Comparative analysis of energy-level splitting of Pr^(3+) doped in LiYF_4 and LiBiF_4 crystals:a complete energy matrix calculation

Based on the combination of Racah＇s group-theoretical consideration with Slater＇s wavefunction, a 91 ×91 complete energy matrix is established in tetragonal ligand field D2d for Pr3＋ ion. Thus, the Stark energy-levels of Pr3＋ ions doped separately in LiYF4 and LiBiF4 crystals are calculated, and our calculations imply that the complete energy matrix method can be used as an effective tool to calculate the energy-levels of the systems doped by rare earth ions. Besides, the influence of Pr3＋ on energy-level splitting is investigated, and the similarities and the differences between the two doped crystals are demonstrated in detail by comparing their several pairs of curves and crystal field strength quantities. We see that the energy splitting patterns are similar and the crystal field interaction of LiYF4：Pr3＋ is stronger than that of LiBiF4：Pr3＋.

EOM-CCSD-Based Neural Network Diabatic Potential Energy Matrix for ^(1)πσ^(*)-Mediated Photodissociation of Thiophenol

A new diabatic potential energy matrix(PEM)of the coupled~^(1)ππ^(*)and~1πσ*states for the~1πσ*-mediated photodissociation of thiophenol was constructed using a neural network(NN)approach.The diabatization of the PEM was specifically achieved by our recent method[Chin.J.Chem.Phys.34,825(2021)],which was based on adiabatic energies without the associated costly derivative couplings.The equation of motion coupled cluster with single and double excitations(EOM-CCSD)method was employed to compute adiabatic energies of two excited states in this work due to its high accuracy,simplicity,and efficiency.The PEM includes three dimensionalities,namely the S-H stretch,C-S-H bend,and C-C-S-H torsional coordinates.The root mean square errors of the NN fitting for the S1 and S2 states are 0.89 and 1.33 me V,respectively,suggesting the high accuracy of the NN method as expected.The calculated lifetimes of the S1 vibronic 00 and31 states are found to be in reasonably good agreement with available theoretical and experimental results,which validates the new EOM-CCSD-based PEM fitted by the NN approach.The combination of the diabatization scheme solely based on the adiabatic energies and the use of EOM-CCSD method makes the construction of reliable diabatic PEM quite simple and efficient.

Generalization for Laplacian energy

Let G be a simple graph with n vertices and m edges. Let λ1, λ2,…, λn, be the adjacency spectrum of G, and let μ1, μ2,…, μn be the Laplacian spectrum of G. The energy of G is E（G） = n∑i=1｜λi｜, while the Laplacian energy of G is defined as LE（G） = n∑i=1｜μi-2m/n｜ Let γ1, γ2, ~ …, γn be the eigenvalues of Hermite matrix A. The energy of Hermite matrix as HE（A） = n∑i=1｜γi-tr（A）/n｜ is defined and investigated in this paper. It is a natural generalization of E（G） and LE（G）. Thus all properties about energy in unity can be handled by HE（A）.

How to Improve Efficiency of Telecom Energy

High OPEX （operation expenditure）, pollution and complicated environment are challenges which telecom operators have to conquer to improve energy efficiency. Integration and flexibility are tendencies for telecom energy solution development, also the keys to improve energy efficiency. With plentiful design and implementation experience, the author provides a three-dimensional energy matrix model to design a perfect energy system, and some practical measures of integration and flexibility are also shared from mechanical to electrical, from single site to whole network.

Theoretical Investigations of the Local Structure Distortion and EPR Parameter for Ni2＋-doped Perovskite Fluorides

By analyzing the optical spectra and electron paramagnetic resonance parameter D, the local structure distortion of （NiF6）4- clusters in AMF3 （A=K, Rb; M=Zn, Cd, Ca） and K2ZnF4 series are studied using the complete energy matrix based on the double spin-orbit coupling parameter model for configuration ions in a tetragonal ligand field. The results indicate that the contribution of ligand to spin-orbit coupling interaction should be considered for our studied systems. Moreover, the relationships between D and the spin-obit coupling coefficients as well as the average parameter and the divergent parameter are discussed.

