Self-consistent Iterative Procedure Research Articles

GradDFT. A software library for machine learning enhanced density functional theory.

Density functional theory (DFT) stands as a cornerstone method in computational quantum chemistry and materials science due to its remarkable versatility and scalability. Yet, it suffers from limitations in accuracy, particularly when dealing with strongly correlated systems. To address these shortcomings, recent work has begun to explore how machine learning can expand the capabilities of DFT: an endeavor with many open questions and technical challenges. In this work, we present GradDFT a fully differentiable JAX-based DFT library, enabling quick prototyping and experimentation with machine learning-enhanced exchange-correlation energy functionals. GradDFT employs a pioneering parametrization of exchange-correlation functionals constructed using a weighted sum of energy densities, where the weights are determined using neural networks. Moreover, GradDFT encompasses a comprehensive suite of auxiliary functions, notably featuring a just-in-time compilable and fully differentiable self-consistent iterative procedure. To support training and benchmarking efforts, we additionally compile a curated dataset of experimental dissociation energies of dimers, half of which contain transition metal atoms characterized by strong electronic correlations. The software library is tested against experimental results to study the generalization capabilities of a neural functional across potential energy surfaces and atomic species, as well as the effect of training data noise on the resulting model accuracy.

Analysis of planar waveguides with a thin overlayer and nonlinear cladding

The aim of this paper is to analyse a planar optical waveguide with Kerr-type nonlinear cladding and a thin linear overlayer separating the cladding and the guiding core. The results of numerical calculations show that the bistable behaviour of the nonlinear waveguides can be easily controlled by changing the parameters of this additional layer. The analysis is based on the self-consistent iteration procedure which provides relevant dispersion characteristics limited to physically stable solutions.

A new approach to regression analysis of censored competing-risks data.

An approximate likelihood approach is developed for regression analysis of censored competing-risks data. This approach models directly the cumulative incidence function, instead of the cause-specific hazard function, in terms of explanatory covariates under a proportional subdistribution hazards assumption. It uses a self-consistent iterative procedure to maximize an approximate semiparametric likelihood function, leading to an asymptotically normal and efficient estimator of the vector of regression parameters. Simulation studies demonstrate its advantages over previous methods.

Combination of electron energy-loss spectroscopy and energy dispersive x-ray spectroscopy to determine indium concentration in InGaN thin film structures

We demonstrate a method to determine the indium concentration, x, of InxGa1-xN thin films by combining plasmon excitation studies in electron energy-loss spectroscopy (EELS) with a novel way of quantification of the intensity of x-ray lines in energy-dispersive x-ray spectroscopy (EDXS). The plasmon peak in EELS of InGaN is relatively broad. We fitted a Lorentz function to the main plasmon peak to suppress noise and the influence from the neighboring Ga 3d transition in the spectrum, which improves the precision in the evaluation of the plasmon peak position. As the indium concentration of InGaN is difficult to control during high temperature growth due to partial In desorption, the nominal indium concentrations provided by the growers were not considered reliable. The indium concentration obtained from EDXS quantification using Oxford Instrument ISIS 300 x-ray standard quantification software often did not agree with the nominal indium concentration, and quantification using K and L lines was inconsistent. We therefore developed a self-consistent iterative procedure to determine the In content from thickness-dependent k-factors, as described in recent work submitted to Journal of Microscopy. When the plasmon peak position is plotted versus the indium concentration from EDXS we obtain a linear relationship over the whole compositional range, and the standard error from linear least-squares fitting shows that the indium concentration can be determined from the plasmon peak position to within Δx = ± 0.037 standard deviation.

Self-consistent iteration procedure in analyzing reflectivity and spectroscopic ellipsometry data of multilayered materials and their interfaces.

For multilayered materials, reflectivity depends on the complex dielectric function of all the constituent layers, and a detailed analysis is required to separate them. Furthermore, for some cases, new quantum states can occur at the interface which may change the optical properties of the material. In this paper, we discuss various aspects of such analysis, and present a self-consistent iteration procedure, a versatile method to extract and separate the complex dielectric function of each individual layer of a multilayered system. As a case study, we apply this method to LaAlO3/SrTiO3 heterostructure in which we are able to separate the effects of the interface from the LaAlO3 film and the SrTiO3 substrate. Our method can be applied to other complex multilayered systems with various numbers of layers.

