Generation of Coulomb matrix elements for the 2D quantum harmonic oscillator

Abstract

The Fock-Darwin states are the natural basis functions for a system of interacting electrons trapped inside a 2D quantum dot. Interaction effects at the mean field level or more elaborate quantum many body descriptions rely on an accurate evaluation of Coulomb matrix elements. In this work, we derive a highly efficient recurrence scheme to compute these elements in Fock-Darwin basis. The algorithm is best implemented on symbolic calculus platforms, preventing the appearance of rounding-off errors. The high speed achieved allows us to calculate all matrix elements in a basis set of several hundred states in very reasonable times (a few hours on a standard computer). We also make use of symmetry to reduce the number of elements to be computed. Finally, we check the reliability of floating-point evaluation for existing analytical expressions. We confirm the stability of a two-folded nonalternating sum for elements involving only the lowest energy levels and put on display the daunting limitations of the expressions commonly used to compute the elements in the general case.