. Validation tests are presented with mesh convergence rates ranging from fourth- to tenth-order.
The potential for density functional calculations to predict the properties of matter under extreme conditions depends crucially upon having a non-empirical approximate free energy functional valid over a wide range of state conditions. Unlike the ground-state case, no such free-energy exchange- correlation (XC) functional exists. We remedy that with systematic construction of a generalized gradient approximation XC free-energy functional based on rigorous constraints, including the free energy gradient expansion. The new functional provides the correct temperature dependence in the slowly varying regime and the correct zero-T, high-T, and homogeneous electron gas limits. Application in Kohn-Sham calculations for hot electrons in a static fcc Aluminum lattice demon- strates the combined magnitude of thermal and gradient effects handled by this functional. Its accuracy in the increasingly important warm dense matter regime is attested by excellent agreement of the calculated deuterium equation of state with reference path integral Monte Carlo results at intermediate and elevated temperatures and by low density Al calculations over a wide T range.
The kinetic reduced-description particle-in-cell simulation technique has been applied to study the nonlinear stage of two-plasmon–decay (TPD) instability in an inhomogeneous plasma driven by crossed laser beams. The TPD instability is found to be a prolific generator of “cavitating” Langmuir turbulence. Langmuir “cavitons”—localized longitudinal electric fields, oscillating near the local electron plasma frequency, trapped in ponderomotive density depressions—collapse to dimensions of a few electron Debye lengths, where the electric field energy is collisionlessly transferred to electron kinetic energy. The resulting hot electrons can attain instantaneous temperatures up to 100 keV with net suprathermal heat flux out of the system of up to a few percent of the input laser energy. Scaling laws for this hot-electron generation by TPD, in regimes motivated by recent experiments on the Omega laser, were presented recently by Vu et al. (H. X. Vu, D. F. DuBois, D. A. Russell, and J. F. Myatt, Phys. Plasmas 19, 102703 (2012)). This paper concentrates on the microscopic mechanisms for hot-electron generation. The spatial distribution of the maxima of the electric field envelope modulus is found to be very spiky, with the distribution of electric field envelope maxima obeying Gaussian statistics. The cavitons are produced in density-depletion trenches produced by the combined ponderomotive interference of the crossed laser beams and the ponderomotive beats of the primary backward-going TPD Langmuir waves (LWs) resulting from the crossed beams. The Langmuir turbulence is strongest in the electron-density region near 0.241× the laser's critical density, where the forward LWs from the crossed-beam TPD are degenerate. Nucleation of cavitons is assisted by the modulation of the electron density in the trenches, which in turn is caused by the beating of the common forward-going LW and the pair of backward-going LWs. The autocorrelation function of the LW envelope field provides a near-universal shape for intense cavitons—in the neighborhood of the local field maxima. The hot-electron temperature is found to be approximately a linear function of the “caviton temperature” determined from the Gaussian distribution of caviton maxima. These diagnostics provide strong evidence for the importance of Langmuir caviton collapse in the generation of hot electrons by TPD. Extended Zakharov model predictions for TPD exhibit the same qualitative phenomena.
The heat conductivity ({lambda}) and the thermal diffusivity (a) of reacting glass batch, or melter feed, control the heat flux into and within the cold cap, a layer of reacting material floating on the pool of molten glass in an all-electric continuous waste glass melter. After previously estimating {lambda} of melter feed at temperatures up to 680 deg C, we focus in this work on the {lambda}(T) function at T > 680 deg C, at which the feed material becomes foamy. We used a customized experimental setup consisting of a large cylindrical crucible with an assembly of thermocouples, which monitored the evolution of the temperature field while the crucible with feed was heated at a constant rate from room temperature up to 1100°C. Approximating measured temperature profiles by polynomial functions, we used the heat transfer equation to estimate the {lambda}(T) approximation function, which we subsequently optimized using the finite-volume method combined with least-squares analysis. The heat conductivity increased as the temperature increased until the feed began to expand into foam, at which point the conductivity dropped. It began to increase again as the foam turned into a bubble-free glass melt. We discuss the implications of this behavior for the mathematical modeling of the cold cap.
The asymmetric Hubbard dimer is used to study the density-dependence of the exact frequency-dependent kernel of linear-response time-dependent density functional theory. The exact form of the kernel is given, and the limitations of the adiabatic approximation utilizing the exact ground-state functional are shown. The oscillator strength sum rule is proven for lattice Hamiltonians, and relative oscillator strengths are defined appropriately. The method of Casida for extracting oscillator strengths from a frequency-dependent kernel is demonstrated to yield the exact result with this kernel. An unambiguous way of labelling the nature of excitations is given. The fluctuation-dissipation theorem is proven for the ground-state exchange-correlation energy. The distinction between weak and strong correlation is shown to depend on the ratio of interaction to asymmetry. A simple interpolation between carefully defined weak-correlation and strong-correlation regimes yields a density-functional approximation for the kernel that gives accurate transition frequencies for both the single and double excitations, including charge-transfer excitations. In conclusion, many exact results, limits, and expansions about those limits are given in the Appendices.
Thermal density functional theory is commonly used in simulations of warm dense matter, a highly energetic phase characterized by substantial thermal effects and by correlated electrons demanding quantum mechanical treatment. Methods that account for temperature dependence, such as Mermin–Kohn–Sham finite-temperature density functional theory and free energy density functional theory, are now employed with more regularity and available in many standard code packages. However, approximations from zero-temperature density functional theory are still often used in temperature-dependent simulations using thermally weighted electronic densities as an input to exchange–correlation functional approximations, a practice known to miss temperature-dependent effects in the exchange–correlation free energy of these systems. Here, in this work, the temperature-dependent adiabatic connection is demonstrated and analyzed using a well-known parameterization of the uniform electron gas free energy. Useful tools based on this formalism for analyzing and constraining approximations of the exchange–correlation at zero temperature are leveraged for the finite-temperature case. Inspired by the Lieb–Oxford inequality, which provides a lower bound for the ground-state exchange–correlation energy, bounds for the exchange–correlation at finite temperatures are approximated for various degrees of electronic correlation.
