High-order Perturbation Research Articles

Theoretical studies on the EPR parameters and local structure of Cu2+ center in [Co(nicotinamide)2(H2O)4](saccharinate)2 crystal

Local structure and electron paramagnetic resonance (EPR) parameters for Cu2+ center in [Co(nicotinamide)2(H2O)4](saccharinate)2 (CoNAS) crystal are theoretically investigated using high-order perturbation formulas for 3d9 ions in rhombically elongated octahedra. In the calculated formulas, the related molecular orbital coefficients are acquired from the superposition model which enables to correlate the crystal field parameters and hence the EPR parameters with the studied Cu2+ center, the admixtures of d-orbitals in the ground state wave function as well as the ligand orbital and spin–orbit coupling contributions are taken into account. Based on the studies, the [CuO4N2]12− cluster of the studied Cu2+ center are found to suffer the axial and perpendicular bond length variations Δz (≈ 0.0612 Å) along z-axis and δr (≈ 0.0360 Å) along x- and y-axes, respectively, due to the Jahn–Teller effect.

Exact Top-Quark Mass Dependence in Hadronic Higgs Production.

The impact of the finite top-quark mass on the inclusive Higgs production cross section at higher perturbative orders has been an open question for almost three decades. In this Letter, we report on the computation of this effect at next-to-next-to-leading order QCD. For the purely gluonic channel, it amounts to +0.62% relative to the result obtained in the Higgs effective field theory approximation. The formally subleading partonic channels overcompensate this shift, leading to an overall effect of -0.26% at a pp collider energy of 13TeV, and -0.1% at 8TeV. This result eliminates one of the main theoretical uncertainties to inclusive Higgs production cross section at the LHC.

A reduced modal subspace approach for damped stochastic dynamic systems

A novel method for characterising and propagating system uncertainty in structures subjected to dynamic actions is proposed, whereby modal shapes, frequencies and damping ratios constitute the random quantities. The latter, defined in the modal subspace rather than the full geometrical space, reduce the number of the random variables and the size of the dynamic problem. A numerical procedure is presented for their identification by calibrating their probabilistic definition in line with the geometrical space. A high-order perturbation technique is proposed for the multi-fidelity response quantification by means of an ad hoc extension of the conventional perturbation method. The approach involves a set of auxiliary deterministic differential equations to be adaptively solved with the piecewise exact method, and moment-cumulant relationships are employed to approximate high-order moments. Finally, a polynomial chaos expansion approach is adopted to complement the second-moment analysis for spectral quantification with the modal subspace reduction. Demonstrated on a multi-storey steel frame with semi-rigid connections and a simply supported bridge subjected to a moving load, the proposed variants exhibit improved performance with respect to the conventional second-order and improved perturbation, as well as increased flexibility, enabling the analyst to decide, on-demand, the level of fidelity, balancing accuracy and computational effort.

A superconvergent B-spline technique for second order nonlinear boundary value problems

In the present work, a high-order numerical scheme based on B-spline functions is developed for solving a class of nonlinear derivative dependent singular boundary value problems (DDSBVP). To derive the method, we first generate a high order perturbation of the original problem by using spline alternate relations. Then, we determine the approximate solution by forcing it to satisfy the resulting perturbed problem at the grid points of the spline. Convergence analysis of the method is established through matrix approach. Four nonlinear examples are considered to demonstrate the accuracy and robustness of the method. The proposed method provides O(h6) superconvergent approximation to the solution of the problem under consideration, where h is the step size. This method produces significantly more accurate results than the two newly developed numerical schemes using the same B-spline functions as used in the present method, namely UCS method and NCS method. Moreover, the computational time of present method is compared with that of NCS method.

