Density functional theory has become an indispensable tool for studying quantum dots due to its ability to predict electronic and optical properties from first principles. The method provides insights into quantum confinement effects, bandgap tuning, and surface interactions that are critical for designing quantum dots with tailored functionalities. By solving the Kohn-Sham equations, DFT approximates the many-body quantum mechanical problem into a tractable form, enabling systematic investigations of nanostructures with varying compositions, sizes, and surface chemistries.

Bandgap engineering is one of the most important applications of DFT in quantum dot research. The bandgap of semiconductor quantum dots exhibits strong size dependence due to quantum confinement, where charge carriers are spatially restricted to dimensions smaller than their excitonic Bohr radius. DFT calculations accurately capture this phenomenon by modeling how electronic states shift as the dot diameter decreases. For CdSe quantum dots, DFT predicts a bandgap increase from 1.74 eV in bulk to 2.5 eV for 2 nm diameter dots, consistent with the quantum confinement effect. The calculations reveal that the valence band maximum, composed primarily of Se 4p orbitals, shifts downward in energy while the conduction band minimum, dominated by Cd 5s states, moves upward as size decreases. This size-dependent tuning enables precise control over optical absorption and emission wavelengths.

Electronic structure calculations using DFT provide detailed information about the density of states, charge distribution, and orbital character in quantum dots. The method shows how quantum confinement transforms continuous bulk energy bands into discrete atomic-like states as dot size reduces. For PbS quantum dots, DFT reveals multiple degenerate states near the band edges that arise from the high symmetry of the rock salt crystal structure. The calculations demonstrate that these degeneracies can be broken through shape anisotropy or surface modifications. DFT also predicts the emergence of surface states that can trap charge carriers and affect optoelectronic performance. By analyzing partial charge densities, researchers can identify whether these states originate from undercoordinated atoms or adsorbates at the quantum dot surface.

Optical properties prediction represents another key application of DFT, despite some inherent limitations. The method calculates dipole-allowed transitions between occupied and unoccupied states, providing insights into absorption spectra. For CdSe quantum dots, DFT correctly predicts the blue shift of absorption onset with decreasing size and reproduces experimental trends in molar extinction coefficients. However, standard DFT implementations using local or semi-local exchange-correlation functionals systematically underestimate bandgaps due to the self-interaction error and lack of derivative discontinuity in the exchange-correlation potential. This limitation can be partially addressed through hybrid functionals that incorporate exact Hartree-Fock exchange or by applying scissor operators based on empirical corrections.

Size-dependent quantization effects are particularly well-suited for DFT investigation. The theory naturally captures how reduced dimensionality affects electronic wavefunctions and energy level spacing. Calculations show that for spherical quantum dots below 5 nm diameter, the energy difference between the highest occupied and lowest unoccupied molecular orbital follows a 1/R² dependence, where R is the dot radius. This relationship holds for various semiconductor materials including CdSe, PbS, and InP. DFT also reveals how quantum confinement affects carrier effective masses, with calculations showing increased masses in smaller dots due to stronger interaction with the confining potential.

Surface passivation plays a critical role in quantum dot properties, and DFT provides valuable insights into these effects. The method can model different passivation schemes, from organic ligands to inorganic shells, and predict their influence on electronic structure. For CdSe quantum dots, DFT shows that thiolate ligands remove surface states from the bandgap region by bonding with undercoordinated Cd atoms. Similarly, calculations demonstrate that ZnS shell growth on CdSe cores reduces non-radiative recombination by passivating Se dangling bonds. DFT also predicts how varying ligand coverage affects charge transport, with fully passivated surfaces showing localized states while partially covered surfaces exhibit delocalized states near band edges.

Case studies of semiconductor quantum dots highlight DFT's capabilities and limitations. For CdSe, DFT calculations reproduce the experimental trend of bandgap versus size and correctly predict the crystal field splitting of valence band states. The method also explains the origin of bright and dark exciton states that affect photoluminescence quantum yield. In PbS quantum dots, DFT reveals unique properties arising from the large dielectric constant and small effective mass, including enhanced dielectric screening and reduced exciton binding energies compared to II-VI semiconductors. The calculations show how PbS quantum dots maintain strong quantum confinement despite their relatively large excitonic Bohr radius due to the pronounced effect of size on electronic states.

DFT faces significant limitations in predicting excitonic effects accurately. The method's single-particle nature cannot properly describe electron-hole interactions that dominate optical processes in quantum dots. While DFT can predict absorption onsets, it fails to capture exciton binding energies that can exceed hundreds of meV in strongly confined systems. This limitation becomes particularly apparent when comparing DFT results with experimental photoluminescence spectra, where excitonic peaks appear at energies below the calculated bandgap. Time-dependent DFT offers some improvement by including electron-hole correlations, but still struggles with charge transfer excitations and multi-exciton generation processes.

The choice of exchange-correlation functional significantly impacts DFT's predictive power for quantum dots. Local density approximation tends to underestimate bandgaps by 30-50%, while generalized gradient approximation provides slightly better results but still shows systematic errors. Hybrid functionals like B3LYP or PBE0 improve agreement with experiment by including exact exchange, but at increased computational cost. Range-separated hybrids such as CAM-B3LYP offer better performance for charge transfer states but require careful parameterization. Recent developments in GW approximation and Bethe-Salpeter equation approaches provide more accurate treatment of excited states but remain computationally prohibitive for large quantum dot systems.

Despite its limitations, DFT continues to provide valuable guidance for quantum dot research through relatively accessible computations. The method enables high-throughput screening of potential quantum dot materials by predicting trends in bandgap, effective masses, and surface chemistry effects. DFT-derived parameters often serve as inputs for more sophisticated many-body calculations or empirical models. As computational resources expand and theoretical methods advance, DFT remains a foundational tool for understanding and designing quantum-confined semiconductor nanostructures with tailored electronic and optical properties.