Zeroth-order Approximation Research Articles

Relativistic and quantum electrodynamics effects on NMR shielding tensors of TlX (X = H, F, Cl, Br, I, At) molecules.

The results of relativistic calculations of nuclear magnetic resonance shielding tensors (σ) for the thallium monocation (Tl+), thallium hydride (TlH), and thallium halides (TlF, TlCl, TlBr, TlI, and TlAt) are presented as obtained within a four-component polarization propagator formalism and a two-component linear response approach within the zeroth-order regular approximation. In addition to a detailed analysis of relativistic effects performed in this work, some quantum electrodynamical (QED) effects on those nuclear magnetic resonance shieldings and other small contributions are estimated. A strong dependence of σ(Tl) on the bonding partner is found, together with a very weak dependence of QED effects with them. In order to explain the trends observed, the excitation patterns associated with relativistic ee (or paramagnetic-like) and pp (or diamagnetic-like) contributions to σ are analyzed. For this purpose, the electronic spin-free and spin-dependent contributions are separated within the two-component zeroth-order regular approximation, and the influence of spin-orbit coupling on involved molecular orbitals is studied, which allows for a thorough understanding of the underlying mechanisms.

Exploration of Free Energy Surface of the Au10 Nanocluster at Finite Temperature.

The first step in comprehending the properties of Au10 clusters is understanding the lowest energy structure at low and high temperatures. Functional materials operate at finite temperatures; however, energy computations employing density functional theory (DFT) methodology are typically carried out at zero temperature, leaving many properties unexplored. This study explored the potential and free energy surface of the neutral Au10 nanocluster at a finite temperature, employing a genetic algorithm coupled with DFT and nanothermodynamics. Furthermore, we computed the thermal population and infrared Boltzmann spectrum at a finite temperature and compared it with the validated experimental data. Moreover, we performed the chemical bonding analysis using the quantum theory of atoms in molecules (QTAIM) approach and the adaptive natural density partitioning method (AdNDP) to shed light on the bonding of Au atoms in the low-energy structures. In the calculations, we take into consideration the relativistic effects through the zero-order regular approximation (ZORA), the dispersion through Grimme's dispersion with Becke-Johnson damping (D3BJ), and we employed nanothermodynamics to consider temperature contributions. Small Au clusters prefer the planar shape, and the transition from 2D to 3D could take place at atomic clusters consisting of ten atoms, which could be affected by temperature, relativistic effects, and dispersion. We analyzed the energetic ordering of structures calculated using DFT with ZORA and single-point energy calculation employing the DLPNO-CCSD(T) methodology. Our findings indicate that the planar lowest energy structure computed with DFT is not the lowest energy structure computed at the DLPN0-CCSD(T) level of theory. The computed thermal population indicates that the 2D elongated hexagon configuration strongly dominates at a temperature range of 50-800 K. Based on the thermal population, at a temperature of 100 K, the computed IR Boltzmann spectrum agrees with the experimental IR spectrum. The chemical bonding analysis on the lowest energy structure indicates that the cluster bond is due only to the electrons of the 6 s orbital, and the Au d orbitals do not participate in the bonding of this system.

Research on spherical harmonics method based on the MOC neutron transport code OpenMOC

In the Method of Characteristics (MOC) neutron transport code OpenMOC, the scattering source term is processed with an isotropic scattering approximation. However, in the actual process of neutron transport, the probability of neutron scattering in various directions is not uniform. By expanding the scattering cross-sections with Legendre polynomials and the angular flux terms with spherical harmonics, anisotropic scattering treatment under the flat source approximation is incorporated in the MOC code OpenMOC. This study selects VERA benchmark problems 1–3, which include different temperatures and burnable poison absorber rods, as well as several problems containing MOX fuel assemblies to validate the higher-order scattering functionality. Additionally, a sensitivity analysis is conducted about the impact of the number of particle histories on simulation time and keff calculation accuracy in OpenMOC. The results show that, compared to the zeroth-order approximation and the transport-corrected approximation, anisotropic scattering treatment achieves higher calculation accuracy in most cases.

