Effective interactions and large-scale diagonalization for quantum dots

Abstract

The widely used large-scale diagonalization method using harmonic oscillator basis functions (an instance of the Rayleigh-Ritz method, also called a spectral method, configuration-interaction method, or ``exact diagonalization'' method) is systematically analyzed using results for the convergence of Hermite function series. We apply this theory to a Hamiltonian for a one-dimensional model of a quantum dot. The method is shown to converge slowly, and the non-smooth character of the interaction potential is identified as the main problem with the chosen basis, while on the other hand its important advantages are pointed out. An effective interaction obtained by a similarity transformation is proposed for improving the convergence of the diagonalization scheme, and numerical experiments are performed to demonstrate the improvement. Generalizations to more particles and dimensions are discussed.

7 figures, submitted to Physical Review B Single reference error fixed

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• Effective-interaction approach to the many-boson problem
• Three interacting atoms in a one-dimensional trap: A benchmark system for computational approaches
• Ab initio computation of circular quantum dots
• Harmonic oscillator eigenfunction expansions, quantum dots, and effective interactions
• Coulomb interaction-driven entanglement of electrons on helium
• Basis-set correction based on density-functional theory: Rigorous framework for a one-dimensional model
• Addition and removal energies of circular quantum dots
• From Divergent Series to Geometry: Resurgence of the Quantum Metric