Ab InitioCalculations for Large Dielectric Matrices of Confined Systems

Calculations for optical excitations in confined systems require knowledge of the inverse screening dielectric function ${ϵ}^{-1}(\mathbf{r},{\mathbf{r}}^{\mathbf{\prime }})$, which plays a crucial role in determining exciton binding energies. We present a new efficient real-space method of inverting and storing large ab initio dielectric matrices of confined systems, which relies on the separability of $ϵ$ matrix in $\mathbf{r}$ and ${\mathbf{r}}^{\mathbf{\prime }}$. The method has allowed, for the first time, full ab initio calculation of ${ϵ}^{-1}(\mathbf{r},{\mathbf{r}}^{\mathbf{\prime }})$ of dimension $N\sim 270000$, and for quantum dots as large as ${\mathrm{S}\mathrm{i}}_{35}{\mathrm{H}}_{36}$. The effective screening in Si quantum dots up to 1.1 nm in diameter is found to be very ineffective with average dielectric constants ranging from 1.1 to 1.4.