The High-Order Perturbation Approximate Solution of the Finite Ultrasonic Wave

To the nonlinear acoustic wave equation, the general used second harmonic solution is not accurate enough because all perturbated expansion equations higher than the second order are ignored during the equation solving process. The purpose of this paper is to obtain a more accurate solution, i.e., the high-order perturbation approximate solution. Firstly, the nonlinear acoustic wave equation is expanded into many inhomogeneous partial differential equations. the low-order harmonic solutions are obtained manually, then we formulate the form of the high-order harmonic solutions according to the properties of the low-order harmonic solutions. Using symbol calculation tool, we finally obtained higher up to the 14th order perturbation special solutions. Odd order solutions contain only odd order harmonics, and even order solutions contain only even order harmonics. The high-order perturbation solution of the second harmonic is finally achieved by summing up all of the second harmonic solution parts. The simulation results show that the relative amplitude (A 2/A 1) of the second harmonic increases and then decreases with the propagation distance, which is in agreement with experimental results. The high-order perturbation approximated solution can compensate for the theory deficiency and can be used to measure the nonlinear parameter with a good precision.

Fine structure of energy levels and electron paramagnetic resonance parameters of Ni2+ doped in CsCdCl3

Complete diagonalization method (CDM) and high-order perturbation method (PTM) with forth-order formulae have been used to explain the optical spectra and ground term 3A2g electron paramagnetic resonance (EPR) parameters g//,g⊥ and zero-field splitting (ZFS) parameter D of Ni2+-doped CsCdCl3 within the framework of molecular orbital (MO) scheme, the calculated results are in good agreement with the experimental data. The trigonal parametersv and v′, Reach parameters B and C and the local defect structure are determined. The effect of vibronic interaction on optical spectra of 3T2g and 3A2g transition has been discussed with a1g and eg vibronic models, and the Huang–Phys parameters and vibronic coupling constant are evaluated.

New slant range model and azimuth perturbation resampling based high-squint maneuvering platform SAR imaging

Strong spatial variance of the imaging parameters and serious geometric distortion of the image are induced by the acceleration and vertical velocity in a high-squint synthetic aperture radar (SAR) mounted on maneuvering platforms. In this paper, a frequency-domain imaging algorithm is proposed based on a novel slant range model and azimuth perturbation resampling. First, a novel slant range model is presented for mitigating the geometric distortion according to the equal squint angle curve on the ground surface. Second, the correction of azimuth-dependent range cell migration (RCM) is achieved by introducing a high-order time-domain perturbation function. Third, an azimuth perturbation resampling method is proposed for azimuth compression. The azimuth resampling and the time-domain perturbation are used for correcting first-order and high-order azimuthal spatial-variant components, respectively. Experimental results illustrate that the proposed algorithm can improve the focusing quality and the geometric distortion correction accuracy of the imaging scene effectively.

Perturbation theory made efficient and effective for predictions of ionization potential and electron affinity.

Ionization potential and electron affinity are essential molecular properties. The most straightforward method is to calculate them by taking the total energy differences of the initial and final states according to the definition. However, it often suffers from a serious convergence problem due to the requirement of the self-consistent field (SCF) calculations for the ionic states with non-Aufbau choices of occupations. In the present work, we have constructed a theoretical framework in view of perturbation theory to bypass the SCF calculations of the ionic states. To address the imbalance issue that arises from the precisely treated neutral ground state followed by the truncated perturbative treatment of the ionic states, an accurate yet effective method has been developed here, which adds back some terms from the higher order perturbations into the lower order to cancel out the most computationally cost terms in the truncated expansion, thus reaching a better convergence with less computation. The validity of the present methodology has been tested out by applying it to the Hartree-Fock (HF) method in combination with the correlation effect described at the second-order Møller-Plesset level in a frozen-orbital approximation. All the derivations in this work are given in a general framework, which are applicable not only to HF but also to a wide range of density functional theory methods from semi-local functionals to hybrid and doubly hybrid functionals.

Approximate Atomic Green Functions.

