The exact Schrieffer–Wolff transformation

The effective s–d spin interaction is derived exactly for the single-impurity Anderson model via a unitary transformation. This unitary transformation was calculated up to infinite order and no restrictions were imposed upon the coefficients of the Hubbard interaction and the hybridization. We also discuss briefly the impact of the obtained result on the magnetic properties of several Kondo compounds. This will shed new light on the understanding of the competition between the Kondo effect and the Ruderman–Kittel–Kasuya–Yosida interaction and reinterpret the Doniach diagram.     

Some special exact solutions in nonlocal Alice-Bob sine-Gordon systems

Three nonlocal Alice-Bob sine-Gordon(ABSG) systems with the parity and time reversal nonlocality and/or space-time exchange nonlocality are investigated. For the common local SG equation, two types of N-soliton solutions and three types of periodic solutions are presented. The multiple solutions, breather solution, double kink solution, and periodic solutions of the ABSG systems are obtained from the symmetry reductions of a coupled local sine-Gordon system.     

Scaled Schrödinger equation and the exact wave function

We propose the scaled Schr?dinger equation and the related principles, and construct a general method of calculating the exact wave functions of atoms and molecules in analytical forms. The nuclear and electron singularity problems no longer occur. Test applications to hydrogen atom, helium atom, and hydrogen molecule are satisfactory, implying a high potentiality of the proposed method.     

Photoconductivity derived from an exact modelling of the uncompensated one-donor model. III—problem of fitting the exact model with approximate models

Abstract

The paper is essentially devoted to a quantitative comparison of the exact (1D) model developed in Part I, with two approximate models indicated by Kittel (1D_a1) and by Van der Ziel (1D_a2). And this is made for identical values of their common parameters. This allows to show that models 1D and 1D_a1 generally present no suitable intervals for fitting. The model of Van der Ziel is more suited, for, a good fitting with the 1D model can be obtained, for relatively reduced G values, and for temperatures bounded upward by the characteristic temperature T ₀ defined previously. The role played by the sub-models, obtained after simplifying the 1D_ai model equations, is studied also.     

Enhancement factors in semi-empirical exchange–correlation functionals

It is demonstrated that the shape of the exchange enhancement factor and whether or not a functional satisfies the uniform density scaling condition can both be controlled, to some extent, in semi-empirical functional development. Specifically, this is achieved by varying the spatial weighting in a fit to high-quality exchange–correlation potentials. The observation is used to highlight the correlation between the exchange enhancement factor and the topography of the helium dimer potential energy curve. It is also used to determine a series of functionals that explicitly satisfy the uniform density scaling condition for the considered r_s values. A modified version of the well-established Hamprecht–Cohen–Tozer–Handy (HCTH-93) functional is also determined, which satisfies this condition. For the systems considered, enforcing the condition has a relatively small effect on covalent bond lengths, thermochemistry, polarizabilities and shielding constants; it improves the potential energy curve of the helium dimer, due to the associated modification of the exchange enhancement factor.     

H-D exchange on graphite–potassium intercalation compounds

Potassium graphite intercalation compounds are able to activate C–H bonds of hydrocarbons at room temperature. In this paper, the hydrogen–deuterium exchange of CHD₃ in the presence of C₈K, C₂₄K and C₃₆K is described.     

Electron–phonon heat exchange in quasi-two-dimensional nanolayers

We study the heat power P transferred between electrons and phonons in thin metallic films deposited on free-standing dielectric membranes. The temperature range is typically below 1 K, such that the wavelengths of the excited phonon modes in the system is large enough so that the picture of a quasi-two-dimensional phonon gas is applicable. Moreover, due to the quantization of the components of the electron wavevectors perpendicular to the metal film’s surface, the electrons spectrum forms also quasi two-dimensional sub-bands, as in a quantum well (QW). We describe in detail the contribution to the electron–phonon energy exchange of different electron scattering channels, as well as of different types of phonon modes. We find that heat flux oscillates strongly with thickness of the film d while having a much smoother variation with temperature (T _e for the electrons temperature and T _ph for the phonons temperature), so that one obtains a ridge-like landscape in the two coordinates, (d, T _e ) or (d, T _ph ), with crests and valleys aligned roughly parallel to the temperature axis. For the valley regions we find P ∝ T _e ^3.5 – T _ph ^3.5 . From valley to crest, P increases by more than one order of magnitude and on the crests P cannot be represented by a simple power law. The strong dependence of P on d is indicative of the formation of the QW state and can be useful in controlling the heat transfer between electrons and crystal lattice in nano-electronic devices. Nevertheless, due to the small value of the Fermi wavelength in metals, the surface imperfections of the metallic films can reduce the magnitude of the oscillations of P vs. d, so this effect might be easier to observe experimentally in doped semiconductors.     

