Harmonic oscillator eigenfunction expansions, quantum dots, and effective interactions

We give a thorough analysis of the convergence properties of the configuration-interaction method as applied to parabolic quantum dots among other systems, including a priori error estimates. The method converges slowly in general, and in order to overcome this, we propose to use an effective two-body interaction well known from nuclear physics. Through numerical experiments we demonstrate a significant increase in accuracy of the configuration-interaction method.

Figure 1

Illustration of ${\mathcal{P}}_{R=4}\left({\mathbb{R}}^2\right)$: (left) Fock-Darwin orbitals. (right) Hermite functions. Basis functions with equal HO energy are shown at same line.

Figure 2

Eigenvalues for $N=4$, $S=0$, and $\lambda =2$ as function of $R$ for $H$ (solid) and $H_{\text{eff}}$ (dashed). $M=0$,1, and 2 are represented by squares, circles, and stars, respectively.

Figure 3

Plot of relative error using the bare interaction for various $N$, $M$, and $S$. Clear $o\left(R^{\alpha }\right)$ dependence in all cases.

Figure 4

Plot of relative error using an effective interaction for various $N$, $M$, and $S$. Clear $o\left(R^{\alpha }\right)$ dependence in all cases, but notice artifacts when $R$ is large, due to errors in most correct eigenvalues. The $R=5$ case does not contain enough data to compute the slopes with sufficient accuracy.