Approximate analytical solution using power series method for the propagation of blast waves in a rotational axisymmetric non-ideal gas

In this paper, the propagation of the blast (shock) waves in non-ideal gas atmosphere in rotational medium is studied using a power series method in cylindrical geometry. The flow variables are assumed to be varying according to the power law in the undisturbed medium with distance from the axis of symmetry. To obtain the similarity solution, the initial density is considered as constant in the undisturbed medium. Approximate analytical solutions are obtained using Sakurai’s method by extending the power series of the flow variables in power of a0U2, where U and a0 are the shock velocity and speed of sound, respectively, in undisturbed fluid. The strong shock wave is considered for the ratio a0U2 which is considered to be a small quantity. With the aid of that method, the closed-form solutions for the zeroth-order approximation is given as well as first-order approximate solutions are discussed. Also, with the help of graphs behind the blast wave for the zeroth-order, a comparison between the numerical solution and the approximate analytical solution are shown. The distributions of flow variables such as density, radial velocity, pressure and azimuthal velocity are analysed. The results for the rotationally axisymmetric non-ideal gas environment are compared to those for the ideal gas atmosphere. The velocity-distance and time-distance curves are also shown to analyse the decaying characteristic of a blast wave.

On the geometry dependence of the nuclear magnetic resonance chemical shift of mercury in thiolate complexes: A relativistic density functional theory study.

Thiolate containing mercury(II) complexes of the general formula [Hg(SR) ] have been of great interest since the toxicity of mercury was recognized. 199Hg nuclear magnetic resonance spectroscopy (NMR) is a powerful tool for characterization of mercury complexes. In this work, the Hg shielding constants in a series of [Hg(SR) ] complexes are therefore investigated computationally with particular emphasis on their geometry dependence. Geometry optimizations and NMR chemical shift calculations are performed at the density functional theory (DFT) level with both the zeroth-order regular approximation (ZORA) and four-component relativistic methods. The four exchange-correlation (XC) functionals PBE0, PBE, B3LYP, and BLYP are used in combination with either Dyall's Gaussian-type (GTO) or Slater-type orbitals (STOs) basis sets. Comparing ZORA and four-component calculations, one observes that the calculated shielding constants for a given molecular geometry have a constant difference of 1070 ppm. This confirms that ZORA is an acceptable relativistic method to compute NMR chemical shifts. The combinations of four-component/PBE0/v3z and ZORA/PBE0/QZ4P are applied to explore the geometry dependence of the isotropic shielding. For a given coordination number, the distance between mercury and sulfur is the key factor affecting the shielding constant, while changes in bond and dihedral angles and even different side groups have relatively little impact.

Stability lobe and chatter prediction based on modified zeroth-order approximation in high-speed ball end milling process

Stability lobe and chatter prediction based on modified zeroth-order approximation in high-speed ball end milling process

Theoretical research on dynamic modeling of rigid–flexible coupling system with double joint folded wing

For meeting the requirements of tactical missiles seeking miniaturized launch devices for storage, transportation, and launch, a tube-launched missile wing is adopted, which folds before launch and quickly unfolds after launch. As a structure installed on the missile body to generate the required aerodynamic force for manipulating the missile, the tube-launched missile wing can effectively stabilize the missile’s flight attitude. At present, most research on the unfolding mechanism of missile folding wings is focused on one-time folding. When the wingspan is large, multiple folding is required to meet the launch requirements of modern tube-launched missiles. Therefore, this article designs a dual-joint folding wing deployment mechanism and studies the rigid–flexible coupling dynamic modeling and related technologies of folding wings based on this structure. Based on the inertial coupling between large-scale rigid body motion and structural flexible deformation, the folding wing breaks through the element convergence of the model and achieves the applicability of the structural model through zero-order approximation model analysis and other technologies. Simulation results show that the hybrid coordinate method can fully and accurately display the vibration information of flexible folding wings. At different speeds, the first-order coupling model is more advanced than the zero-order coupling model. In addition, increasing rotational speed, increasing wing thickness, and reducing wing span length can effectively increase the fundamental frequency of wing flutter. The structural design of folding wings has shown important reference significance.