In atomic and many-particle physics, Green functions often occur as propagators to formally represent the (integration over the) complete spectrum of the underlying Hamiltonian. However, while these functions are very crucial to describing many second- and higher-order perturbation processes, they have hardly been considered and classified for complex atoms. Here, we show how relativistic (many-electron) Green functions can be approximated and systematically improved for few- and many-electron atoms and ions. The representation of these functions is based on classes of virtual excitations, or so-called excitation schemes, with regard to given bound-state reference configurations, and by applying a multi-configuration Dirac-Hartree-Fock expansion of all atomic states involved. A first implementation of these approximate Green functions has been realized in the framework of Jac, the Jena Atomic Calculator, and will facilitate the study of various multi-photon and/or multiple electron (emission) processes.

Dynamical behavior of a stochastic multigroup staged-progression HIV model with saturated incidence rate and higher-order perturbations

In this paper, we analyze a higher-order stochastically perturbed multigroup staged-progression model for the transmission of HIV with saturated incidence rate. We obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a suitable stochastic Lyapunov function. In addition, we make up adequate conditions for complete eradication and wiping out the infectious disease. In a biological interpretation, the existence of a stationary distribution implies that the disease will prevail and persist in the long term. Finally, examples and numerical simulations are introduced to validate our theoretical results.

Dynamics of a stochastic SICA epidemic model for HIV transmission with higher-order perturbation

In this paper, we propose and study a higher order stochastically perturbed SICA epidemic model for HIV transmission. We obtain sufficient conditions for the existence and uniqueness of an ergodic stationary distribution of positive solutions to the system by establishing a series of suitable Lyapunov functions. In a biological point of view, the existence of a stationary distribution indicates that the infectious disease will be prevalent and persistent in the population. Moreover, we make up adequate conditions for complete eradication and wiping out of the infectious disease.

The cosmic neutrino background as a collection of fluids in large-scale structure simulations

A significant challenge for modelling the massive neutrino as a hot dark matter is its large velocity dispersion. In this work, we investigate and implement a multi-fluid perturbation theory that treats the cosmic neutrino population as a collection of fluids with a broad range of bulk velocities. These fluids respond linearly to the clustering of cold matter, which may be linear and described by standard linear perturbation theory, or non-linear, described using either higher-order perturbation theory or N-body simulations. We verify that such an alternative treatment of neutrino perturbations agrees closely with state-of-the-art neutrino linear response calculations in terms of power spectrum and bispectrum predictions. Combining multi-fluid neutrino linear response with a non-linear calculation for the cold matter clustering, we find for a reference νΛCDM cosmology with neutrino mass sum ∑ m ν = 0.93 eV an enhancement of the small-scale neutrino power by an order of magnitude relative to a purely linear calculation. The corresponding clustering enhancement in the cold matter, however, is a modest ∼ 0.05%. Importantly, our multi-fluid approach uniquely enables us to identify that the slowest-moving 25% of the neutrino population clusters strongly enough to warrant a non-linear treatment. Such a precise calculation of neutrino clustering on small scales accompanied by fine-grained velocity information would be invaluable for experiments such as PTOLEMY that probe the local neutrino density and velocity in the solar neighbourhood.

Domain walls without a potential

We show that domain walls, or kinks, can be constructed in simple scalar theories where the scalar has no potential. These theories belong to a class of k-essence where the Lagrangian vanishes identically when one lets the derivatives of the scalar vanish. The domain walls we construct have positive energy and stable quadratic perturbations. As particular cases, we find families of theories with domain walls and their quadratic perturbations identical to the ones of the canonical Mexican hat or sine-Gordon scalar theories. We show that canonical and non canonical cases are nevertheless distinguishable via higher order perturbations or a careful examination of the energies. In particular, in contrast to the usual case, our walls are local minima of the energy among the field configuration having some fixed topological charge, but not global minima.

