Finite Element Calculations of the Electron-Electron Coulomb Repulsions in a Quantum Dot Dimer

1. Silicon devices that incorporate a few impurities forming arrays are ideal materials for the solid-state implementation of quantum technologies. The nearly-atomic precision on the impurity distribution provides remarkable tunability of the device properties, resulting in their potential application in quantum simulations. A model Hamiltonian of interest is the extended Hubbard model, and a crucial part of it is the Coulomb electron-to-electron repulsion term, U. Coulomb interactions can be efficiently computed from the wavefunction of the bound electrons and by solving the Poisson Equation using the Finite Element Method (FEM). An investigation into FEM's numerical properties is necessary to understand its limitations in studying the dopant array system. We present results on two-Dimensional FEM calculations of U as a function of impurity separation distance, 2 ≤ D ≤ 10 au, for an impurity model where the molecular hydrogen wavefunctions describe the electron ground and excited states. We also present results on the numerical stability of one-dimensional FEM, finding that while linear elements are sufficient to compute the electric potential for the Gaussian electron density, they are insufficient to compute the correct electric potential with a hydrogen 1S orbital wavefunction electron density, even with an adaptive mesh. Consequently, we conclude that to obtain the true potential due to hydrogen-like electron charge distributions, nonlinear elements are required.