Revision of the thermodynamics of the proton in gas phase.

Proton transfer is ubiquitous in various physical/chemical processes, and the accurate determination of the thermodynamic parameters of the proton in the gas phase is useful for understanding and describing such reactions. However, the thermodynamic parameters of such a proton are usually determined by assuming the proton as a classical particle whatever the temperature. The reason for such an assumption is that the entropy of the quantum proton is not always soluble analytically at all temperatures. Thereby, we addressed this matter using a robust and reliable self-consistent iterative procedure based on the Fermi-Dirac formalism. As a result, the free proton gas can be assumed to be classical for temperatures higher than 200 K. However, it is worth mentioning that quantum effects on the gas phase proton motion are really significant at low temperatures (T ≤ 120 K). Although the proton behaves as a classical particle at high temperatures, we strongly recommend the use of quantum results at all temperatures, for the integrated heat capacity and the Gibbs free energy change. Therefore, on the basis of the thermochemical convention that ignores the proton spin, we recommend the following revised values for the integrated heat capacity and the Gibbs free energy change of the proton in gas phase and, at the standard pressure (1 bar): ΔH0→T = 6.1398 kJ mol(-1) and ΔG0→T = -26.3424 kJ mol(-1). Finally, it is important noting that the little change of the pressure from 1 bar to 1 atm affects notably the entropy and the Gibbs free energy change of the proton.

Thermodynamics of the Electron Revisited and Generalized.

The accurate evaluation of redox potentials in various media and the ability of electron transfer in some biological or chemical reactions are subject to the determination of the accurate gas phase thermodynamic data of the electron. These data are also useful to describe with a high accuracy the movement of the electron in a stellar core. However, these data were not available at all temperatures, and the available data were not sufficiently accurate. I addressed this matter using a robust and reliable self-consistent iterative procedure which determines the entropy of a gas phase free electron and, thereafter, allows the calculation of its heat capacity, enthalpy, and free energy. Extremely accurate analytic expressions of the aforementioned thermodynamic parameters were provided at all temperatures. The thermodynamic parameters of the gas phase electron are now known at all temperatures (integer or noninteger) in the standard atmosphere with a high accuracy. Analytic expressions proposed for the thermodynamic parameters are highly advisable where iteratively computed data are unavailable. Note that at room temperature (T = 298.15 K), the values recommended for the thermodynamic parameters of the gas phase electron are S = 22.6432 J mol(-1) K(-1), CP = 17.1062 J mol(-1) K(-1), ΔH = 3.1351 kJ mol(-1), and ΔG = -3.6160 kJ mol(-1).

Spin Structures of the Hubbard Clusters and the Pseudogap

The spin and charge structures formed in the Hubbard model for a finite two-dimensional cluster have been studied in the mean field approximation. The self-consistent iterative procedure reduces an uncorrelated initial spin distribution into stable structures with spectral properties typical of stripes. It is shown that the density of states of the system for any doping has a sharp minimum at the Fermi level, the pseudogap. The pinning of the gap at the Fermi level is characteristic not only of a superconducting state, but also typical of a normal state of spin glasses. Our results support conception of the PG state as a state with frozen locally nematic spin structures of antiferromagnetic domains.

Spin structures and the nature of the pseudogap in cuprates

The spin and charge structures formed in a Hubbard model for a finite two-dimensional cluster have been studied in the mean field approximation. The self-consistent iterative procedure reduces an uncorrelated initial spin distribution into stable structures with characteristic spectral properties. It has been shown that the density of states of the system for any doping has a sharp minimum at the Fermi level, the pseudogap. This means that the pinning of the gap at the Fermi level is not an exclusive property of a superconducting state, but is also typical of a normal state of spin glasses.

Fitting parton distribution data with multiplicative normalization uncertainties

The extraction of robust parton distribution functions with faithful errors requires a careful treatment of the uncertainties in the experimental results. In particular, the data sets used in current analyses each have a different overall multiplicative normalization uncertainty that needs to be properly accounted for in the fitting procedure. Here we consider the generic problem of performing a global fit to many independent data sets each with a different overall multiplicative normalization uncertainty. We show that the methods in common use to treat multiplicative uncertainties lead to systematic biases. We develop a method which is unbiased, based on a self-consistent iterative procedure. We then apply our generic method to the determination of parton distribution functions with the NNPDF methodology, which uses a Monte Carlo method for uncertainty estimation.