Original and modified non-perturbative renormalization group equations of the BMW scheme at the arbitrary order of truncation

We consider the non-perturbative renormalization group (RG) equations, obtained as approximations of the exact Wetterich RG flow equation within the Blaizot–Mendez–Wschebor (BMW) truncation scheme. For the first time, we derive explicit RG flow equations for the scalar model at the arbitrary order of truncation. Moreover, we consider original, as well as modified, approximations, used to obtain a set of closed equations. We compare these equations at the s = 2 order of truncation with those recently derived in J. Phys. A: Math. Theor. 53, 415002 (2020) within a new truncation scheme and find a striking similarity. Namely, the first-order equations of the latter scheme, those of the original BMW scheme, and those of the modified BMW scheme (at s = 2) differ only in one term. We solved these equations by a recently proposed and tested method of semi-analytic approximations. Thus, the critical exponents η, ν, and ω were evaluated, recovering also the known results of the original BMW scheme. In addition, we estimated the subleading correction-to-scaling exponent ω2 for the three equations considered. To the best of our knowledge, this exponent has not yet been extracted from the Wetterich equation beyond the local potential (the zeroth order) approximation. Our current estimate for the 3D Ising model is ω2 = 2.02 (40), where the error bars include the expected truncation error in the BMW scheme.

Generating and grading 34 optimised norm-conserving Vanderbilt pseudopotentials for actinides and super-heavy elements in the PseudoDojo

In the last decades, material discovery has been a very active research field driven by the need to find new materials for many different applications. This has also included materials with heavy elements, beyond the stable isotopes of lead, as most actinides exhibit unique properties that make them useful in various applications. Furthermore, new heavy elements beyond actinides, collectively referred to as super-heavy elements (SHEs), have been synthesized, filling previously empty space of Mendeleev periodic table. Their chemical bonding behaviour, of academic interest at present, would also benefit of state-of-the-art modelling approaches.In particular, in order to perform first-principles calculations with planewave basis sets, one needs corresponding pseudopotentials. In this work, we present a series of scalar- and fully-relativistic optimised norm-conserving Vanderbilt pseudopotentials (ONCVPs) for thirty-four actinides and super-heavy elements, for three different exchange-correlation functionals (PBE, PBEsol and LDA). The scalar-relativistic version of these ONCVPs is tested by comparing equations of states for crystals, obtained with abinit 9.6, with those obtained by all-electron zeroth-order regular approximation (ZORA) calculations, without spin-orbit coupling, performed with the Amsterdam Modelling Suite band code. Δ-Gauge and Δ1-Gauge indicators are used to validate these pseudopotentials. This work is a contribution to the PseudoDojo project, in which pseudopotentials for the whole periodic table are developed and systematically tested. The pseudopotential files are available on the PseudoDojo web-interface pseudo-dojo.org in psp8 and UPF2 formats, both suitable for abinit, the latter being also suitable for Quantum ESPRESSO.

A hierarchy of asymptotic models for a fluid-loaded elastic layer

A hierarchy of asymptotic models governing long-wave low-frequency in-plane motion of a fluid-loaded elastic layer is established. In contrast to a layer with traction-free faces, modelled by Neumann boundary conditions, a fluid-loaded one assumes more involved conditions along the interfaces, dictating a special asymptotic scaling. The latter corresponds to a fluid-borne bending wave, controlled by elastic stiffness of the layer and fluid inertia. In this case, the transverse inertia of the layer and fluid compressibility do not appear at zero-order approximation. The first-order approximation is associated with a Kirchhoff plate, immersed into incompressible fluid. In the studied free vibration setup, the fluid compressibility has to be taken into account only at third order, along with elastic rotary inertia. Transverse shear deformation enters the second-order approximation along with a few other corrections. The conventional impenetrability condition has to be also refined at second order. Dispersion relations corresponding to the developed asymptotic models are compared with the polynomial expansions of the full dispersion relation, obtained from the plane-strain problem of linear elasticity.

