Electronic and other elementary excitations play a central role in understanding and controlling the properties of nanostructures. Yet at present simulation methods are inadequate to describe these properties, being limited to either relatively small molecules or unit cells. In this proposal, theoretical methods to treat elementary excitations will be developed that merge molecular and solid state methods to address the special challenges of large disordered nanostructures. This work will adapt molecular methods to treat large numbers of particles, and solid state methods to treat systems with disorder, and combine each with embedding methods that treat extended environmental effects. Underlying all of these simulation methods are common mathematical kernels from numerical linear algebra and optimization theory that will be addressed by close interactions between simulation scientists and mathematicians. Furthermore, the problem of extending methods that describe electronic excitations to nanostructures is fundamentally one of treating electronic interactions of differing strength at different levels of resolution. This is accomplished by physical approximations above, but is essentially a more fundamental challenge. Thus, an additional central element of the proposal is close interaction between physical scientists and mathematicians to develop new multiresolution approaches to describing the behavior of electrons in nanomaterials. The work described here will specifically advance modeling of optical response, charge transport, coupling between radiation and nanomotion in nanostructures of all kinds, as well as providing broader impacts from the improvements in electronic structure simulation methodology and fundamental algorithms in applied mathematics.