Investigations of the electron paramagnetic resonance parameters and the tetragonal local structure for (VCl_6)^(4-) coordination complex in MCl:V^(2+) (M=Na,K,Rb) systems

By simulating the electron paramagnetic resonance （EPR） and optical spectra on the basis of the 120 × 120 complete energy matrix, this paper determines the local lattice structure parameters R1 and R2 for MCl：V2＋ （M=Na, K, Rb） systems at 77K, 195 K and RT （room temperature 295 K or 302 K）, respectively. The theoretical results indicate that there exists a compressed distortion in MCl：V2＋ systems. Meanwhile, it finds that the structure parameters R1, R2 and ｜△R｜（ = R1 - R2） increase with the rising temperature. Subsequently, from the analysis it concludes that the relation of EPR parameter D vs. △R is approximately linear. Finally, the effects of orbital reduction factor k on the g factors for the three systems have been discussed.

Theoretical investigations of the local distortion and electron paramagnetic resonance parameter for CdCl_2:V^(2+) and CsMgX_3:V_^(2+)(X=Cl,Br) systems

This paper systematically investigates the local distortion and electron paramagnetic resonance （EPR） parameter for CdCl2：V^2＋ and CsMgX3：V^2＋ （X=Cl, Br） systems on the basis of the complete energy matrix, in which not only the contributions due to the spin-orbit coupling of the central ions but also that of the ligands are considered. To describe the difference of overlapping between d-orbits and p orbit, two spin-orbit coupling coefficients are introduced. By simulating the crystal field parameter and EPR parameter, the local distortion parameters are studied and the relationships between the EPR parameter and the spin-orbit coupling coefficients as well as divergent parameter are discussed. These results show that the local structures exhibit compression distortion for CdCl2：V^2＋ and elongation distortions for CsMgX3：V^2＋ （X：Cl, Br）, respectively. It notes that the empirical formula R≈RH T （ri - rh）/2 is not suitable for CdCl2：V^2＋ and CsMgX3：V^2＋ （X=Cl, Br） systems. The contributions of ligand to spin-orbit coupling interaction cannot be neglected for strong covalent systems, especially for V^2＋ doped in CsMgBr3：V^2＋.

IMPROVED COVARIANCE DRIVEN BLIND SUBSPACE IDENTIFICATION METHOD

An improved covariance driven subspace identification method is presented to identify the weakly excited modes. In this method, the traditional Hankel matrix is replaced by a reformed one to enhance the identifiability of weak characteristics. The robustness of eigenparameter estimation to noise contamination is reinforced by the improved Hankel matrix, in combination with component energy index （CEI） which indicates the vibration intensity of signal components, an alternative stabilization diagram is adopted to effectively separate spurious and physical modes. Simulation of a vibration system of multiple-degree-of-freedom and experiment of a frame structure subject to wind excitation are presented to demonstrate the improvement of the proposed blind method. The performance of this blind method is assessed in terms of its capability in extracting the weak modes as well as the accuracy of estimated parameters. The results have shown that the proposed blind method gives a better estimation of the weak modes from response signals of small signal to noise ratio （SNR）and gives a reliable separation of spurious and physical estimates.

Dualism of the Heisenberg and Schrodinger Approaches to the Quantum States Entering a One-Dimensional Electron Gas

According to quantum mechanics, the commutation property of the energy Hamiltonian with the momentum operator should give the definite values not only for energy but also for the momentum quantum levels. A difficulty provided by the standing-like boundary conditions of the electron gas is that the Hamiltonian eigenfunctions are different than eigenfunctions of the momentum operator. In results the electron momenta are obtained from the correspondence rule between the classical and quantum mechanics given by Landau and Lifshits. As a consequence the statistics of solutions representing not only the energy values but also the electron momenta should be taken into account. In the Heisenberg picture of quantum mechanics, the momenta are easily obtained because the electron oscillators are there directly considered. In fact, the Hamiltonian entering the Heisenberg method can be defined in two different ways each giving the set of the electron energies known from the Schr?dinger’s approach.

