First-Principles Simulation of Electronic Structure for Two-Dimensional Atoms: A Study of Hydrogen and Helium

Abstract

This study presents a first-principles investigation into the electronic properties and behaviour of two-dimensional (2D) quantum systems, including the Hydrogen atom, the Helium atom, and the Quantum Harmonic Oscillator, with the primary objective of understanding their fundamental electronic structure and response to external electric fields. The methodology employed centers on grid-based numerical simulations, leveraging the Imaginary Time Evolution (ITE) method for determining quantum ground states and the Hartree Self-Consistent Field (SCF) approximation to handle electron-electron interactions in Helium, all implemented using the efficient split-operator Fourier transform technique on a real-space grid. Results for the 2D Hydrogen atom accurately reproduce the expected spherically symmetric ground state, and quantitative analysis of the weak-field Stark effect yields a static dipole polarizability (e.g., 0.1940 a.u.) in good agreement with theoretical values, while strong-field simulations qualitatively illustrate field ionization via quantum tunnelling. For the two-electron 2D Helium atom, the Hartree SCF method successfully converges to a self-consistent solution, demonstrating electron polarization in external fields, and discussion of the calculated static dipole polarizability (e.g., 0.1151 a.u.) in comparison to more accurate values highlights the limitations of mean-field approximations in capturing electron correlation effects; estimation of the correlation energy (e.g., -4.352920 a.u.) quantitatively underscores the significant contribution of instantaneous electron interactions. The simulation of the 2D Quantum Harmonic Oscillator serves as a validation and visual demonstration of the ITE method's convergence process. In conclusion, this work demonstrates the successful application of grid-based computational techniques to fundamental 2D quantum systems, providing valuable insights into their properties, field responses, and the importance of electron correlation, thereby serving as a foundation for further exploration in computational quantum physics.