Regular Approximation Research Articles

Global weak solutions for the two-component Novikov equation

<p style='text-indent:20px;'>The two-component Novikov equation is an integrable generalization of the Novikov equation, which has the peaked solitons in the sense of distribution as the Novikov and Camassa-Holm equations. In this paper, we prove the existence of the <inline-formula><tex-math id="M1">$ H^1 $</tex-math></inline-formula>-weak solution for the two-component Novikov equation by the regular approximation method due to the existence of three conserved densities. The key elements in our approach are some a priori estimates on the approximation solutions.</p>

Regular approximation of linear Hamiltonian operators with two singular endpoints

In this paper we investigate regular approximation of linear Hamiltonian operators with two singular endpoints. For any given singular self-adjoint Hamiltonian operator, its spectrum can be approximated with the eigenvalues of a sequence of regular operators. The inherited restriction operators are constructed by the limit-circle solutions and the limit-point solutions, and proved to be self-adjoint. Via the abstract operator theory, we obtain the strongly resolvent convergence of the inherited regular operators and spectral inclusion of spectrum. Moreover for the case with both endpoints being limit circle, spectral exactness of spectrum is obtained by Hilbert-Schmidt norm convergence. In the case of Friedrichs extension of singular Hamiltonian systems, spectral exactness also holds due to variational principle of eigenvalues.

Formation of Al + (C 6 H 6 ) 13 : The Origin of Magic Number in Metal–Benzene Clusters Determined by the Nature of the Core

The identification of highly abundant, “magic” species in the mass spectra of clusters have proven to be valuable in nanoscience, leading to the discovery of new stable species such as fullerenes a...

Comparative Experimental and Theoretical Study of the C and O K-Edge X-ray Absorption Spectroscopy in Three Highly Popular, Low Spin Organoiron Complexes: [Fe(CO)5], [(η5-C5H5)Fe(CO)(μ-CO)]2, and [(η5-C5H5)2Fe

The unoccupied electronic structures of three closed-shell, highly popular organoiron complexes ([Fe(CO)5], [(η5-C5H5)Fe(CO)(μ-CO)]2, and [(η5-C5H5)2Fe]; 0, I, and II, respectively) have been investigated both experimentally and theoretically by combining original gas-phase X-ray absorption spectroscopy (XAS) outcomes recorded at the C and O K-edge with results of scalar relativistic time-dependent density functional calculations carried out within the zeroth order regular approximation. Experimental evidence herein discussed complement the Fe L2,3-edges XAS ones we recently recorded, modeled, and assigned for the same complexes (Carlotto et al. Inorg. Chem. 2019, 58, 5844). The first-principle simulation of the C and O K-edge features allowed us to univocally identify the electronic states associated to the ligand-to-metal charge transfer (LMCT) transitions both in I and in II. At variance to that, LMCT transitions with sizable oscillator strengths do not play any role in determining neither the C nor the O K-edge spectral pattern of 0. The higher π-acceptor capability of the CO ligand, regardless of its terminal or bridging coordination, with respect to [(η5-C5H5)]- is herein ultimately confirmed.

13C and 207Pb NMR Chemical Shifts of Dirhodio- and Dilithioplumbole Complexes: A Quantum Chemical Assessment.

Density functional theory (DFT) and zeroth-order regular approximation DFT calculations were performed to investigate the electronic structures and 13C and 207Pb nuclear magnetic resonance (NMR) chemical shifts of metal-coordinated plumboles, namely, monorhodioplumbole ([Rh-plumbole]-), dirhodioplumbole (Rh2-plumbole), and dilithioplumbole (Li2-plumbole), which have a five-membered ring containing lead. The molecular orbital correlation diagram and extended transition state-natural orbitals for chemical valence analysis of the [Rh-plumbole]- and Rh2-plumbole complexes showed that the plumbole is primarily a π-donor, with π-donation being dominant in the Rh2-plumbole complex. The present calculations show that the Pb-Cα internuclear distances are longer in the Rh2-plumbole complex than in [Rh-plumbole]- because of the combined effect of strong π-donation and weak π-back-donation in the Rh2-plumbole complex. The calculated 207Pb and 13Cα NMR chemical shifts agree with the experimental trends reasonably well. The influences of the relativistic effect, role of the functional, effect of the solvent, and dependence of the exact exchange admixture on the calculated 207Pb and 13Cα NMR chemical shifts were investigated. The NMR chemical shift trend of the 207Pb atom in the complexes originates from the paramagnetic and spin-orbit contributions. NMR component analysis revealed that the upfield shift of the 13Cα atoms of the [Rh-plumbole]- and Rh2-plumbole complexes compared to that of the Li2-plumbole complex is mainly due to the decrease in the paramagnetic term.

