Fundamental Gaps in Finite Systems from Eigenvalues of a Generalized Kohn-Sham Method

We present a broadly applicable, physically motivated, first-principles approach to determining the fundamental gap of finite systems from single-electron orbital energies. The approach is based on using a range-separated hybrid functional within the generalized Kohn-Sham approach to density functional theory. Its key element is the choice of a range-separation parameter such that Koopmans’ theorem for both neutral and anion is obeyed as closely as possible. We demonstrate the validity, accuracy, and advantages of this approach on first, second and third row atoms, the oligoacene family of molecules, and a set of hydrogen-passivated silicon nanocrystals. This extends the quantitative usage of density functional theory to an area long believed to be outside its reach.

Figure 1

${\mathrm{BNL}}^{*}$ HOMO-LUMO gaps (computed using the aug-cc-pVTZ basis set), compared with experimental fundamental gaps [43]. The value of $\gamma$, determined by minimizing $J$, is indicated near each point. Inset: the deviation from experiment of GKS HOMO-LUMO gaps based on ${\mathrm{BNL}}^{*}$ (this work) and MCY3 [12].

Figure 2

Left: ${\mathrm{BNL}}^{*}$ HOMO-LUMO gaps, compared with gaps from experimental (vertical) ionization potentials (IP) [32] and best estimates of vertical electron affinities [33], for the oligoacenes ${\mathrm{C}}_{2+4n}{\mathrm{H}}_{4+2n}$, $n=1$ (benzene) to 6 (hexacene). The value of $\gamma$, determined by minimizing $J$, is indicated near each point. Right: ${\mathrm{BNL}}^{*}$ HOMO and LUMO energies compared to GW [36] and experimental [38] IP and EA of hydrogen terminated Si nanocrystals, as a function of diameter. The values of the tuned range parameter are shown in red. In both systems the cc-pVTZ basis set was used. Geometries were obtained from a B3LYP calculation for the oligoacenes and from Ref. 44 for the Si nanocrystals.