The matrix representation of the many-electron Hamoltonian operator in a crystal

Document Type : Full length research Paper

Authors

Faculty of Physics, Shahid Bahonar University of Kerman, Kerman, Iran

Abstract

In the present work, we show how to visualize a many-electron Hamiltonian in the matrix form. This procedure is very important for finding eigenvalues of energy and eigenvectors. Here, the many-particle Hamiltonian in a crystal includes electron-nucleus effects and electron kinetic energy (one-point operators) as well as electron-electron mutual repulsion (two-point operators). It is thoroughly discussed how to reduce a many-particle problem to a two-particle problem. Starting from the antisymmetric wave functions as space bases, the Hamiltonian matrix was obtained in a graphical representation-based way. Each step of forming the components of the Hamiltonian matrix is explained instructively. Results show that the kinetic energy of electrons and the electron-nucleus interaction construct the diagonal components of the Hamiltonian matrix. In representing electron-electron interactions, we employ diagonal elements that do not require the exchange of wave functions over two particles, but off-diagonal components incorporate the effect of exchanging wave functions over the particles. In this study, since we were looking for an exact solution, we used a three-electron system, but the method used can be extended to larger systems. Also, the code written for the Hamiltonian matrix creation process can be provided to researchers in this field.

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References

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Journal of Research on Many-body Systems
Volume 13, Issue 1 - Serial Number 36
spring 2023
May 2023
Pages 65-77

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History

  • Receive Date: 13 August 2022
  • Revise Date: 03 January 2023
  • Accept Date: 06 February 2023

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