Ab Initio Self-Consistent X-Ray Absorption Fine Structure Analysis for Metalloproteins

X-ray absorption fine structure is a powerful tool for probing the structures of metals in proteins in both crystalline and noncrystalline environments. Until recently, a fundamental problem in biological XAFS has been that ad hoc assumptions must be made concerning the vibrational properties of the amino acid residues that are coordinated to the metal to fit the data. Here, an automatic procedure for accurate structural determination of active sites of metalloproteins is presented. It is based on direct multiple-scattering simulation of experimental X-ray absorption fine structure spectra combining electron multiple scattering calculations with density functional theory calculations of vibrational modes of amino acid residues and the genetic algorithm differential evolution to determine a global minimum in the space of fitting parameters. Structure determination of the metalloprotein active site is obtained through a self-consistent iterative procedure with only minimal initial information.

New quantitative "reading" of dielectric spectra of complex biological systems

Dielectric spectroscopy data from measurements on four insect species from 200 MHz to 20 GHz over a temperature range from 10 to 70degC are described through reduction of numerous micromotions to a few collective motions, recognition of a low-frequency dispersion through ratio presentation format, application of a self-consistent iteration procedure, a separation procedure for types of collective motions, and application of the eigen-coordinates method for calculation of seven fitting parameters. Use of such fitting parameters to describe the total system relaxation processes quantitatively should enable practical uses of dielectric spectroscopy for measuring desired qualities or characteristics of biological and other complex materials. This general approach based on the reduction procedure and the use of different formats, including the original fitting procedure, allows more precise determination of the corresponding fitting function and description of dielectric spectra of complex systems. The application of this general methodology can be considered as the basic goal and motivation for this paper

The effect of quark exchange in A = 3 mirror nuclei and neutron/proton structure functions ratio

By using the quark-exchange formalism, realistic Faddeev wave functions and Fermi motion effect, we investigate deep inelastic electron scattering from A=3 mirror nuclei in the deep-valence region. The initial valence quarks in put are taken from the GRV's next-to-leading order calculations on $F_2^p(x,Q^2)$ which give very good fit to the available data in the $(x,Q^2)$-plane. It is shown that the free neutron to proton structure functions ratio can be extracted from corresponding EMC ratios for $^3He$ and $^3H$ mirror nuclei by using self-consistent iteration procedure and the results are in good agreement with other theoretical models as well as present available experimental data, especially the one expected form the proposed 11 GeV Jefferson Laboratory.

Recognition of the “fractional” kinetics in complex systems: Dielectric properties of fresh fruits and vegetables from 0.01 to 1.8 GHz

Based on the new theoretical approach and the self-consistent iteration procedure for calculating the limiting values ( ε( ∞) and ε(0)) of the frequency-dependent permittivity, it becomes possible to recognize the fitting function for the function ε(j ω) measured for a complex system representing plant tissues of fresh fruits and vegetables in the frequency range (10 7–1.8×10 9 Hz). The recognized fitting function is common for all of a set of nine fruits and vegetables (apple, avocado, banana, cantaloupe, carrot, cucumber, grape, orange, and potato) and contains seven fitting parameters. These parameters are varied for different fruits and vegetables, and their behavior with respect to temperature is different but nevertheless exhibits some common features. This fitting function containing power-law exponents and confirming the existence of relaxation processes described in terms of fractional kinetic equations for some complex biological systems can be used for practical purposes to construct a desired calibration curve with respect to quality factors, as for example, moisture content or degree of maturity. The discovered common “universality” in dielectric behavior of such complex materials as plant tissues opens a possibility to use dielectric spectroscopy as a nondestructive method of control in analysis of electrical behavior (measured in the form of complex permittivity or impedance) for other complex materials.

