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

Document Type : Full length research Paper


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


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.


Main Subjects

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