New exact solutions of Burgers’ type equations with conformable derivative

In this paper, the new exact solutions for some nonlinear partial differential equations are obtained within the newly established conformable derivative. We use the first integral method to establish the exact solutions for time-fractional Burgers’ equation, modified Burgers’ equation, and Burgers–Korteweg–de Vries equation. We report that this method is efficient and it can be successfully used to obtain new analytical solutions of nonlinear FDEs.     

New exact solutions of Einstein—Maxwell equations for magnetostatic fields

The symmetry reduction method based on the Fréchet derivative of differential operators is applied to investigate symmetries of the Einstein-Maxwell field equations for magnetostatic fields, which is a coupled system of nonlinear partial differential equations of the second order. The technique yields invariant transformations that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations. Some of the reduced systems are further studied to obtain the exact solutions.     

New explicit exact solutions to a nonlinear dispersive－dissipative equation

Using the first-integral method, we obtain a series of new explicit exact solutions such as exponential function solutions, triangular function solutions, singular solitary wave solution and kink solitary wave solution of a nonlinear dispersive－dissipative equation, which describes weak nonlinear ion-acoustic waves in plasma consisting of cold ions and warm electrons.     

Antiferromagnetic exchange coupling in amorphous Fe–Sc alloys

Results of ⁵⁷Fe Mössbauer, AC and DC susceptibility, grazing incidence X-ray diffraction, resistivity and Rutherford-backscattering measurements on the amorphous alloys ${\text{Fe}}_{100 - x} {\text{Sc}}_x (8 \leqslant x \leqslant 70)$ give for the first time convincing evidence for the antiferromagnetic exchange coupling between Fe-moments. The antiferromagnetic coupling between Fe-moments is for $(8 \leqslant x \leqslant 70)$ limited to certain regions (magnetic clusters) of the sample. The nature of the coupling is of the Heisenberg type. For $8 \leqslant x < 20$ , the magnetic coupling between Fe-moments is across nonmagnetic Sc-atoms. The conduction electrons mediate an indirect magnetic interaction between the Fe-moments. The magnetic exchange coupling between Fe-moments across Sc-atoms is negative. The antiferromagnetic coupling between Fe-moments can be explained by the Ruderman-Kittel-Kasuya-Yosida, RKKY, interaction by taking into account the damped oscillatory behavior of the RKKY interactions.     

A method for constructing exact solutions and application to Benjamin Ono equation

By using an improved projective Riccati equation method, this paper obtains several types of exact travelling wave solutions to the Benjamin Ono equation which include multiple soliton solutions, periodic soliton solutions and Weierstrass function solutions. Some of them are found for the first time. The method can be applied to other nonlinear evolution equations in mathematical physics.     

Replica field theories,painlevé transcendents,and exact correlation functions

Exact solvability is claimed for nonlinear replica sigma models derived in the context of random matrix theories. Contrary to other approaches reported in the literature, the framework outlined does not rely on traditional "replica symmetry breaking" but rests on a previously unnoticed exact relation between replica partition functions and Painlevé transcendents. While expected to be applicable to matrix models of arbitrary symmetries, the method is used to treat fermionic replicas for the Gaussian unitary ensemble (GUE), chiral GUE (symmetry classes A and AIII in Cartan classification) and Ginibre's ensemble of complex non-Hermitian random matrices. Further applications are briefly discussed.     

The coupling constant averaged exchange–correlation energy density

ABSTRACT

The exchange–correlation energy, central to density-functional theory, may be represented in terms of the coupling constant averaged (CCA) exchange–correlation energy density. We present an approach to calculate the CCA energy density using accurate ab initio methods and its application to simple atomic systems. This function provides a link between intrinsically non-local, many-body electronic structure methods and simple local and semi-local density-functional approximations (DFAs). The CCA energy density is resolved into separate exchange and correlation terms and the features of each compared with those of quantities commonly used to construct DFAs. In particular, the more complex structure of the correlation energy density is found to exhibit features that align well with those present in the Laplacian of the density, suggesting its role as a key variable to be used in the construction of improved semi-local correlation functionals. The accurate results presented in this work are also compared with those provided by the Laplacian-dependent Becke–Roussel model for the exchange energy.     

Painlev property of the modified C-KdV equation and its exact solutions

In this paper,the Painlev’e properties of the modified C-KdV equation are verified by using the W-K algorithm.Then some exact soliton solutions are obtained by applying the standard truncated expansion method and the nonstandard truncated expansion method with the help of Maple software,respectively.