arXiv subjects

Evgeny Z. Liverts

Publications and source records attributed to Evgeny Z. Liverts.

15 recordsLinked to original sources

arXiv2024

Two-electron atomic systems. A simple method for calculating the ground state near the nucleus. Some applications

A simple method of variational calculations of the electronic structure of a two-electron atom/ion, primarily near the nucleus, is proposed. The method as a whole consists of a standard solution of a generalized matrix eigenvalue equation, all matrix elements of which are reduced to a numerical calculation of one-dimensional integrals. Distinctive features of the method are: The use of the hyperspherical coordinate system. The inclusion of logarithms of the hyperspherical radius $R$ in the basis functions, similar to the Fock expansion. Using a special basis function including the leading angular Fock coefficients to provide the correct behavior of the wave function near the nucleus. The main numerical parameters characterizing the properties of the helium atom and a number of helium-like ions near the nucleus are calculated and presented in tables. Among others, the specific coefficients $a_{21}$ of the Fock expansion, which can only be calculated using a wave function with the correct behavior near the nucleus, are presented in table and graphs.

physics.atom-ph↗

arXiv2022

Fock expansion for two-electron atoms. High order angular coefficients

The Fock expansion, which describes the properties of two-electron atoms near the nucleus, is studied. The angular Fock coefficients $ψ_{k,p}(α,θ)$ with the maximum possible value of subscript $p$ are calculated on examples of the coefficients with $5\leq k \leq 10$. The presented technique makes it possible to calculate such angular coefficients for any arbitrarily large $k$. The mentioned coefficients being leading in the logarithmic power series representing the Fock expansion may be indispensable for the development of simple methods for calculating the helium-like electronic structure. The theoretical results obtained are verified by other suitable methods. The Wolfram Mathematica is used extensively.

physics.atom-ph↗

arXiv2021

Analytic scattering parameters at low energies

The scattering of two and more particles at low energies is described by the so called effective-range expansion. The leading terms of this expansion are the scattering length and effective range. The analytic expressions for both of the aforementioned scattering parameters are presented for the inverse-power potential and the Woods-Saxon potential. A technique for calculating the approximate scattering parameters is proposed. Approximate analytic formulas representing the scattering length and effective range are obtained for the Yukawa potential. The corresponding figures demonstrate a few interesting features of the effective range. All analytic formulas, both exact and approximate, were verified by comparing with the corresponding results obtained by direct numerical calculations. Wolfram Mathematica is heavily used. The presented results can be used with advantage in the fields of nuclear physics, atomic and molecular physics, quantum chemistry and many others.

nucl-th↗

arXiv2021

Electrons on sphere in the helium-like atomic systems

The properties of a special configuration of a helium-like atomic system, when both electrons are on the surface of a sphere of radius $r$, and angle $θ$ characterizes their positions on sphere, are investigated. Unlike the previous studies, $r$ is considered as a quantum mechanical variable but not a parameter. It is important that the "electrons-on-sphere" and the "collinear" configuration are coincident in two points. For $θ=0$ one obtains the state of the electron-electron coalescence, whereas the angle $θ=π$ characterizes the $\textbf{e-n-e}$ configuration when the electrons are located at the ends of the diameter of sphere with the nucleus at its center. The Pekeris-like method representing a fully three-body variational technique is used for the expedient calculations. Some interesting features of the expectation values representing the basic characteristic of the "electrons-on-sphere" configuration are studied. The unusual properties of the expectation values of the operators associated with the kinetic and potential energy of the two-electron atom/ion possessing the "electrons-on-sphere" configuration are found. Refined formulas for calculations of the two-electron Fock expansion by the Green's function approach are presented. The analytic wave functions of high accuracy describing the "electrons-on-sphere" configuration are obtained. All results are illustrated in tables and figures.

physics.atom-ph↗

arXiv2021

Accurate exponential representations for the ground states of the collinear two-electron atomic systems

In the framework of the study of helium-like atomic systems possessing the collinear configuration, we propose a simple method for computing compact but very accurate wave functions describing the relevant $S$ state. It is worth noting that the considered states include the well-known states of the electron-nucleus and electron-electron coalescences as a particular case. The simplicity and compactness imply that the considered wave functions represent a linear combinations of few single exponentials. We have calculated such model wave functions for the ground state of helium and the two-electron ions with nucleus charge $1 \leq Z \leq 5$. The parameters and the accompanying characteristics of these functions are presented in tables for number of exponential from 3 to 6. The accuracy of the resulting wave functions are confirmed graphically. The specific properties of the relevant codes by Wolfram Mathematica are discussed. An example of application of the compact wave functions under consideration is reported.