Studies of local structures for Cu2+ centers in M2Zn(SO4)2·6H2O (M = NH4 and Rb) crystals

Abstract The electron paramagnetic resonance (EPR) parameters (g factor g i , and hyperfine structure constants A i , with i = x, y, z) and local structures for Cu2+ centers in M2Zn(SO4)2·6H2O (M = NH4 and Rb) are theoretically investigated using the high order perturbation formulas of these EPR parameters for a 3d 9 ion under orthorhombically elongated octahedra. In the calculations, contribution to these EPR parameters due to the admixture of d-orbitals in the ground state wave function of the Cu2+ ion are taken into account based on the cluster approach, and the required crystal-field parameters are estimated from the superposition model which enables correlation of the crystal-field parameters and hence the studied EPR parameters with the local structures of the Cu2+ centers. Based on the calculations, the Cu–H2O bonds are found to suffer the axial elongation ratio δ of about 3 and 2.9% along the z-axis, meanwhile, the planar bond lengths may experience variation ratio τ (≈3.8 and 1%) along x- and y-axis for Cu2+ center in (NH4)2Zn(SO4)2·6H2O and Rb2Zn(SO4)2·6H2O, respectively. The theoretical results show good agreement with the observed values.

First-Order Error-Adapted Eigen Perturbation for Real-Time Modal Identification of Vibrating Structures

Abstract A new computationally efficient error adaptive first-order eigen-perturbation technique for real-time modal identification of linear vibrating systems is proposed. The existence of error terms in the approximation of the eigenvalue problem of response covariance matrix in a perturbative framework often hinders the convergence of response-only modal identification. In the proposed method, the error in first-order eigen-perturbation is incorporated using a feedback, formulated by exploiting the generalized eigenvalue decomposition of the real-time covariance matrix of streaming response data. Since the incorporation of the higher-order perturbation terms in the total perturbation is mathematically challenging, the proposed feedback approach provides a computationally efficient framework yet in a more elegant manner. A new criterion for the quality of updated eigenspace is proposed in the present work utilizing the concept of diagonal dominance. Numerical case studies and validation using a standard ASCE benchmark problem have shown applicability of the proposed approach in faster estimation of real-time modal properties and anomaly identification with minimal number of initially required batch data. The applicability of the proposed approach toward real-time under-determined modal identification problems is demonstrated using a real-time decentralized framework. The advantage of rapidly converging online mode-shapes is demonstrated using a passive vibration control problem, where a multi-tuned-mass-damper (MTMD) for a multi-degrees-of-freedom system is tuned online. An extension for online retuning of the detuned MTMD system further demonstrates the fidelity of the proposed algorithm in online passive control.

Instability of Dust–Acoustic Waves in Plasmas with Two-Temperature Nonthermal Ions

Theoretical and numerical investigations are carried out for the instability of the dust–acoustic waves (DAWs) under the transverse perturbation in a magnetized dusty plasma with two-temperature nonthermal ions. By the reductive perturbation technique, the Zakharov–Kuznetsov (ZK) equation and modified ZK equation of the DAWs are derived. Under the higher-order transverse perturbations, the instability growth rate of the soliton solutions of the ZK equations is studied. It is found that the most unstable wave mode exists in system. It is also noted that solitary waves are unstable when the ratio of the perturbation wave length to solitary wave width in the experimental coordinate is larger than a certain critical value.

On the Method of Correct Determination of Eigenvalues of a Truncated Hamiltonian Matrix on the Example of a Morse Oscillator

A method of precise determination of the eigenvalues of energies of the molecular Hamiltonian based on high-order perturbation theory has been developed and implemented for a diatomic molecule. The proposed method makes it possible not only to obtain the energy values, but also to estimate the accuracy of their prediction and the applicability limits for the employed model. Numerical calculations have been performed for the extended Morse oscillator with corrections for the potential functions up to the sixth power of the Morse coordinate. The results obtained are compared with the results of calculation for the model of the truncated Hamiltonian matrix. The possibilities of application of the method are analyzed compared to other approaches to the determination of the potential functions for polyatomic molecules.

