All‐electron relativistic computations on the low‐lying electronic states,bond length,and vibrational frequency of CeF diatomic molecule with spin–orbit coupling effects

Ab initio all‐electron computations have been carried out for Ce⁺ and CeF, including the electron correlation, scalar relativistic, and spin–orbit coupling effects in a quantitative manner. First, the n‐electron valence state second‐order multireference perturbation theory (NEVPT2) and spin–orbit configuration interaction (SOCI) based on the state‐averaged restricted active space multiconfigurational self‐consistent field (SA‐RASSCF) and state‐averaged complete active space multiconfigurational self‐consistent field (SA‐CASSCF) wavefunctions have been applied to evaluations of the low‐lying energy levels of Ce⁺ with [Xe]4f¹5d¹6s¹ and [Xe]4f¹5d² configurations, to test the accuracy of several all‐electron relativistic basis sets. It is shown that the mixing of quartet and doublet states is essential to reproduce the excitation energies. Then, SA‐RASSCF(CASSCF)/NEVPT2 + SOCI computations with the Sapporo(‐DKH3)‐2012‐QZP basis set were carried out to determine the energy levels of the low‐lying electronic states of CeF. The calculated excitation energies, bond length, and vibrational frequency are shown to be in good agreement with the available experimental data. © 2018 Wiley Periodicals, Inc.     

Accurate spectroscopic constants of the lowest three electronic states in halonitrenes with multireference configuration interaction

Internally contracted multireference configuration interaction (icMRCI) calculations of the ground state (X³Σ⁻), the first excited state (a¹Δ) as well as the second excited state (b¹Σ⁺) have been performed for a series of halogenated nitrenes NXs (X = Cl, Br, and I). Accurate spectroscopic constants of these lowest three electronic states of each NX were obtained in this work using MRCI methods with aug‐cc‐pVXZ (X = T, Q, 5) basis sets and complete basis set (CBS) limit. In addition, various corrections, including the Davidson correction, scalar relativistic effect, core‐valence correlation, and spin‐orbit coupling effect, have been studied in calculating spectroscopic constants, especially for heavy‐atom nitrenes. Comparisons have been made with previous computational and experimental results where available. The icMRCI + Q calculations presented in this work provide a comprehensive series of results at a consistent high level of theory for all of the halogenated nitrenes.     

Tauberian theorems and stability of solutions of the Cauchy problem

Let be a bounded, strongly measurable function with values in a Banach space , and let be the singular set of the Laplace transform in . Suppose that is countable and uniformly for , as , for each in . It is shown that

as , for each in ; in particular, if is uniformly continuous. This result is similar to a Tauberian theorem of Arendt and Batty. It is obtained by applying a result of the authors concerning local stability of bounded semigroups to the translation semigroup on , and it implies several results concerning stability of solutions of Cauchy problems.

     

On the determination of the potential function from given orbits

The paper deals with the problem of finding the field of force that generates a given (N ? 1)-parametric family of orbits for a mechanical system with N degrees of freedom. This problem is usually referred to as the inverse problem of dynamics. We study this problem in relation to the problems of celestial mechanics. We state and solve a generalization of the Dainelli and Joukovski problem and propose a new approach to solve the inverse Suslov’s problem. We apply the obtained results to generalize the theorem enunciated by Joukovski in 1890, solve the inverse Stäckel problem and solve the problem of constructing the potential-energy function U that is capable of generating a bi-parametric family of orbits for a particle in space. We determine the equations for the sought-for function U and show that on the basis of these equations we can define a system of two linear partial differential equations with respect to U which contains as a particular case the Szebehely equation. We solve completely a special case of the inverse dynamics problem of constructing U that generates a given family of conics known as Bertrand’s problem. At the end we establish the relation between Bertrand’s problem and the solutions to the Heun differential equation. We illustrate our results by several examples.     