Linguistic neutrosophic matrix energy and its application in multiple criteria group decision-making

Matrix energy is an important representation tool of collective information.Then,it is not applied to various fuzzy and linguistic environments.To compensate for this gap,this article aims to extend the matrix energy to propose the energy of a linguistic neutrosophic matrix(LNM)for solving a multiple criteria group decision-making(MCGDM)problem,which fully contains LNMs of decisionmaker weights,criteria weights,and alternative evaluations.To realize the objective,this study first presents the energy of LNM in view of the true matrix energy,the false matrix energy,and the indeterminate matrix energy.Then,a MCGDM technique is established in view of the LNM energy method in a LNM circumstance.Finally,the developed MCGDM technique using the LNM energy is used to solve the hospital location choice problem in the full LNM scenario.Meanwhile,the decision results indicate the validity and usability of the established MCGDM technique.

Analysis of the nonlinear dynamic characteristics of two-phase flow based on an improved matrix pencil method

Gas–liquid two-phase flow is complex and has uncertainty in phase interfaces, which make the two-phase flow look very complicated. Even though the flow behavior(e.g. coalescence, crushing and separation) of single bubble or bubble groups in the liquid phase looks random, combining some established characteristics and methodologies can find regularities among the randomness. In order to excavate the nonlinear dynamic characteristics of gas–liquid two-phase flow, the authors developed an improved matrix pencil(IMP) method to analyze the pressure difference signals of the two-phase flow. This paper elucidates the influence of signal length on MP calculation results and the anti-noise-interference ability of the MP method. An IMP algorithm was applied to the fluctuation signals of gas–liquid two-phase flow to extract the mode frequency and damping ratio, which were combined with the component energy index(CEI) entropy to identify the different flow patterns. It is also found that frequency, damping ratio, CEI entropy and stability diagram together not only identify flow patterns, but also provide a new way to examine and understand the evolution mechanism of physical dynamics embedded in flow patterns. Combining these characteristics and methods, the evolution of the nonlinear dynamic physical behavior of gas bubbles is revealed.

Optical and magnetic properties of transition-metal ions in tetrahedral and octahedral compounds

This paper presents the complete energy matrix of the 3d2system containing the electron-electron interaction, the ligand-fieldinteraction, the spin-orbit coupling interaction, and the Zeeman interaction, in which the optical spectra and g-factor of V3+andTi2+ions in the series of tetrahedral AIIBVI(AII=Zn, Cd, BVI=S, Se, Te) semiconductor materials are determined. In the inves-tigation of the optical and magnetic properties of these transition-metal ions in the tetrahedral coordination complexes, wecompared the data obtained from the transition-metal ions in the tetrahedral coordination complexes with those obtained fromthe corresponding ions in the octahedral ones, and found that the tetrahedral complexes have weaker crystal-field strength, in-verse energy level ordering and stronger covalence effect.

Effect of interface debonding on matrix multicracking evolution of fiber-reinfroced ceramic-matrix composites

An analytical methodology was developed to investigate the effect of fiber/matrix interface debonding on matrix multicracking evolution of fiber-reinforced CMCs(ceramic-matrix composites).The Budiansky-Hutchinson-Evans shear-lag model was adopted to analyse the micro-stress field of the damaged composites.The critical matrix strain energy criterion,which presupposes the existence of an ultimate or critical matrix strain energy with matrix,was obtained to simulate the matrix multicracking evolution of CMCs.With the increase of the applied stress,the matrix multicracking and fiber/matrix interface debonding occurred to dissipate the additional energy entered into the composites.The fiber/matrix interface debonded length under matrix multicracking evolution was obtained by treating the interface debonding as a particular crack propagation problem.The conditions for no-debonding and debonding during the evolution of matrix multicracking were discussed in terms of two interfacial properties,i.e.,the interface shear stress and interface debonded toughness.When the fiber/matrix interface was bonded,the matrix multicracking evolution was much more intense compared with the interface debonding;when the fiber/matrix interface was debonded,the matrix crack density increased with the increasing of interface shear stress and interface debonded energy.The theoretical results were compared with experimental data of unidirectional SiC/CAS(calcium alumina silicate),SiC/CAS-Ⅱand SiC/borosilicate composites.