Density functional theory has become an indispensable tool for investigating point defects in nanostructures due to its balance between accuracy and computational efficiency. The method provides atomic-scale insights into defect formation mechanisms, charge states, and kinetic processes that govern the behavior of quantum dots, nanowires, and two-dimensional materials. Unlike bulk systems, nanostructures exhibit pronounced quantum confinement and surface effects that significantly alter defect properties.

Three primary types of point defects dominate nanostructure behavior. Vacancies occur when atoms are missing from lattice sites, interstitials form when atoms occupy non-lattice positions, and substitutional defects arise when impurity atoms replace host atoms. The formation energy represents the thermodynamic stability of these defects and follows the general equation:

E_form = E_defect - E_pristine - Σn_iμ_i + q(E_Fermi + E_VBM)

where E_defect and E_pristine are total energies of defective and pristine systems, n_i represents the number of atoms added or removed, μ_i denotes chemical potentials, q is the charge state, E_Fermi is the Fermi level, and E_VBM is the valence band maximum.

In quantum dots, quantum confinement causes defect formation energies to become size-dependent. For CdSe quantum dots smaller than 3 nm, selenium vacancy formation energies decrease by up to 1.2 eV compared to bulk values due to relaxed structural constraints. Interstitial defects show even more pronounced size effects, with formation energy reductions exceeding 2 eV in sub-2 nm particles. The increased surface-to-volume ratio allows more facile defect relaxation.

Nanowires exhibit anisotropic defect behavior depending on orientation. In [001]-oriented ZnO nanowires, oxygen vacancies form more readily along the growth axis than in radial directions, with formation energy differences reaching 0.8 eV for diameters below 5 nm. Substitutional doping shows similar anisotropy - aluminum incorporation at zinc sites occurs preferentially at surface sites in sub-10 nm wires, with formation energies 0.5 eV lower than bulk-like interior positions.

Two-dimensional materials demonstrate unique defect physics due to their reduced dimensionality. In monolayer MoS2, sulfur vacancies create in-gap states 0.3-0.8 eV below the conduction band minimum, depending on local strain environment. The formation energy of these vacancies decreases by approximately 0.4 eV under 2% tensile strain. Interstitial defects in graphene show exceptionally high formation energies exceeding 5 eV due to sp2 bond disruption, explaining their rarity in experiments.

Charge transition levels represent thermodynamic thresholds where defect charge states change. These levels critically influence electronic properties and follow the formalism:

ε(q/q') = (E_defect^q - E_defect^q')/(q' - q) - E_VBM

For silicon vacancies in diamond quantum dots, the (-/0) transition level occurs 1.7 eV above the valence band maximum, shifting upward by 0.3 eV in 3 nm dots compared to bulk diamond. In GaN nanowires, the (2+/+) transition level for nitrogen vacancies moves 0.4 eV closer to the conduction band in 4 nm diameter wires.

Defect-mediated recombination processes govern optoelectronic performance. In CdTe quantum dots, tellurium vacancies introduce trap states that enhance non-radiative recombination by 3 orders of magnitude compared to defect-free dots. The Shockley-Read-Hall model describes this process through:

τ_nr = (σ_n v_th N_t)^-1

where σ_n is capture cross-section, v_th is thermal velocity, and N_t is trap concentration. For selenium vacancies in MoSe2 monolayers, first-principles calculations yield σ_n values of 2×10^-15 cm^2, explaining observed photoluminescence quenching.

The Nudged Elastic Band method enables calculation of diffusion barriers for defect migration. This technique identifies minimum energy paths between initial and final configurations through a series of images. For oxygen vacancy diffusion in TiO2 nanowires, the method reveals anisotropic barriers of 0.7 eV along [110] directions versus 1.2 eV along [001] in 5 nm wires. In silicon quantum dots, interstitial diffusion barriers decrease from 1.5 eV in bulk to 0.9 eV in 3 nm dots due to surface proximity.

Nitrogen-vacancy centers in nanodiamonds demonstrate the complex interplay between defects and nanostructure geometry. DFT calculations show the NV^- center's zero-phonon line shifts by 3 meV per nanometer of diamond size reduction below 10 nm. The vacancy formation energy decreases by 1.8 eV in 2 nm particles compared to bulk, while nitrogen incorporation becomes 0.7 eV more favorable at surface sites. These effects combine to enhance NV center concentration in small nanodiamonds.

Chalcogen vacancies in transition metal dichalcogenides illustrate defect engineering possibilities. In WS2 monolayers, sulfur vacancies create unpaired electrons that can be passivated by oxygen adsorption, with binding energies of 2.1 eV predicted by DFT. The vacancies also modify catalytic properties - calculations show hydrogen adsorption free energy decreases by 0.3 eV at vacancy sites, improving hydrogen evolution reaction activity.

Defect interactions become crucial at high concentrations. In silicon nanowires below 5 nm diameter, vacancy clusters containing 3-4 vacancies become more stable than isolated vacancies by 0.5 eV per vacancy. Similarly, in MoS2 bilayers, sulfur vacancy pairs exhibit 0.9 eV lower formation energy than single vacancies at separations below 1 nm.

Temperature effects can be incorporated through ab initio molecular dynamics or thermodynamic integration. For gold nanoparticles below 2 nm, simulations reveal that interstitial defects become mobile above 300 K, with diffusion coefficients increasing by 6 orders of magnitude between 300-500 K. In carbon nanotubes, vacancy migration barriers decrease by 0.2 eV at 600 K due to lattice vibrations.

Recent methodological advances improve defect modeling accuracy. Hybrid functionals like HSE06 correct band gap underestimation that affects defect level positions. For ZnO nanowires, PBE+U calculations place oxygen vacancy levels 1.2 eV higher than standard PBE, matching experimental observations. Spin-polarized calculations prove essential for transition metal defects - in Fe3O4 nanoparticles, oxygen vacancy formation energies differ by 0.7 eV between spin configurations.

Defect engineering through DFT-guided design has enabled numerous applications. In silicon quantum dots, controlled phosphorus substitution at specific sites can enhance quantum yield by 40% according to computational predictions. For TiO2 nanowire photocatalysts, nitrogen doping at oxygen vacancy sites reduces the band gap by 0.5 eV while maintaining favorable charge carrier lifetimes.

The continued development of computational approaches promises deeper understanding of defects in nanostructures. Machine learning potentials trained on DFT data enable larger-scale defect simulations without sacrificing accuracy. These methods have successfully predicted defect migration pathways in 10 nm silicon nanoparticles that would be prohibitively expensive with conventional DFT. Coupled with experimental validation, such computational tools will further establish defect engineering as a cornerstone of nanotechnology development.