Time-dependent density functional theory calculations for the Stokes shift in hydrogenated silicon clusters

Experimental evidence suggests that the Stokes shift may be considerable in silicon clusters. Multiple theoretical methods have been used to study this problem, with varying results: predicted Stokes shifts can differ in energy by several electron volts, and predicted minimum-energy structures can have either relaxed cores or relaxed outer shells. Here we present the lowest energy configuration for excited states and the Stokes shift for a series of silicon clusters in two separate ways. First, by energy minimization using density functional theory and changing the electronic occupation such that an electron is moved from the highest occupied molecular orbital to the lowest unoccupied molecular orbital; and second, by total-energy minimization using time-dependent density functional theory to calculate the energy of the first electronic transition with non-negligible oscillatory strength.

Figure 1

Schematic illustration of the Stokes shift. The Stokes shift is defined as the difference between the absorption, $\left(1\right)\to \left(2\right)$, and emission, $\left(3\right)\to \left(4\right)$, energies: ${\Omega }_1^0-{\Omega }_1^1$.

Figure 2

Structures of hydrogenated Si clusters.

Figure 3

Stokes shifts calculated by promoting an electron from the HOMO to the LUMO. Our results (PARSEC) are shown along with the previous ABINIT (Ref. 16) and JEEP (Ref. 15) calculations.

Figure 4

Left panel: ground-state structures of ${\text{Si}}_m{\text{H}}_n$ clusters. Right panel: lowest energy structures of ${\text{Si}}_m{\text{H}}_n$ clusters when an electron is promoted from the HOMO to the LUMO within DFT-LDA. For clarity, only the Si atoms are shown for larger clusters.

Figure 5

Stokes shift calculated by minimizing the total $\left(\text{LDA}+\text{TDLDA}\right)$ energy. Our results (TDLDA) are shown along with the previous QMC (Ref. 15) and TDDFTB (Ref. 17) calculations.

Figure 6

Left panel: ground-state structures of ${\text{Si}}_m{\text{H}}_n$ clusters. Right panel: lowest energy structures of ${\text{Si}}_m{\text{H}}_n$ clusters obtained by TDLDA total-energy minimization. For clarity, only the Si atoms are shown for the larger clusters.