Effective interactions, large-scale diagonalization, and one-dimensional 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. PACS numbers: 73.21.La, 31.15.Pf

Harmonic oscillator eigenfunction expansions, quantum dots, and effective interactions

“…It generalizes and refines the findings of a recent study of one-dimensional systems 2 and is applicable to, for example, nuclear systems 3 and quantum chemical calculations 4 as well. We demonstrate the estimates with calculations in the d = 2 dimensional case for N Յ 5 electrons, paralleling computations in the literature.…”

Three interacting atoms in a one-dimensional trap: a benchmark system for computational approaches

“…However, this path may be unpractical for more fermions, due to the complexity of two-body interaction matrix elements represented in this basis. Some authors proposed effective-interaction schemes or similar techniques borrowed by nuclear physics [85][86][87][88]. The interested reader may consult the recent review [89].…”

Addition and removal energies of circular quantum dots

“…In the present work, we focus on the latter. As derived by Kvaal 72,73 , the asymptotic convergence of quantum dot observables in a finite harmonic oscillator basis can be approximately described by a power law model:…”