Plasmonic nanoparticles, particularly gold (Au) and silver (Ag), exhibit unique optical properties due to their localized surface plasmon resonances (LSPRs). These resonances arise from the collective oscillation of conduction electrons when excited by electromagnetic radiation, leading to phenomena such as near-field enhancement, hot carrier generation, and strong light absorption. Density functional theory (DFT) provides a quantum mechanical framework for understanding these effects at the atomic and electronic levels, complementing classical electrodynamic approaches like Mie theory or finite-difference time-domain (FDTD) simulations.

DFT calculates the electronic structure of plasmonic nanoparticles by solving the Kohn-Sham equations, which approximate the many-body Schrödinger equation using an exchange-correlation functional. For plasmonic systems, the electron density response to an external field is critical. Time-dependent DFT (TDDFT) extends this framework to excited states, enabling the prediction of LSPRs by modeling the frequency-dependent polarizability of the nanoparticle. The plasmon resonance energy depends on the nanoparticle's size, shape, and dielectric environment, all of which influence the electron density distribution.

A key challenge in DFT modeling of plasmonic nanoparticles is accurately describing the size-dependent dielectric function. Bulk dielectric functions, such as those from the Drude model, fail at the nanoscale due to quantum confinement and surface effects. For nanoparticles below 10 nm, the electron mean free path becomes comparable to the particle dimensions, leading to increased surface scattering and damping. DFT accounts for these effects by explicitly treating the electronic states, but computational cost limits simulations to clusters of a few hundred atoms. Hybrid functionals, such as PBE0 or HSE, improve the description of excited states but require significant resources.

Hot carrier generation, a process where plasmon decay produces high-energy electrons and holes, is another area where DFT provides insights. The non-radiative decay of plasmons can generate carriers with energies above the Fermi level, which are relevant for photocatalysis and photodetection. DFT calculations reveal the distribution of these carriers and their relaxation pathways, which depend on the nanoparticle's band structure and coupling to phonons. For example, in Au nanoparticles, hot electrons are primarily generated near the interband transition threshold (~2.4 eV), while Ag nanoparticles exhibit stronger plasmonic efficiency in the visible range due to lower damping.

Near-field enhancement, the dramatic increase in electric field intensity near the nanoparticle surface, is crucial for surface-enhanced spectroscopy and sensing. DFT can predict the spatial distribution of this enhancement by analyzing the induced charge density under external excitation. However, classical methods often outperform DFT for larger nanoparticles due to computational constraints. A combined approach, where DFT informs the local dielectric response and classical methods handle the far-field optics, is increasingly used.

Nanoparticle dimers introduce additional complexity due to plasmon coupling. When two nanoparticles approach within a few nanometers, their plasmons hybridize, forming bonding and antibonding modes. DFT captures this interaction by modeling the charge transfer and induced dipole moments between particles. The gap distance plays a critical role: for sub-nanometer separations, quantum tunneling effects dominate, which classical models cannot describe. Substrate effects further complicate the picture, as the dielectric environment modifies the plasmon resonance and near-field distribution. DFT simulations of nanoparticles on graphene or metal oxides reveal charge redistribution at the interface, influencing both the resonance energy and hot carrier dynamics.

Despite its strengths, DFT has limitations in modeling plasmonic systems. The computational expense restricts simulations to small nanoparticles or simplified geometries, making it impractical for systems larger than 5 nm. Additionally, standard exchange-correlation functionals often underestimate plasmon damping and fail to capture long-range electron correlations critical for extended systems. Advances in orbital-free DFT and machine learning potentials aim to bridge this gap, enabling larger-scale simulations while retaining quantum accuracy.

In summary, DFT provides a powerful tool for studying plasmonic nanoparticles at the quantum level, offering insights into LSPRs, hot carrier generation, and near-field effects. While challenges remain in scaling to larger systems and improving dielectric function descriptions, the method is indispensable for understanding the fundamental physics of plasmonics. Combined with classical approaches, DFT helps design nanoparticles for applications in sensing, energy conversion, and nanophotonics.