Solute drag and dynamic phase transformations in moving grain boundaries

A discrete model and the regular solution approximation are applied to describe the effect of grain boundary motion on grain boundary phase transformations in a binary alloy. The model predicts all thermodynamic properties of the grain boundary and the solute drag force, and permits a broad exploration of the parameter space and different dynamic regimes. The grain boundary phases continue to exist in the moving grain boundary and show a dynamic hysteresis loop, a dynamic critical line, and other features that are similar to those for equilibrium phases. Grain boundary motion strongly affects the relative stability of the phases and can even stabilize phases that are absolutely unstable under equilibrium conditions. Grain boundary phase transformations are accompanied by drastic changes in the boundary mobility. The results are analyzed in the context of non-equilibrium thermodynamics. Unresolved problems and future work are discussed.

Theoretical Search for the Highest Valence States of the Coinage Metals: Roentgenium Heptafluoride May Exist.

We present here a relativistic density functional theory investigation of the penta- and heptavalent states of gold and roentgenium, employing the ZORA (zeroth order regular approximation to the Dirac equation) Hamiltonian, including spin-orbit coupling at the two-component level, and large all-electron relativistic Slater-type quadruple-ζ quadruple polarization (ZORA-STO-QZ4P) basis sets. Unsurprisingly, our calculations confirm the stability of the experimentally known complexes AuF6- and Au2F10 with respect to decomposition to trivalent Au products and F2. The calculations also predict that RgF6- and Rg2F10 should be even more stable with respect to an analogous decomposition pathway. Like an earlier DFT study ( Inorg. Chem. 2007, 46 (13), 5338-5342), our calculations rule out the true heptavalent Au complex AuF7 as a stable species, preferring instead a Cs AuF5···F2 formulation. Remarkably, our calculations confirm a D5 h pentagonal-bipyramidal structure of RgF7 as the global minimum, at an energy of approximately half an electron volt below the RgF5···F2 form.

Stabilized dimensional factorization preconditioner for solving incompressible Navier-Stokes equations

In this paper, we propose a stabilized dimensional factorization (SDF) preconditioner for saddle point problems arising from the discretization of Navier-Stokes equations. The idea is based on regularization, block factorization and selective approximation. The spectral properties of the preconditioned matrix are analyzed in details. Based on the analysis, we prescribe a reasonable choice of the regularization matrix W in the preconditioner. By using the connection with the RDF preconditioner, we determine the relaxation parameter α for the problems discretized by uniform grids and stretched grids, respectively. Finally, numerical experiments on the finite element discretizations of both steady and unsteady incompressible flow problems show that the SDF preconditioner is more efficient and robust than the RDF preconditioner, which has been illustrated very competitive with some existing preconditioners.

Systematic study of relativistic and chemical enhancements of P,T -odd effects in polar diatomic radicals

Polar diatomic molecules that have, or are expected to have a $^2\Sigma_{1/2}$-ground state are studied systematically with respect to simultaneous violation of parity $\mathcal{P}$ and time-reversal $\mathcal{T}$ with numerical methods and analytical models. Enhancements of $\mathcal{P,T}$-violating effects due to an electric dipole moment of the electron (eEDM) and $\mathcal{P,T}$-odd scalar-pseudoscalar nucleon-electron current interactions are analyzed by comparing trends within columns and rows of the periodic table of the elements. For this purpose electronic structure parameters are calculated numerically within a quasi-relativistic zeroth order regular approximation (ZORA) approach in the framework of complex generalized Hartree-Fock (cGHF) or Kohn-Sham (cGKS). Scaling relations known from analytic relativistic atomic structure theory are compared to these numerical results. Based on this analysis, problems of commonly used relativistic enhancement factors are discussed. Furthermore the ratio between both $\mathcal{P,T}$-odd electronic structure parameters mentioned above is analyzed for various groups of the periodic table. From this analysis an analytic measure for the disentanglement of the two $\mathcal{P,T}$-odd electronic structure parameters with multiple experiments in dependence of electronic structure enhancement factors is derived.

Uncovering Heavy Actinide Covalency: Implications for Minor Actinide Partitioning.

