Effective interactions, large-scale diagonalization, and one-dimensional quantum dots

The widely used large-scale diagonalization method using harmonic oscillator basis functions (an instance of the Rayleigh-Ritz method [S. Gould, Variational Methods for Eigenvalue Problems: An Introduction to the Methods of Rayleigh, Ritz, Weinstein, and Aronszajn (Dover, New York, 1995)], 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 nonsmooth 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.

Figure 1

Two-body model space defined by a cut in energy. The two-body state has quantum numbers $n_1$ and $n_2$, the sum of which does not exceed $N_{\max }$.

Figure 2

(Top) Coefficients $\mid c_n\mid$ of the function $\exp (-x^2∕2)∕(\mid x\mid +\delta )$ for $\delta ∊[0.01,2]$, $n=0,2,\dots ,5000$. (Bottom) Estimated decay rate $\alpha$ of $\mid c_n\mid$, i.e., slope of the graphs in the left panel.

Figure 3

(Left) Symmetric double-well potential created with the Gaussian perturbation $A\exp [-C{(x-\mu )}^2]$, with $A=4$, $\mu =0$, and $C=2$. (Right) Asymmetric double-well potential created with the Gaussian perturbation with $A=4$, $\mu =0.75$, and $C=2$, and single-well potential using $C=0.25$.

Figure 4

(Top) Coefficients of the exact ground states of the Hamiltonians ${\widehat{H}}_{t,i}$. For ${\widehat{H}}_i$, only even-numbered coefficients are nonzero and, thus, displayed. The almost straight line indicates approximately $o\left(n^{-1.57}\right)$ decay of the coefficients around $n=600$, and $o\left(n^{-1.73}\right)$ around $n=5000$ (cf. Fig. 2). For the ${\widehat{H}}_t$ ground state, we clearly have spectral convergence. (Bottom) The error in the ground state energies and wave functions when using the CI method. For ${\widehat{H}}_i$, we have $o\left(n^{-1.24}\right)$ decay for the energy error, and $o\left(n^{-1.20}\right)$ decay for the wave function error, both evaluated at $n=600$. For ${\widehat{H}}_t$, we clearly have spectral convergence. A full two-particle CI calculation is superimposed, showing that the error in the interaction part of the Hamiltonian (2) completely dominates. Here, the error in the energy is $o\left(N_{\max }^{-1.02}\right)$ at $n=70$, while for ${\widehat{H}}_i$ alone, we have $o\left(N_{\max }^{-1.01}\right)$.

Figure 5

(Top) Plot of ground state coefficients of ${\widehat{H}}^{\prime }$ and ${\widehat{H}}_{\mathrm{eff}}$. (Bottom) Pointwise error (in relative coordinate $x$) of effective Hamiltonian ground state ${\psi }_{\mathrm{eff}}\left(x\right)$.

Figure 6

Ground state energy relative error for a two-particle simulation using the confinement potential $V\left(x\right)=x^2∕2+4\exp [-2{(x-0.75)}^2]$. For the CI method without effective interactions, we obtain $\alpha \approx -1.02$, while the effective interactions give $\alpha \approx -2.57$. The electron density is superimposed on the potential plot.

Figure 7

Estimates of $\alpha$ for CI calculations with (bottom) and without (top) effective interactions.