Mathematical Methods for Evaluating Second‐Order Three‐Body Interactions between Atoms or Ions with Gaussian Wavefunctions

1967 ◽  
Vol 8 (6) ◽  
pp. 1266-1270 ◽  
Author(s):  
Shimshon Zimering
Keyword(s):  
Second Order ◽  
Mathematical Methods ◽  
Three Body

scholarly journals First order normalization in the generalized photo gravitational non-planar restricted three body problems

2015 ◽  
Vol 3 (2) ◽  
pp. 46
Author(s):  
Nirbhay Kumar Sinha
Keyword(s):  
Oblate Spheroid ◽  
Second Order ◽  
Elliptic Motion ◽  
First Order ◽  
Short Period ◽  
Order Part ◽  
Three Body

<p>In this paper, we normalised the second-order part of the Hamiltonian of the problem. The problem is generalised in the sense that fewer massive primary is supposed to be an oblate spheroid. By photogravitational we mean that both primaries are radiating. With the help of Mathematica, H<sub>2</sub> is normalised to H<sub>2</sub> = a<sub>1</sub>b<sub>1</sub>w<sub>1</sub> + a<sub>2</sub>b<sub>2</sub>w<sub>2</sub>. The resulting motion is composed of elliptic motion with a short period (2p/w<sub>1</sub>), completed by an oscillation along the z-axis with a short period (2p/w<sub>2</sub>).</p>

scholarly journals VISUALIZATION OF SECOND ORDER CURVES IN GEOGEBRA WHEN TEACHING MATHEMATICS TO STUDENTS OF VOCATIONAL EDUCATION INSTITUTIONS

2021 ◽  
Vol 57 (3) ◽  
pp. 134-146
Author(s):  
D.V. Bochkareva ◽  
Keyword(s):  
Mathematics Education ◽  
Vocational Education ◽  
Cognitive Abilities ◽  
Creative Thinking ◽  
Statistical Processing ◽  
Education Institution ◽  
Second Order ◽  
Educational Process ◽  
Teaching Mathematics ◽  
Mathematical Methods

Problem statement. This article is devoted to such a topical issue as the use of the dynamic mathematics system (DMS) GeoGebra in teaching mathematics. DMSs are gradually being introduced into mathematics education, but the question is raised about the effectiveness of their application. The purpose of the article is to study the influence of GeoGebra on the quality of the solving of tasks on the topic «Second order curves» by 2nd year students studying in the specialty “Applied Informatics (by branches)” at NovosibirskProfessional and Pedagogical Vocational Education Institution. The research methodology consisted in the analysis and synthesis of scientific literature on the chosen topic, pedagogical experiment, mathematical methods of processing the results of this experiment and observation of the participants in the pedagogical process.The obtained results of the research indicate that the use of the GeoGebra program has a positive effect on the academic performance of students. Statistical processing of the obtained data shows that there are no differences in the selected criteria between the group that worked in the GeoGebra and the group that worked without computer programs. However, the use of DMSmakes it possible to visually demonstrate the material, draw the attention of students to the academic discipline, promotes the development of creative thinking, and allows students to experiment with mathematical objects on their own. Conclusion. Due to the widespread use of dynamic systems in mathematics education, the study of their influence on the development of various cognitive abilities and personal qualities of students is of practical and scientific interest. The materials of the work can be used in the educational process when teaching and learning the course EN.01 Mathematics at the level of secondary vocational education and distributed to various topics and sections of mathematical disciplines at other levels of education.

Temperature Dependence of Second and Higher Order Elastic Constants of Orientationally Disordered (NH4I)x (KI)1-x Mixed Crystals

2014 ◽  
Vol 1047 ◽  
pp. 65-70 ◽  
Author(s):  
Alpana Tiwari
Keyword(s):  
Elastic Constants ◽  
Body Force ◽  
Coupling Effect ◽  
Second Order ◽  
Mixed Crystals ◽  
Elastic Behaviour ◽  
Coupling Term ◽  
Third Order ◽  
Third Order Elastic Constants ◽  
Three Body

We have incorporated the translational rotational (TR) coupling effects in the framework of three body force shell model (TSM) to develop an extended TSM (ETSM). This ETSM has been applied to reveal the second order elastic constants (C11, C12and C44) in the dilute regimes 0≤ x ≤ 0.50 as a function of temperature for 10K≤T≤300K. The anomalous elastic behaviour in C44below 100 K has been depicted well by ETSM results in the orientationally disordered (NH4I)x(KI)1-xmixed crystals. In order to present a visual comparison of the TR-coupling effect on second order elastic constants, we have evaluated the SOECs with and without TR coupling term in ETSM. Besides third order elastic constants have also been studied and discussed for concentration range 0≤x≤0.50 as a function of temperature for 10K≤T≤300K.

Second-order relativistic corrections for the S(L=0) states in one- and two-electron atomic systems

2005 ◽  
Vol 83 (1) ◽  
pp. 1-21
Author(s):  
Alexei M Frolov ◽  
Catalin C Mitelut ◽  
Zheng Zhong
Keyword(s):  
Second Order ◽  
Electron Systems ◽  
Specific Interest ◽  
Atomic Systems ◽  
Expectation Value ◽  
Relativistic Corrections ◽  
Alternative Approach ◽  
Compensation Technique ◽  
Three Body ◽  
Regular Operators

An analytical approach is developed to compute the first- (~α2) and second-order (~α4) relativistic corrections in one- and two-electron atomic systems. The approach is based on the reduction of all operators to divergent (singular) and nondivergent (regular) parts. Then, we show that all the divergent parts from the differentmatrix elements cancel each other. The remaining expression contains only regular operators and its expectation value can be easily computed. Analysis of the S(L = 0) states in such systems is of specific interest since the corresponding operators for these states contain a large number of singularities. For one-electron systems the computed relativistic corrections coincide exactly with the appropriate result that follows from the Taylor expansion of the relativistic (i.e., Dirac) energy. We also discuss an alternative approach that allows one to cancel all singularities by using the so-called operator-compensation technique. This second approach is found to be very effective in applications of more complex systems, such as helium-like atoms and ions, H+2-like ions, and some exotic three-body systems.

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