Using state of the art time-dependent density functional theory and multireference second-order perturbation theory, we have accurately calculated (within 0.3 eV) in real space the optical gap of small silicon quantum dots, with diameters up to 25 Å. Our results, which support the quantum confinement hypothesis, are in excellent agreement with recent and earlier experimental data on oxygen-free samples and the conclusions of Wilcoxon et al. [Phys. Rev. B 60, 2704 (1999)]. We have found that the diameter of the smallest dot, which could emit photoluminescence in the visible region of the spectrum, is around 22 Å. Our work can resolve existing controversies and bridge diverse experimental and theoretical results.

The equivalence between localized states in disordered $d$-dimensional systems and bound states in local potential wells (or local potential fluctuations) in $d$-dimensional space is tested against numerical data. The agreement is found to be satisfactory. A new criterion for the onset of localization is deduced. It reduces to Mott's criterion at the center of the band but it differs qualitatively from Mott's picture when the mobility edge is near the band edges.

We considered a muffin-tin periodic potential (-δ inside the spheres and zero outside). Using the augmented-plane-wave method we studied systematically the location of gaps for positive energy and for various values of δ and the sphere radius a. Our results are applicable to the problem of classical wave propagation in composite media and relevant to the problem of optical localization.

Extensive numerical results using finite-size-scaling methods for two-dimensional (2D) and three-dimensional (3D) disordered systems are presented for eigenenergies E different from zero. Single-parameter scaling law is satisfied for all of our results in both 2D and 3D systems. In 2D we find only localized states, however, the localization length does not have its maximum at the band center. In 3D, the dependence of the mobility edge on the strength of the diagonal order is obtained. These results are found to be in satisfactory agreement with the predictions of the coherent-potential approximation and the potential-well analogy.

Various structural possibilities for the Si₂C₄ and Si₄C₂ clusters are investigated by employing a basis set of triple‐zeta plus polarization quality; electron correlation is generally accounted for by second‐order Mo/ller–Plesset and, in certain instances, by higher‐order perturbation (CASPT2) approaches. The building‐up principle recently suggested from an analysis of Si₃C₃ clusters is found to be fully operative for Si₂C₄ and Si₄C₂ clusters. A comparison of the structure and stability of various geometrical arrangements in the series C₆, Si₂C₄, Si₃C₃, Si₄C₂, and Si₆ shows that linear and planar structures become rapidly less stable if carbons are replaced by silicons and that the three‐dimensional bipyramidal forms become less favorable as soon as silicons are exchanged by carbons in the parent Si₆ structure. The effects can be rationalized in qualitative terms based on differences in silicon and carbon bonding.