A procedure for calculating the many-particle Bohm quantum potential

In a recent work, M.Kohout (Int. J. Quant. Chem. 87 (2002) 12) raised the important question of how to make a correct use of Bohm's approach for defining a quantum potential. In this work, by taking into account Kohout's results, we propose a general self-consistent iterative procedure for solving this problem.

Non-linear optical properties of quasi-one-dimensional periodic systems: consistent theoretical treatment within the crystal orbital method

The concepts of the interaction of a homogeneous electric field with a periodic quasi-one-dimensional system are discussed. The translational periodicity in the presence of the electric field is demostrated using a transformation of the Hamilton operator. The periodic perturbation Hamiltonian is introduced into the ab initio Hartree–Fock crystal orbital method and the generalized hermitian eigenvalue problem is solved with a self-consistent iteration procedure. Calculations of the static polarizability of an infinite water chain as a model and of polyyne are performed and compared with results reported in the literature.

Incorporating Protein Environments in Density Functional Theory: A Self-Consistent Reaction Field Calculation of Redox Potentials of [2Fe2S] Clusters in Ferredoxin and Phthalate Dioxygenase Reductase

An approach to calculating molecular electronic structures of active-site clusters in the presence of protein environments has been developed. The active-site cluster is treated by density functional theory. The protein field, together with the reaction field arising mainly from solvent, is obtained from a finite-difference solution to the Poisson−Boltzmann equation with three dielectric regions, and then these are coupled to the density functional calculation by a self-consistent iterative procedure. The method is applied to compute redox potentials of ferredoxin from Anabaena 7120 and phthalate dioxygenase reductase (PDR) from Pseudomonas cepacia, both having similar [Fe2S2(SR)4] active-site clusters. The calculated redox potentials, −1.007 V and −0.812 V in 0.05 M ionic strength for ferredoxin and PDR, respectively, deviate significantly from experimental values of −0.440 and −0.174 V. However, the calculated data reproduce the experimental trend fairly well. The calculated redox potential for PDR is 195 mV more positive than that for ferredoxin, comparing very well with the experimental value of 266 mV. The energy decomposition scheme reveals that the protein field plays a key role in differentiating the redox potentials of these two proteins.

Rotational and Translational Motions of Trapped Methane. Incoherent Inelastic Neutron Scattering of Methane Hydrate

The temperature dependence of the rotational and translational motion of methane encaged in a deuterated clathrate hydrate has been studied with incoherent inelastic neutron scattering. The methane molecules were found to behave almost like free rotors even at 6 K. The most interesting observation is that, unlike methane deposited in rare-gas matrices, the rotational modes do not converge to the classical rotation−diffusion limit at high temperature. The spectral features indicate that the methane interacts strongly with the host lattice vibrations and its motion is damped at high temperature. The methane vibrational density of states obtained from the experiment using a self-consistent iterative procedure to correct for multiexcitation and multiple scattering effect is found to be in good agreement with molecular dynamics calculations.

Statistical Mechanics of a Multipolar Gas: A Lattice Field Theory Approach

A recently proposed lattice field theory approach to the statistical mechanics of a classical Coulomb gas (J. Chem. Phys. 1992, 97, 5653) is generalized to treat particles with arbitrary electric multipole moments. Explicit development of the mean-field approximation is given for the case of mobile dipoles and monopoles (ions) surrounding an arbitrary collection of fixed charges embedded in macroions. In particular, a modified Poisson−Boltzmann (PB) equation is derived in which the mobile dipoles provide an effective spatially dependent dielectric constant. This equation implies a self-consistent iteration procedure by which the monopole and dipole densities are simultaneously determined in terms of single scalar (PB) field. Moreover, this equation can be derived from a minimum principle; an annealing strategy for computing the PB field is thereby suggested. In addition, explicit mean-field expressions for thermodynamic free energies are obtained as simple functionals of the PB field. Connection to the well-known Langevin dipole model is made. A numerical application to a system consisting of two parallel plates is presented in order to illustrate the utility of the formulation.

Correction of the drift mobility measurements in GaAs MESFETS

Abstract Drift mobility and electron concentration values in GaAs MESFETs obtained by capacitance-conductance measurements are corrected by means of a self-consistent iterative procedure. The final results are compared with the experimental Hall mobility data for the same wafer. It is found that the agreement is good for temperatures T > 175 K.