Failure of standard approximations of the exchange coupling in nanostructures

We calculate the exchange coupling for a double dot system using a numerically exact technique based on finite-element methods and an expansion in two-dimensional Gaussians. 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 of standard approximation schemes (e.g., Heitler-London, Hund-Mulliken, Hubbard), and they show that the standard methods do not have reliable predictive power even for simple model systems. Their value in modeling more realistic quantum-dot structures is thus cast in serious doubt.