Accurate Exponential Representations for the Ground State Wave Functions 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 linear combinations of a 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≤Z≤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.

Global Wave Velocity Change Measurement of Rock Material by Full-Waveform Correlation

Measuring accurate wave velocity change is a crucial step in damage assessment of building materials such as rock and concrete. The anisotropy caused by the generation of cracks in the damage process and the uncertainty of the damage level of these building materials make it difficult to obtain accurate wave velocity change. We propose a new method to measure the wave velocity change of anisotropic media at any damage level by full-waveform correlation. In this method, the anisotropy caused by the generation of cracks in the damage process is considered. The accuracy of the improved method is verified by numerical simulation and compared with the existing methods. Finally, the proposed method is applied to measure the wave velocity change in the damage process of rock under uniaxial compression. We monitor the failure process of rock by acoustic emission (AE) monitoring system. Compared with the AE ringing count, the result of damage evaluation obtained by the proposed method is more accurate than the other two methods in the stage of increasing rock heterogeneity. These results show that the proposed method is feasible in damage assessment of building materials such as rock and concrete.

Exact propagation of the isolated waves model described by the three coupled nonlinear Maccari’s system with complex structure

In this paper, the three nonlinear Maccari’s-system (TNLMS) which describes how isolated waves are propagated in a finite region of space is studied. New accurate wave solutions of this model are obtained for the first time using the Riccati–Bernoulli Sub-ODE method (RBSOM) that treats the problem for which the balance rule fails. The efficiency of this method for constructing these exact solutions has been demonstrated. The obtained results give an accurate interpretation of the propagation of these isolated waves. With an appropriate values for the physical parameters, some representative wave structures are graphically presented.

A multivibrator and oscillator trainer kit for an emerging pedagogy in engineering

Implementation of a complex circuit with hardware components in a conventional way seems to be quite difficult for the students. Resultant wave-shapes may not be accurate and determined values of different parameters may not be exact. Due to inaccurate outcomes and time-consuming process in a conventional way, researchers despond to continue their approach. In this paper, a trainer kit has been developed which is just like a laboratory with necessary components and aims to implement multivibrators (Bistable, Monostable, and Astable) and RC phase shift oscillators in the laboratory. A variable power supply is constructed in the trainer kit that provides 9 V dc to the multivibrators and 15 V dc to the oscillator. Potentiometer, voltage regulator, different active and passive components such as resistors, capacitor, op-amp, transistor, and numerous jumper wires are introduced in this kit. The trainer kit ascertains an easy implementation by eliminating complexity and assures accurate wave-shapes at the oscilloscope within a short time. To validate the proposed system, a survey program has been introduced among many students and faculty members. The outcome of the survey is the demonstration of student and instructor views and opinions which acknowledges the trainer kit as an efficient and compatible working platform.

Hartree–Fock implementation using a Laguerre-based wave function for the ground state and correlation energies of two-electron atoms

An implementation of the Hartree–Fock (HF) method using a Laguerre-based wave function is described and used to accurately study the ground state of two-electron atoms in the fixed nucleus approximation, and by comparison with fully correlated (FC) energies, used to determine accurate electron correlation energies. A variational parameter A is included in the wave function and is shown to rapidly increase the convergence of the energy. The one-electron integrals are solved by series solution and an analytical form is found for the two-electron integrals. This methodology is used to produce accurate wave functions, energies and expectation values for the helium isoelectronic sequence, including at low nuclear charge just prior to electron detachment. Additionally, the critical nuclear charge for binding two electrons within the HF approach is calculated and determined to be Z HF C =1.031 177 528. This article is part of the theme issue ‘Modern theoretical chemistry’.