This dissertation studies the computational modeling for nanostructures in response to external electromagnetic fields. Light-matter interactions on nanoscale are at the heart of nano-optics. To fully characterize the optical interactions with nanostructures quantum electrodynamics (QED) must be invoked, however, the required extremely intense computation and analysis prohibit QED from applications in nano-optics. To avoid the expensive computations and be able to seize the essential quantum effects a semiclassical model is developed. The wellposedness of the model partial differential equations is established. Emphasis is placed on the optical interactions with an individual nanostructure, excitons and biexcitons effects and finite-size effects are investigated.

The crucial step of our model is to couple the electromagnetic fields with the motion of the excited particles to yield a new dielectric constant which contains quantum effects of interest. A novel feature of the dielectric constant is the wavevector-dependence which leads to a multi-wave propagation inside the medium. Additional boundary conditions are proposed to deal with this situation. We proceed with incorporating this dielectric constant to Maxwell's equations, and by solving a scattering problem the quantum effects can be captured in the scattered spectra.