Across the actinide period, the stability of the trivalent oxidation state predominates in the heavy actinides, making their chemical nature close to that of rare earth elements. The resemblance in their chemistry poses difficulties in separating heavy actinides from lanthanides, which is a vital separation in the minor actinide partitioning process. Actinide contraction has conventionally implied electrostatic actinide-ligand interactions among the heavy actinides. The present study challenges this conventional understanding and reveals increasing covalency in the actinide-ligand bond across Am to Cf. Complexes of Am, Cm, Bk, and Cf have been examined for their electronic structure with a focus on the nature of their interactions with different ligands within the framework of density functional theory, where the relativistic effects have been incorporated by using zero-order regular approximation and spin-orbit coupling. The choice of ligands selected for this study facilitates the effect of the donor atom as well as denticity to be accounted for. Hence, heavy actinide complexes of the N- and O-donor ligand dipicolinic acid, S and O mixed donor ligands of the Cyanex type, and an octadentate ligand N, N, N' N'-tetrakis[(6-carboxypyridin-2-yl)methyl]ethylenediamine have been optimized and evaluated. In each case energy decomposition analysis has been used to explicitly decompose the metal-ligand interaction energy into components which have then been analyzed. Irrespective of the hard-soft characteristics of donor atoms or the denticity of the ligands, steadily increased covalency has been observed across Am to Cf. Inspection of the ligand highest energy occupied molecular orbitals and metal orbitals sheds light on the origin of the unexpected covalency. An overall increase in bonding and also the orbital contribution along the Am-Cf series is clearly due to the enhancement in covalency, which is complementary to the orbital degeneracy induced covalency proposed very recently by Batista and co-workers.

A sparse regular approximation lemma

We introduce a new variant of Szemeredi's regularity which we call the sparse regular approximation lemma (SRAL). The input to this is a graph $G$ of edge density $p$ and parameters $\epsilon, \delta$, where we think of $\delta$ as a constant. The goal is to construct an $\epsilon$-regular partition of $G$ while having the freedom to add/remove up to $\delta |E(G)|$ edges. As we show here, this weaker variant of the regularity already suffices for proving the graph removal and the hypergraph regularity lemma, which are two of the main applications of the (standard) regularity lemma. This of course raises the following question: can one obtain quantitative bounds for SRAL that are significantly better than those associated with the regularity lemma? Our first result answers the above question affirmatively by proving an upper bound for SRAL given by a tower of height $O(\log 1/p)$. This allows us to reprove Fox's upper bound for the graph removal lemma. Our second result is a matching lower bound for SRAL showing that a tower of height $\Omega(\log 1/p)$ is unavoidable. We in fact prove a more general multicolored lower bound which is essential for proving lower bounds for the hypergraph regularity lemma.

Additive atomic approximation for relativistic effects: A two-component Hamiltonian for molecular electronic structure calculations.

An approximate relativistic two-component Hamiltonian for use in molecular electronic structure calculations is derived in the form of a sum of fixed atom-centered kinetic and spin-orbit operators added to the non-relativistic Hamiltonian. Starting from the well-known zeroth-order regular approximation, further steps are taken to get rid of its nonlinearity in the potential, ending up with a simple formulation with easily computable integrals that can seamlessly work with any traditional electronic structure method. Molecular tests show a good accuracy of this approximation.

English

It is known that the nonlinear nonhomogeneous backward Cauchy problem ut(t) + Au(t) = f (t, u(t)), 0 ⩽ t < τ with u(τ) = φ, where A is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on φ and f, that a solution of the above problem satisfies an integral equation involving the spectral representation of A, which is also ill-posed. Spectral truncation is used to obtain regularized approximations for the solution of the integral equation, and error analysis is carried out with exact and noisy final value φ. Also stability estimates are derived under appropriate parameter choice strategies. This work extends and generalizes many of the results available in the literature, including the work by Tuan (2010) for linear homogeneous final value problem and the work by Jana and Nair (2016b) for linear nonhomogeneous final value problem.

Wong–Zakai approximation for the stochastic Landau–Lifshitz–Gilbert equations

In this work we study stochastic Landau–Lifshitz–Gilbert equations (SLLGEs) in one dimension, with non-zero exchange energy only. Firstly, by introducing a suitable transformation, we convert the SLLGEs to a highly nonlinear time dependent partial differential equation with random coefficients, which is not fully parabolic. We then prove that there exists a pathwise unique solution to this equation and that this solution enjoys the maximal regularity property. Following regular approximation of the Brownian motion and using reverse transformation, we show existence of strong solution of SLLGEs taking values in a two-dimensional unit sphere S2 in R3. The construction of the solution and its corresponding convergence results are based on Wong–Zakai approximation.

Synthesis and Characterization of Tris-chelate Complexes for Understanding f-Orbital Bonding in Later Actinides.

An isostructural family of f-element compounds (Ce, Nd, Sm, Gd; Am, Bk, Cf) of the redox-active dioxophenoxazine ligand (DOPOq; DOPO = 2,4,6,8-tetra- tert-butyl-1-oxo-1 H-phenoxazin-9-olate) was prepared. This family, of the form M(DOPOq)3, represents the first nonaqueous isostructural series, including the later actinides berkelium and californium. The lanthanide derivatives were fully characterized using 1H NMR spectroscopy and SQUID magnetometry, while all species were structurally characterized by X-ray crystallography and electronic absorption spectroscopy. In order to probe the electronic structure of this new family, CASSCF calculations were performed and revealed these systems to be largely ionic in contrast to previous studies, where berkelium and californium typically have a small degree of covalent character. To validate the zeroth order regular approximation (ZORA) method, the same CASSCF analysis using experimental structures versus UDFT-ZORA optimized structures does not exhibit sizable changes in bonding patterns. This shows that UDFT-ZORA combined with CASSCF could be a useful first approximation to predict and investigate the structure and electronic properties of actinides and lanthanides that are difficult to synthesize or characterize.

