Quantum Monte Carlo calculations of the optical gaps of Ge nanoclusters using core-polarization potentials

We report the results of a study of the ground-state and excitation energies of Ge atoms, molecules, and clusters as large as ${\mathrm{Ge}}_{29}{\mathrm{H}}_{36}$, using quantum Monte Carlo (QMC) for the valence electrons. QMC is one of the most accurate many-body methods; however, its accuracy is limited by the way the core is treated, which is especially important for atoms with shallow cores such as Ge. Here we treat the Ge core with a relativistic Hartree-Fock pseudopotential plus a core polarization potential (CPP) to take into account core-valence correlation at a many-body level. The CPP is found to be important for accurate calculations of the total energy, and for excitations of atoms and small molecules; however, there are only small changes in the lowest optical excitations of larger clusters, which can be understood in terms of the nature of the excitations. The results are compared with previous QMC calculations for the corresponding Si molecules and clusters. In addition, we find that QMC gaps are higher than those found in recent time-dependent density functional studies, by amounts similar to that previously found for Si systems.

Figure 1

(Color online) The ${\mathrm{Ge}}_{29}{\mathrm{H}}_{36}$ nanocluster with diameter approximately $12\mathrm{\AA }$.

Figure 2

(a) Comparison of our present $\mathrm{DMC}+\mathrm{CPP}$ results for the lowest gaps of the ${\mathrm{Ge}}_n{\mathrm{H}}_m$, $(n+m)⩽65$, nanoclusters (●) to the TDLDA results of Nesher et al. (Ref. 20) (◆). The TDLDA results are “effective gaps” as described in the text. (b) Comparison of our DMC results for ${\mathrm{Ge}}_n{\mathrm{H}}_m$ (●) to the DMC results for the corresponding Si nanoclusters by Porter et al. (Ref. 6) (▵) and Williamson et al. (Ref. 7) (▿). We have also included our own DMC results (엯) for ${\mathrm{Si}}_5{\mathrm{H}}_{12}$ and ${\mathrm{Si}}_{10}{\mathrm{H}}_{16}$.