Coordination of Mercury(II) in Water Promoted over Hydrolysis in Solvated Clusters [Hg(H2O)1-6](aq)2+: Insights from Relativistic Effects and Free Energy Analysis.

Understanding the nature of the interaction between mercury(II) ions, Hg2+, and water molecules is crucial to describe the stability and chemical behavior of structures formed during solvation, as well as the conditions that favor the Hg2+ coordination or inducing water hydrolysis. In our study, we explored exhaustively the potential energy surface of Hg2+ with up to six water molecules. We analyzed electronic and Gibbs free energies, binding, and nuclear magnetic resonance parameters. We used the zeroth-order regular approximation Hamiltonian, including scalar and spin-orbit relativistic corrections for free energy calculations and geometry optimizations to explore the interplay between electron correlation and relativistic effects. We analyzed intermolecular interactions with energy decomposition analysis, quantum theory of atoms in molecules, and natural bond orbital. Additionally, we used the four-component Dirac Hamiltonian to compute solvent effect on the magnetic shielding and J-coupling constants. Our results revealed that the water hydrolysis by Hg2+ requires a minimum of three water molecules. We found that the interaction between Hg2+ and water molecules is an orbital interaction due to relativistic effects and the most stable structures are opened-shape clusters, reducing the number of oxygen-mercury contacts and maximizing the formation of hydrogen bonds among water molecules. In these types of clusters, Hg2+ promotes the water hydrolysis over coordination with oxygen atoms. However, when we considered the change associated with the transfer of a cluster from the ideal gas to a solvated system, our solvation free energy analysis revealed that closed-shape clusters are more favorable, maximizing the number of oxygen-mercury contacts and reducing the formation of hydrogen bonds among water molecules. This finding suggests that, under room conditions, the coordination of Hg2+ is more favorable than hydrolysis. Our results have significant implications for understanding Hg2+ behavior in water, helping to develop targeted strategies for mercury remediation and management, and contributing to advancements in the broader field of environmental chemistry.

Low-Profile Multibeam Antenna Based on Modulated Metasurface

Linearly polarized (LP) multi-port multi-beam antennas (MBAs) based on modulated metasurface (MTS) are presented in this paper. A key challenge for MTS to generate multiple independent beams is the mutual interference of different impedance modulations. This interference can lead to reduced directivities of main beams and high side lobe levels (SLLs). A full-wave based optimization for a large number of beams is significantly time consuming and practically ineffective. The source locations are optimized here by employing the aperture field calculated from a zeroth-order approximation (ZOA) of the currents on the surface. This provides a good preliminary accuracy without any full-wave analysis. The relevant design method is verified by two examples of 3-beam and 7-beam MTS antennas. Furthermore, the 7-beam MTS antenna is fabricated and successfully measured. The measurements show a measured realized gain of 20 dBi at 14.10 GHz with a beam coverage (10-dB overlap) up to ±36°. It is seen that the same performance requires larger areas with conventional summation of apertures.

