2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/139131We calculate the exchange coupling for a double dot system using a novel, numerically exact yet efficient technique, based on finite-element methods. Specifically, we evaluate the exchange coupling both for a quasi-one and a two-dimensional system, also including an applied magnetic field. Our numerical results provide a stringent test for standard approximation schemes (e.g., Heitler-London, Hund-Mulliken, Hubbard), and they show that the standard methods do not have reliable predictive power for real quantum-dot systems.7 pages, 4 figuresMesoscale and Nanoscale Physicstext