Fundamental gaps of quantum dots on the cheap

We show that the fundamental gaps of quantum dots can be accurately estimated at the computational effort of a standard ground-state calculation supplemented with a non-self-consistent step of negligible cost, all performed within density-functional theory at the level of the local-density approximation.

Figure 1

EXX-KLI results for the fundamental gaps computed according to $G_{\mathrm{E},\mathrm{N}}$ [Eq. (1)], $G_{\varepsilon ,\mathrm{N}}$ [Eq. (2)], and $G_{\mathrm{\Delta },\mathrm{N}}$ [Eq. (3) together with Eq. (22)]. For each $N$, the bars from left to right correspond to $\omega =0.50$, 1.50, and 2.50, respectively, and $\alpha =1.05$ [see Eq. (25)].

Figure 2

Fundamental gaps $G_{\mathrm{\Delta },N}$ obtained with the exchange-only KLI and LDA approximations, respectively, for elliptic quantum dots [Eq. (25) with $\alpha =1.05]$ with $N=12$ electrons and varying confinement strength $\omega$. The contributions of the Kohn-Sham gap [Eq. (4)] are marked by shaded open boxes. The remaining part is given by the derivative discontinuity, that is, Eq. (22) and Eq. (23) in the case of KLI and LDA, respectively. All the numerical results are given in Table 1.

Figure 3

Same as Fig. (2) but for a fixed value of the confinement strength $\omega =0.5$ and varying number of electrons $N$.

Figure 4

Fundamental gaps including correlations for parabolic quantum dots [Eq. (25) with $\alpha =1]$ with a fixed confinement strength of $\omega =0.35$ and variable number of electrons $N$. $G_{E,N}^{\mathrm{MB}}$ is the full configuration interaction value from Ref. [19]; $G_{\epsilon ,N}^{\mathrm{LDA}}$ is obtained from Eq. (2) at the LDA level; $G_{\mathrm{\Delta },N}^{\mathrm{LDA}}$ from Eq. (24). See also Table 2.