Single atoms show discrete energy states. Bulk semiconductors show energy bands. So one would expect the band gap behaviour to change with the size of the solid. For example one can tune the color of the emission of colloidal quantum dots just by varying their size.
Brus developed a model to predict the band gap energy of nanocrystallites, quantum dots and other small spherical structures with radius R. Roughly speaking, the approach is to start with the bulk value of the energy gap, add a "particle-in-a-box" like term for electrons and holes using an effective mass approximation for the masses of both and subtracting a term, which corresponds to the Coulomb-attraction between electrons and holes. The particle-in-a-box-term scales with 1/R² and the attraction term scales with 1/R, so in small structures the 1/R²-term dominates.
However this approach uses effective masses, which do not depend on the size of the structure. Therefore the results are ok for structures of intermediate size, but are rather wrong for very small structures.