Molecular QTAIM Topology Is Sensitive to Relativistic Corrections.

The topology of the molecular electron density of benzene dithiol gold cluster complex Au4 -S-C6 H4 -S'-Au'4 changed when relativistic corrections were made and the structure was close to a minimum of the Born-Oppenheimer energy surface. Specifically, new bond paths between hydrogen atoms on the benzene ring and gold atoms appeared, indicating that there is a favorable interaction between these atoms at the relativistic level. This is consistent with the observation that gold becomes a better electron acceptor when relativistic corrections are applied. In addition to relativistic effects, here, we establish the sensitivity of molecular topology to basis sets and convergence thresholds for geometry optimization.

Shear modulus of isotropic ferrogels

Abstract We present results of theoretical study of magnetorheological effect in ferrogels with magnetizable spherical particles chaotically distributed in a current gel. To avoid intuitive constructions with uncontrolled accuracy and adequacy, the analysis is done in the frames of the mathematically regular pair approximation. Our results demonstrate non monotonic increase of the composite shear modulus with the applied magnetic field. This effect is stronger for the systems with the soft gel, than for the relatively rigid ones.

On the role of polynomials in RBF-FD approximations: III. Behavior near domain boundaries

Radial basis function generated finite difference (RBF-FD) approximations generalize grid-based regular finite differences to scattered node sets. These become particularly effective when they are based on polyharmonic splines (PHS) augmented with multi-variate polynomials (PHS+poly). One key feature is that high orders of accuracy can be achieved without having to choose an optimal shape parameter and without having to deal with issues related to numerical ill-conditioning. The strengths of this approach were previously shown to be especially striking for approximations near domain boundaries, where the stencils become highly one-sided. Due to the polynomial Runge phenomenon, regular FD approximations of high accuracy will in such cases have very large weights well into the domain. The inclusion of PHS-type RBFs in the process of generating weights makes it possible to avoid this adverse effect. With that as motivation, this study aims at gaining a better understanding of the behavior of PHS+poly generated RBF-FD approximations near boundaries, illustrating it in 1-D, 2-D and 3-D.

Solute Drag and Dynamic Phase Transformations in Moving Grain Boundaries

A discrete model and the regular solution approximation are applied to describe the effect of grain boundary motion on grain boundary phase transformations in a binary alloy. The model predicts all thermodynamic properties of the grain boundary and the solute drag force, and permits a broad exploration of the parameter space and different dynamic regimes. The grain boundary phases continue to exist in the moving grain boundary and show a dynamic hysteresis loop, a dynamic critical line, and other features that are similar to those for equilibrium phases. Grain boundary motion strongly affects the relative stability of the phases and can even stabilize phases that are absolutely unstable under equilibrium conditions. Grain boundary phase transformations are accompanied by drastic changes in the boundary mobility. The results are analyzed in the context of non-equilibrium thermodynamics. Unresolved problems and future work are discussed.

Calculating transition dipole moments of phosphorescent emitters for efficient organic light-emitting diodes.

The out-coupling of light from an organic light-emitting diode, and thus its efficiency, strongly depends on the orientation of the transition dipole moment (TDM) of the emitting molecules with respect to the substrate surface. Despite the importance of this quantity, theoretical investigations of the direction of the TDM of phosphorescent emitters based on iridium(iii) complexes remain limited. One challenge is to find an appropriate level of theory able to accurately predict the direction of the TDM. Here, we report relativistic time-dependent density functional theory (TDDFT) calculations of the TDM, emission energies and lifetimes for both the ground-state (S0) and triplet (T1) excited-state geometries of fac-tris(2-phenylpyridyl)iridium(iii) (Ir(ppy)3), using the two-component zero-order regular approximation (ZORA) or including spin-orbit coupling (SOC) perturbatively using the simpler one-component (scalar) formulation. We show that the one- and two-component approaches give similar emission energies and overall radiative lifetimes for each individual geometry. Use of the S0 geometry leads to two of the excited triplet substates being degenerate, with the degeneracy lifted for the T1 geometry, with the latter matching experiment. Two-component calculations using the T1 geometry give results for the direction of the TDM more consistent with experiment than calculations using the S0 geometry. Finally, we show that adding a dielectric medium does not affect the direction of TDM significantly, but leads to better agreement with the experimentally measured radiative lifetime.