Asymptotic approximations for semilinear parabolic convection-dominated transport problems in thin graph-like networks

We consider time-dependent convection-diffusion problems with high Péclet number of order O(ε−1) in thin three-dimensional graph-like networks consisting of cylinders that are interconnected by small domains (nodes) with diameters of order O(ε). On the lateral surfaces of the thin cylinders and the boundaries of the nodes we account for solution-dependent inhomogeneous Robin boundary conditions which render the associated initial-boundary problem to be nonlinear. The strength of the inhomogeneity is controlled by an intensity factor of order O(εα), α∈R. The asymptotic behavior of the solution is studied as ε→0, i.e., when the diffusion coefficients are eliminated and the thin three-dimensional network is shrunk into a graph. Three qualitatively different cases are discovered in the asymptotic behavior of the solution depending on the value of the intensity parameter α: α=1, α>1, and α<1. We construct the asymptotic approximation of the solution, which provides us with the hyperbolic limit model for ε→0 for the first two cases, and prove the corresponding uniform pointwise estimates and energy estimates. As the main result, we derive uniform pointwise estimates for the difference between the solutions of the convection-diffusion problem and the zero-order approximation that includes the solution of the corresponding hyperbolic limit problem.

Numerical simulation of 3D fracture propagation problem with reproducing kernel peridynamic method

The non-ordinary state-based peridynamic (NOSB-PD) method has significant advantages in solving crack propagation problems due to its non-local characteristics, but it also faces problems such as numerical oscillations and boundary effects. In order to solve the above defects, the paper first discusses the NOSB-PD method within the Galerkin framework. After that, the peridynamic differential operator (PDDO) approximation was introduced by extending the calculation of the deformation gradient tensor F in NOSB-PD. Since the zero-order approximations in the PDDO approximation and the reproducing kernel (RK) approximation are identical, the distinctions between the two approximations have been thoroughly analysed. Although the two approximations show the same consistency, only the RK approximation satisfies the integrable condition. Therefore, a RK-PD coupling method is proposed in the paper, and an implicit iterative algorithm suited to the RK-PD method is presented as well. At last, several numerical examples demonstrate that the proposed method does not suffer from numerical oscillations and can be used to solve the three-dimensional crack propagation problems.

A new boson expansion theory utilizing a norm operator

Abstract We propose a new boson expansion method using a norm operator. The small parameter expansion, in which the boson approximation becomes the zeroth-order approximation, requires the double commutation relations between phonon operators that are not closed between the phonon excitation modes adopted as boson excitations. This results in an infinite expansion regardless of whether the type of the boson expansion is Hermitian or non-Hermitian. The small parameter expansion does not hold when the commutation relations are closed. The norm operator is expressed as a function of the number operator in the physical subspace, which enables us to obtain substantially a finite boson expansion regardless of the Hermitian or non-Hermitian type. We also point out the problems of the conventional boson expansion methods. The normal-ordered linked-cluster expansion theory has failed to refute Marshalek’s claim that KT-1 and KT-2 are of chimerical boson expansion. The Dyson boson expansion theory does not have exceptional superiority over other types. Previous studies using the boson expansion methods should be re-examined.

Robust Homography Estimation from Local Affine Maps

The corresponding point coordinates determined by classic image matching approaches define local zero-order approximations of the global mapping between two images. But the patches around keypoints typically contain more information, which may be exploited to obtain a first-order approximation of the mapping, incorporating local affine maps between corresponding keypoints. Several methods have been proposed in the literature to compute this first-order approximation. In this paper we present several modifications of the RANSAC (RANdom SAmple Consensus) algorithm, that uses affine approximations and a-contrario procedures to improve the homography estimation between a pair of images. The a-contrario methodology provides a definition of the soundness of an estimation and allows for adaptive thresholds for inlier/outlier discrimination. These approaches outperform the state-of-the-art for different choices of image descriptors and image datasets, and permit to increase the probability of success in identifying image pairs in challenging matching databases.