Analogies between Jahn–Teller and Rashba spin physics

In developing physical theories, analogical reasoning has been found to be very powerful, as attested by a number of important historical examples. An analogy between two apparently different phenomena, once established, allows one to transfer information and bring new concepts from one phenomenon to the other. Here, we discuss an important analogy between two widely different physical problems, namely, the Jahn–Teller distortion in molecular physics and the Rashba spin splitting in condensed matter physics. By exploring their conceptual and mathematical features and by searching for the counterparts between them, we examine the orbital texture in Jahn–Teller systems, as the counterpart of the spin texture of the Rashba physics, and put forward a possible way of experimentally detecting the orbital texture. Finally, we discuss the analogy by comparing the coexistence of linear Rashba + Dresselhaus effects and Jahn–Teller problems for specific symmetries, which allow for nontrivial spin and orbital textures, respectively.     

A new uniqueness criterion for the number of periodic orbits of Abel equations

A solution of the Abel equation such that x(0)=x(1) is called a periodic orbit of the equation. Our main result proves that if there exist two real numbers a and b such that the function aA(t)+bB(t) is not identically zero, and does not change sign in [0,1] then the Abel differential equation has at most one non-zero periodic orbit. Furthermore, when this periodic orbit exists, it is hyperbolic. This result extends the known criteria about the Abel equation that only refer to the cases where either A(t)?0 or B(t)?0 does not change sign. We apply this new criterion to study the number of periodic solutions of two simple cases of Abel equations: the one where the functions A(t) and B(t) are 1-periodic trigonometric polynomials of degree one and the case where these two functions are polynomials with three monomials. Finally, we give an upper bound for the number of isolated periodic orbits of the general Abel equation , when A(t), B(t) and C(t) satisfy adequate conditions.     

On the determination of the basin of attraction of a periodic orbit in two-dimensional systems

We consider the general nonlinear differential equation with x∈R² and develop a method to determine the basin of attraction of a periodic orbit. Borg's criterion provides a method to prove existence, uniqueness and exponential stability of a periodic orbit and to determine a subset of its basin of attraction. In order to use the criterion one has to find a function W∈C¹(R²,R) such that LW(x)=W^′(x)+L(x) is negative for all x∈K, where K is a positively invariant set. Here, L(x) is a given function and W^′(x) denotes the orbital derivative of W. In this paper we prove the existence and smoothness of a function W such that LW(x)=−μ‖f(x)‖. We approximate the function W, which satisfies the linear partial differential equation W^′(x)=〈∇W(x),f(x)〉=−μ‖f(x)‖−L(x), using radial basis functions and obtain an approximation w such that Lw(x)<0. Using radial basis functions again, we determine a positively invariant set K so that we can apply Borg's criterion. As an example we apply the method to the Van-der-Pol equation.     

Study of the ferrodistorsive orbital ordering in NaNiO2 by neutron diffraction and submillimeter wave ESR

NaNiO₂ has been studied by neutron-powder diffraction, magnetic susceptibility and submillimeter wave ESR. The monoclinic structure at room temperature is characterised by a ferrodistorsive orbital ordering due to the Jahn-Teller (JT) effect of the Ni³⁺ ions in the low spin state. NaNiO₂ undergoes a structural transition at around 480 K, above which the orbital ordering disappears. The high temperature phase is rhombohedral with the layered -NaFeO₂ structure ( space group). The magnetic susceptibility exhibits hysteresis and we observe a change of the Curie-Weiss law parameters above the JT transition. The anisotropy of the g-factor at 200 K can be attributed to the JT effect which favours the orbital occupation. Finally, the interplay between the magnetic and structural properties of NaNiO₂ and Li_1-xNi_1+xO₂ is discussed. Received 29 May 2000     

一类扩张了的软弹簧型Duffing方程的紊动性态

本文用Melnikov函数方法讨论了一类扩张了的软弹簧型Duffing方程(k=1,2,3,…)在周期激励下的紊动现象.给出了出现二阶同宿切的条件.文中所采用的方法对于不能给出并宿轨道的显式的系统的研究是非常有用的.     

一类二维流场模型的周期解与浑沌解的共存性*

本文研究Kelvin-Stuart猫眼流在周期扰动下的动力学行为,运用Melnikov方法确定出振动型周期轨产生偶数阶次谐分枝、旋转型周期轨产生任意阶次谐分枝的条件,并进一步发现周期解与浑沌解共存的复杂现象.