physics.atom-ph↗

arXiv2020

The collinear helium atom and two-electron ions

Collinear configurations of the helium-like atomic systems, relevant, e.g., for the quasifree mechanism of the double photoionization of helium, are studied, parameterized by the single scalar parameter $-1\leq λ\leq1$ ("collinear parameter") where $λ=0$ corresponds to the electron-nucleus ($\textbf{e-n}$) coalescence and $λ=1$ corresponds to the electron-electron ($\textbf{e-e}$) coalescence. In general, $λ>0$ corresponds to the \textbf{n-e-e} configuration, and $λ<0$ to the \textbf{e-n-e} configuration. Simple mathematical representations of the expectation values of the Dirac delta function relevant for the collinear configurations are derived and calculated from fully three-body dynamics without approximation for the two-electron atomic wave functions with nuclear charge $1\leq Z\leq5$. Simple formulas for calculating the expectation values of the kinetic and potential energy operators in collinear configurations are derived. Unusual physical properties of the \textbf{n-e-e} collinear configurations found for certain ranges of $λ$ are presented. The first few angular Fock coefficients for collinear configurations are derived as functions of $λ$. Highly accurate model wave functions describing the ground states of the two-electron atoms with collinear arrangement of the particles are constructed. All results are illustrated by tables and figures.

physics.atom-ph↗

arXiv2020

Small recoil momenta double ionization of He and two-electron ions by high energy photons

We calculate various differential and double differential characteristics of ionization by a single photon for H$^-$, He and for the two-electron ions with Z=3,4,5 in the region of the so-called quasi-free mechanism (QFM) domination. We employ highly accurate wave functions at the electron-electron coalescence line where coordinates of both ionized electrons coincide. We trace the Z dependence for the double differential distribution. For all considered targets we discuss the dependence of the photoelectron energy distribution on the photon energy. Our calculation demonstrated the rapid decrease of QFM contribution with increase of the difference in energy of two outgoing electrons, and with decrease of the angle between two outgoing momenta. As a general feature, we observe the decrease of QFM contribution with nuclear charge growth.

physics.atom-ph↗

arXiv2016

Two-particle atomic coalescences: Boundary conditions for the Fock coefficient components

The exact values of the presently determined components 26of the angular Fock coefficients at the two-particle coalescences were obtained and systematized. The Green Function approach was successfully applied to simplify the most complicated calculations. The boundary conditions for the Fock coefficient components in the hyperspherical angular coordinates, which follows from the Kato cusp conditions for the two-electron wave function in the natural interparticle coordinates, were derived. The validity of the obtained boundary conditions was verified on examples of all the presently determined components. The additional boundary conditions are not arising from the Kato cusp conditions were obtained as well. The Wolfram Mathematica was used intensively.

physics.atom-ph↗

arXiv2016

The exact calculation of the edge components of the angular Fock coefficients

The present paper constitutes the development of our previous work devoted to calculations of the angular Fock coefficients, $ψ_{k,p}(α,θ)$. The explicit analytic representations for the edge components $ψ_{k,0}^{(0)}$ and $ψ_{k,0}^{(k)}$ with $k\leq8$ are derived. The methods developed enable such a calculation for arbitrary $k$. The single-series representation for subcomponent $ψ_{3,0}^{(2e)}$ missed in the previous paper is developed. It is shown how to express some of the complicated subcomponents through the hypergeometric and the elementary functions, as well. Using the specific operator of the Wolfram Mathematica, the simple explicit representations for some complex mathematical objects under consideration are obtained.

physics.atom-ph↗

arXiv2016

Subcomponents $ψ_{4,1}^{(2d)}$ and $ψ_{3,0}^{(2c)}$ of the angular Fock coefficients: Explicit analytic representation

The Fock expansion [1] describes the $S$-state wave function of the two-electron atomic system in the vicinity of the triple coalescence point. The present work constitutes the additional appendix to our paper [2] devoted to refinement and further development of calculation of the angular Fock coefficients of the Fock expansion. We derive the explicit analytic expressions for the complicated subcomponents $ψ_{4,1}^{(2d)}$ and $ψ_{3,0}^{(2c)}$ of the angular Fock coefficients. An unusual method of using the Wolfram Mathematica was applied.

