Semiconductor materials can be synthesized as nanocrystals: quantum dots, quantum rods, nanowires, etc. When the size of these nanocrystals are in a few nanometers, their electronic structures will change due to quantum confinement effects. Besides, due to the confined space, the electronhole and electronelectron interactions will also change. This can lead to larger exciton binding energy, enhanced Auger effect, or multiexciton generation. The calculated band gaps, optical absorption spectrum, and multiexciton effects and exciton binding energies can be compared with experiments. Besides, in a confined environment, the electronphonon interaction is also different from the bulk counter part, which will change the carrier dynamics, including hot carrier cooling and phonon assisted carrier transport.
We have developed several methods to calculate the electronic structure of inorganic nanocrystals. All our calculations are based on plane wave as the basis functions for the single electron wave functions. The first method is the folded spectrum method (FSM), which can be used to solve a few eigen states near the band gap edge of a nanosystem for a given single particle Hamiltonian. The FSM avoids the calculation of thousands of occupied state below the band gap. There are two methods to generate the single particle Hamiltonian. One is the empirical pseudopotential method (EPM), which represent the total potential in the single particle Hamiltonian as a summation of the individual atomic potentials, while these atomic potentials are fitted to experimental band structures. The second approach, which we have used extensively, is the charge patching method (CPM). This method generates the atomic charge motifs (which are the charge density belongs to one atom in a given bonding environment) from small prototype calculations, then use such motifs to generate the charge density of a nanosystems without doing selfconsistent calculations. The Poisson equation is used to generate the DFT like potential from the charge density. Some empirical correction on the s, p, d nonlocal pseudopotentials, or GW correction can be used to correct the DFT band gap error. These methods can be used to calculate nanosystems with a few thousand to tens of thousands of atoms.
For embedded quantum dots (e.g., InAs in GaAs), or semiconductor devices, we have also developed a linear combination of bulk band (LCBB) method. This method uses the EPM as its Hamitlonian, but instead of using the original plane wave basis to expand the electron wave function, it uses the bulk Bloch states to expand the wave function, as a result, a few thousand basis functions can often be enough to describe the nanostructure states. The LCBB method can be used to calculate systems with millions of atoms.
Finally, for large system (e.g, >10,000 atoms) selfconsistent DFT calculations, we have a linear scaling three dimensional fragment (LS3DF) method. This is a divideandconquer DFT method for large systems. It can be used to relax the atoms, and calculate the dipole moments etc.
A few examples are given here for the systems calculated using these methods.
The CBM state in a CdTe quantum dot calculated using CPM.

The electron wave function in a GaAlAs alloy with 2 million atoms.

The optical absorption spectrum for a ZnO/ZnS core/shell nanowire. It can be used to reduce their optical absorption band gap, hence make it possible for solar cell.

The change of the VBM nature of a CdSe quantum rod when its aspect ratio increases.

The QD band gaps versus the QD size

The CdSe quantum dot excited states and comparison with experiment

CdTe/CdSe nanojunction and interface exciton

ZnO nanorod, and its surface dipole moment as calculated by the LS3DF method.

A CdS/CdSe core/shell nanorod, where the electron is localized on the CdSe shell, the electron is localized in CdS core.