Electron correlation effects on the g factor of lithiumlike ions

We present the systematic QED treatment of the electron correlation effects on the $g$ factor of lithiumlike ions for the wide range of nuclear charge number $Z=14$--82. The one- and two-photon exchange corrections are evaluated rigorously within the QED formalism. The electron-correlation contributions of the third and higher orders are accounted for within the Breit approximation employing the recursive perturbation theory. The calculations are performed in the framework of the extended Furry picture, i.e., with inclusion of the effective local screening potential in the zeroth-order approximation. In comparison to the previous theoretical calculations, the accuracy of the interelectronic-interaction contributions to the bound electron $g$ factor in lithiumlike ions is substantially improved.

Perturbation Theory Treatment of Spin–Orbit Coupling. III: Coupled Perturbed Method for Solids

A previously proposed noncanonical coupled-perturbed Kohn-Sham density functional theory (KS-DFT)/Hartree-Fock (HF) treatment for spin-orbit coupling is here generalized to infinite periodic systems. The scalar-relativistic periodic KS-DFT/HF solution, obtained with a relativistic effective core potential, is taken as the zeroth-order approximation. Explicit expressions are given for the total energy through third-order, which satisfy the 2N + 1 rule (i.e., requiring only the first-order perturbed wave function for determining the energy through third-order). Expressions for additional second-order corrections to the perturbed wave function (as well as related one-electron properties) are worked out at the uncoupled-perturbed level of theory. The approach is implemented in the Crystal program and validated with calculations of the total energy, electronic band structure, and density variables of spin-current DFT on the tungsten dichalcogenide hexagonal bilayer series (i.e., WSe2, WTe2, WPo2, WLv2), including 6p and 7p elements as a stress test. The computed properties through second- or third-order match well with those from reference two-component self-consistent field (2c-SCF) calculations. For total energies, E(3) was found to consistently improve the agreement against the 2c-SCF reference values. For electronic band structures, visible differences w.r.t. 2c-SCF remained through second-order in only the single-most difficult case of WLv2. As for density variables of spin-current DFT, the perturbed electron density, being vanishing in first-order, is the most challenging for the perturbation theory approach. The visible differences in the electron densities are, however, largest close to the core region of atoms and smaller in the valence region. Perturbed spin-current densities, on the other hand, are well reproduced in all tested cases.

Semianalytic stability algorithm in the frequency domain for interrupted milling

The construction of stability diagrams of interrupted milling cases is generally carried out by means of time-consuming numerical methods in either frequency or time domain, since the period doubling lobes arising under interrupted cutting are omitted by the time-averaged parametric scan of the traditional zeroth order approximation. This paper presents a novel seminalytic method in the frequency domain for interrupted milling. Taking advantage of their analytical chatter frequency distribution, this method adds the period doubling lobes calculated in a single frequency scan to the existing Hopf limits of the zeroth order solution. This, together with intelligent selection of scanning frequency ranges and truncation to the minimum set of harmonics, allows very fast calculation of the stability charts of interrupted milling cases while retaining the analytical basis and advantages of the zeroth order algorithm. The method accurately describes the stability limits at period doubling dominated zones and improves the existing zeroth order solution, but exhibits slight inaccuracies in the prediction of Hopf boundaries due to mode interaction and harmonic truncation effects.

All-electron ZORA triple zeta basis sets for the elements Cs–La and Hf–Rn

Segmented all-electron basis set of triple zeta valence quality plus polarization functions (TZP) for the elements of the fifth row to be used together with the zero-order regular approximation (ZORA) is carefully constructed. To correctly describe electrons distant from atomic nuclei, the basis set is augmented with diffuse functions giving rise to a set designated as ATZP-ZORA. At the ZORA-B3LYP theoretical level, these sets are used to calculate the ionization energy and mean dipole polarizability of some atoms, bond length, dissociation energy, and harmonic vibrational frequency of diatomic molecules. Then, these results are compared with the theoretical and experimental data found in the literature. Even considering that our sets are relatively compact, they are sufficiently accurate and reliable to perform property calculations involving simultaneously electrons from the inner shell and outer shell. The performances of the ZORA and second-order Douglas–Kroll–Hess Hamiltonians are evaluated and the results are also discussed.