We show approximations of the non-additive non-interacting kinetic energy (NAKE) as an explicit functional of the density are the basis of several electronic structure methods that provide improved computational efficiency over standard Kohn-Sham calculations. However, within most fragment-based formalisms, there is no unique exact NAKE, making it difficult to develop general, robust approximations for it. When adjustments are made to the embedding formalisms to guarantee uniqueness, approximate functionals may be more meaningfully compared to the exact unique NAKE. We use numerically accurate inversions to study the exact NAKE of several rare-gas dimers within partition density functional theory, a method that provides the uniqueness for the exact NAKE. We find that the NAKE decreases nearly exponentially with atomic separation for the rare-gas dimers. We compute the logarithmic derivative of the NAKE with respect to the bond length for our numerically accurate inversions as well as for several approximate NAKE functionals. We show that standard approximate NAKE functionals do not reproduce the correct behavior for this logarithmic derivative and propose two new NAKE functionals that do. The first of these is based on a re-parametrization of a conjoint Perdew-Burke-Ernzerhof (PBE) functional. The second is a simple, physically motivated non-decomposable NAKE functional that matches the asymptotic decay constant without fitting.
We study the resistivity scaling in nanometer-sized metallic wires due to surface roughness and grain-boundaries, currently the main cause of electron scattering in nanoscaled interconnects. The resistivity has been obtained with the Boltzmann transport equation, adopting the relaxation time approximation of the distribution function and the effective mass approximation for the conducting electrons. The relaxation times are calculated exactly, using Fermi's golden rule, resulting in a correct relaxation time for every sub-band state contributing to the transport. In general, the relaxation time strongly depends on the sub-band state, something that remained unclear with the methods of previous work. The resistivity scaling is obtained for different roughness and grain-boundary properties, showing large differences in scaling behavior and relaxation times. Our model clearly indicates that the resistivity is dominated by grain-boundary scattering, easily surpassing the surface roughness contribution by a factor of 10.
Surface diffusion of water monomer, dimer, and trimer on the (111) terrace, (221) and (322) stepped, and (763) and (854) kinked surfaces of platinum was studied by density functional theory using the PW91 approximation to the energy functional. Monomer diffusion on the terrace is facile, with an activation barrier of 0.20 eV, while dimer and trimer diffusions are restricted due to their high activation barriers of 0.43 and 0.48 eV, respectively. During monomer diffusion on the terrace the O–Pt distance increases by 0.54 Å, about 23% of the initial distance of 2.34 Å. The calculated rate of monomer diffusion hops is in good agreement with the onset temperature of diffusion measurements of Daschbach et al., J. Chem. Phys., 120 (2004) 1516. Alternative monomer diffusion pathways, in which the molecule rolls or flips, were also found. These pathways have diffusion barriers of 0.22 eV. During dimer diffusion on the terrace, the donor molecule rises 0.4 Å at the saddle point, while the acceptor rises by only 0.03 Å. Monomer diffusion up to steps and kinks, with activation barriers of 0.11–0.13 eV, facilitate chain formation on top of step edges. The energy landscape of monomer diffusion from terrace to step to kink sites is downhill with a maximum activation barrier of 0.26 eV. A model for water adsorption is presented inwhichmonomer diffusion leads to concurrent formation of terrace clusters and population of steps/kinks, the latter consistent with the STMmeasurements ofMorgenstern et al., Phys. Rev. Lett., 77 (1996) 703.
Catalyst confinement within microporous media provides the opportunity to site isolate reactive intermediates, enforce intermolecular functionalization chemistry by co-localizing reactive intermediates and substrates in molecular-scale interstices, and harness non-covalent host–guest interactions to achieve selectivities that are complementary to those accessible in solution. As part of an ongoing program to develop synthetically useful nitrogen-atom transfer (NAT) catalysts, we have demonstrated intermolecular benzylic amination of toluene at a Ru 2 nitride intermediate confined within the interstices of a Ru2-based metal–organic framework (MOF), Ru 3 (btc) 2 X 3 (btc = 1,3,5-benzenetricarboxylate, i.e., Ru-HKUST-1 for X = Cl). Nitride confinement within the extended MOF lattice enabled intermolecular C–H functionalization of benzylic C–H bonds in preference to nitride dimerization, which was encountered with soluble molecular analogues. Detailed study of the kinetic isotope effects (KIEs, i.e., k H /k D ) of C–H amination, assayed both as intramolecular effects using partially labeled toluene and as intermolecular effects using a mixture of per-labeled and unlabeled toluene, provided evidence for restricted substrate mobility on the time scale of interstitial NAT. Analysis of these KIEs as a function of material mesoporosity provided approximate experimental values for functionalization in the absence of mass transport barriers. Here, we disclose a combined experimental and computational investigation of the mechanism of NAT from a Ru 2 nitride to the C–H bond of toluene. Computed kinetic isotope effects for a H-atom abstraction (HAA)/radical rebound (RR) mechanism are in good agreement with experimental data obtained for C–H amination at the rapid diffusion limit. These results provide the first detailed analysis of the mechanism of intermolecular NAT to a C–H bond, bolster the use of KIEs as a probe of confinement effects on NAT within MOF lattices, and provide mechanistic insights unavailable by experiment because rate-determining mass transport obscured the underlying chemical kinetics.