Adjoint-Based Calculation of Parametric Thermoacoustic Maps of an Industrial Combustion Chamber

Abstract The aim of this study is to efficiently calculate parametric thermoacoustic maps of typical combustion chambers. Two configurations are considered: an academic configuration based on a Rijke tube, and an industrial combustion chamber, which is the core of a recently developed microturbine for power generation. Such maps can be understood as the collection of loci of thermoacoustic eigenfrequencies obtained under systematic variations of some defined parameters, while considering the Helmholtz equation as the thermoacoustic model of interest. In this study we consider variations on two parameters: the gain n and time-delay τ associated with a generic flame response model. We also show the feasibility of the proposed approach when considering more realistic flame responses. A straight-forward way to calculate such a thermoacoustic map is by solving the Helmholtz equation, and, thus, the corresponding nonlinear eigenvalue problem (NLEVP), one time per parameter combination. With that approach, the nonlinear eigenvalue problem needs to be solved hundreds or thousands of times if an adequate resolution of the thermoacoustic map is sought. Such a strategy may be computationally unaffordable. In order to overcome this difficulty, this work utilizes an adjoint-based, high-order perturbation method. The actual eigenvalue problem is only solved once at a baseline point. After applying the perturbation equations at that point, a polynomial rational function—the Padé approximant—is obtained to estimate the eigenfrequency drift that results for a small or large perturbation in the flame response. It is demonstrated, for both academic and industrial test cases, that the obtained maps are accurate. Additionally, it is shown that these maps reveal a large variety of thermoacoustic features, such as stability boundaries, intrinsic thermoacoustic modes, and exceptional points. The numerical costs for such calculations are negligible even for the industrial combustion chamber investigated.

An investigation of a series of layered Zn-1,4-diazabicyclo[2.2.2]octane metal–organic frameworks: local structure and adsorption properties of mixed-metal centre

The mixed-metal pillar-layered metal–organic frameworks of Zn(bdc)(DABCO)0.5, Zn0.5Cu0.5(bdc)(DABCO)0.5 and Cu(bdc)(DABCO)0.5 (bdc = 1,4-benzenedicarboxylate, DABCO = 1,4-diazabicyclo[2.2.2]octane) are investigated for their local structures and gas adsorption properties. According to the obtained electron paramagnetic resonance (EPR) spectra, the distorted structures around Cu2+ are proposed to be tetragonally and orthorhombically elongated [CuO4N] with the well fitted high-order perturbation formulae of the EPR parameters. Due to the doped Cu2+, the adsorption isotherms of industrially relevant gases (CO2, CO, CH4 and N2) and lower alkanes (CH4, C2H6, C3H8, C4H10 and C5H12) are different, especially at different temperatures. By combining the structural properties and adsorption isotherms, a comprehensive study suggests that the ZnDABCO series can be a controllable tool in gas storage and separation.

Running Coupling Constant from Position-Space Current-Current Correlation Functions in Three-Flavor Lattice QCD.

In this Letter, we provide a determination of the coupling constant in three-flavor quantum chromodynamics (QCD), α_{s}^{MS[over ¯]}(μ), for MS[over ¯] renormalization scales μ∈(1,2) GeV. The computation uses gauge field configuration ensembles with O(a)-improved Wilson-clover fermions generated by the Coordinated Lattice Simulations (CLS) consortium. Our approach is based on current-current correlation functions and has never been applied before in this context. We convert the results perturbatively to the QCD Λ parameter and obtain Λ_{MS[over ¯]}^{N_{f}=3}=342±17 MeV, which agrees with the world average published by the Particle Data Group and has competing precision. The latter was made possible by a unique combination of state-of-the-art CLS ensembles with very fine lattice spacings, further reduction of discretization effects from a dedicated numerical stochastic perturbation theory simulation, combining data from vector and axial-vector channels, and matching to high-order perturbation theory.