physics.atom-ph↗

arXiv2015

Angular Fock coefficients. Fixing the errors, and further development

The angular coefficients $ψ_{k,p}(α,θ)$ of the Fock expansion characterizing the S-state wave function of the two-electron atomic system, are calculated in hyperspherical angular coordinates $α$ and $θ$. To solve the problem the Fock recurrence relations separated into the independent individual equations associated with definite power $j$ of the nucleus charge $Z$, are applied. The "pure" $j$-components of the angular Fock coefficients, orthogonal to of the hyperspherical harmonics $Y_{kl}$, are found for even values of $k$. To this end, the specific coupling equation is proposed and applied. Effective techniques for solving the individual equations with simplest nonseparable and separable right-hand sides are proposed. Some mistakes/misprints made earlier in representations of $ψ_{2,0}$, were noted and corrected. All $j$-components of $ψ_{4,1}$ and the majority of components and subcomponents of $ψ_{3,0}$ are calculated and presented for the first time. All calculations were carried out with the help of the Wolfram \emph{Mathematica}.

physics.atom-ph↗

arXiv2014

Analytic matrix elements for the two-electron atomic basis with logarithmic terms

The two-electron problem for the helium-like atom/ions in $S$-state is considered. The basis containing the integer powers of $\ln r$, where $r$ is a radial variable of the Fock expansion, is studied. In this basis, the analytic expressions for the matrix elements of the corresponding Hamiltonian are presented. These expressions include only special functions presented by the built-in $Mathematica$ codes, what enables very fast and accurate computation of the matrix elements. The decisive contribution of the correct logarithmic terms to the behavior of the two-electron wave function in the vicinity of the triple-coalescence point is reaffirmed.

math-ph↗

arXiv2012

About disposition of energy levels

The unique properties of central potential of the form $-βe^{-r}r^γ$ were studied using the recently developed critical parameter technique. The particular cases of $γ=0$ and $γ=-1$ yield, respectively, the exponential and Yukawa potentials widely used in the atomic, molecular and nuclear physics. We found different behavior of the energy levels of this potential for three different ranges of the value of $γ$. For $γ\geq0$ it was found that the energy of bound states with the same principal quantum number $N$ decreases with increasing angular momentum $\ell$. The Gaussian and Woods-Saxon potentials also show this behavior. On the contrary, for $-2\leqγ\leq-1$ increasing $\ell$ gives a higher energy, resembling the Hulthen potential. However, a potential with $-1<γ<0$ possesses mixed properties, which give rise to several interesting results. For one, the order of energy levels with different quantum numbers is not preserved when varying the parameter $β$. This leads to a quantum degeneracy of the states, and in fact, for a given value of $γ$ we can find the values $β_{thr}$ for which two energy levels with different quantum numbers coincide. Another interesting phenomena is the possibility, for some values of $γ$ in this range, for two new energy levels with different quantum numbers to appear simultaneously when $β$ reaches their common critical value.

physics.comp-ph↗

arXiv2011

Transition states and the critical parameters of central potentials

Transition states or quantum states of zero energy appear at the boundary between the discrete part of the spectrum of negative energies and the continuum part of positive energy states. As such, transition states can be regarded as a limiting case of a bound state with vanishing binding energy, emerging for a particular set of critical potential parameters. In this work we study the properties of these critical parameters for short range central potentials. To this end we develop two exact methods and also utilize the first and second order WKB approximations. Using these methods we have calculated the critical parameters for several widely used central potentials. The general analytic expressions for the asymptotic representations of the critical parameters were derived for cases where either the orbital quantum number $l$ or the number $n$ of bound states approaches infinity. The above mathematical models enable us to answer the following physical (quantum mechanical) questions: i) what is the number of bound states for a given central potential and given orbital quantum number $l$; ii) what is the maximum value of $l$ which can provide a bound state for the given central potential; iii) what is the order of energy levels for the given form of the central potential. It is revealed that the ordering of energy levels depends on the potential singularity at the origin.

quant-ph↗

arXiv2001

Solving the Quantum Chemistry Equations and High-Temperature-Superconductivity Problem

The conventional technique for solving the equations of quantum chemistry (of solid state) is extended unconventionally to the structures possessing certain symmetries. This proposal concerns changing the way for selection of occupied orbitals, allowing, in turn, to release the unoccupied electronic states located lower than the ground state Fermi level of a specific system. Such states can be treated as 'spectral holes'. Application of this technique, in particular, when calculating the electronic structure of the HTSC-compound $YBa_2Cu_3O_{7-x}$ (0 0). One could easily connect the obtained results to the high-temperature superconductivity of Little's linear chains, as well as Ginzburg's two-dimensional layers and even to BCS-model.

physics.chem-ph↗