Plasmonic excitations in nanostructures arise from the collective oscillation of conduction electrons when interacting with electromagnetic fields. At length scales approaching the sub-nanometer regime, quantum mechanical effects become dominant, necessitating advanced theoretical approaches beyond classical electrodynamics. Classical models, such as Maxwell's equations coupled with the Drude-Lorentz dielectric function, provide a reasonable description of plasmonic behavior in larger nanostructures but fail to capture critical quantum phenomena at atomic scales. Quantum mechanical methods, including time-dependent density functional theory (TD-DFT) and ab initio techniques, are essential for accurately modeling plasmonic excitations in this regime.

Classical descriptions of plasmons rely on the local response approximation (LRA), which assumes that the induced charge density at a given point depends only on the electric field at that same point. This approximation breaks down at sub-nanometer separations, where electron tunneling and nonlocal screening effects become significant. For instance, in metallic dimers with gaps below 1 nm, classical models overestimate field enhancements by orders of magnitude because they neglect the quantum mechanical nature of electron transport. Nonlocal effects, where the electron response depends on the electric field over a finite spatial range, must be incorporated to correct these inaccuracies.

Time-dependent density functional theory (TD-DFT) is a powerful quantum mechanical framework for modeling plasmonic excitations. TD-DFT extends density functional theory (DFT) to time-dependent systems, enabling the calculation of dynamic polarizabilities and optical absorption spectra. The method treats electron-electron interactions through an exchange-correlation potential, which can be approximated at various levels of accuracy. For plasmonic systems, adiabatic local density approximation (ALDA) is often used, but more advanced functionals, such as the adiabatic generalized gradient approximation (AGGA) or hybrid functionals, improve the description of electron correlation and excitation energies. TD-DFT captures phenomena like plasmon damping due to electron-electron scattering and Landau damping, which are absent in classical models.

Ab initio methods, such as the random phase approximation (RPA) and many-body perturbation theory (GW-BSE), provide even higher accuracy for plasmonic excitations. The RPA describes the dielectric response by summing over all possible electron-hole excitations, while the GW approximation corrects single-particle energies by accounting for dynamic screening. The Bethe-Salpeter equation (BSE) then incorporates electron-hole interactions, crucial for excitonic effects that can hybridize with plasmons in certain materials. These methods are computationally demanding but necessary for systems where electron correlation and nonlocal screening play pivotal roles.

Quantum corrections to classical models are often employed to bridge the gap between full quantum treatments and computationally efficient classical approaches. The quantum corrected model (QCM) introduces an effective electron density-dependent dielectric function that accounts for electron spill-out and tunneling. Similarly, the hydrodynamic model (HDM) extends the LRA by including a nonlocal term proportional to the gradient of the induced charge density. These corrections are particularly useful for systems like metallic nanoparticles or thin films, where full ab initio calculations may be prohibitively expensive.

Electron tunneling is a quintessential quantum effect that classical models entirely miss. In sub-nanometer gaps, electrons can tunnel between adjacent nanostructures, leading to charge transfer plasmons (CTPs) and modified resonance frequencies. TD-DFT and ab initio methods naturally incorporate tunneling by explicitly treating the electronic wavefunctions and their overlap. For example, in gold nanoparticle dimers with sub-nanometer gaps, tunneling-induced damping redshifts the plasmon resonance and reduces field enhancement compared to classical predictions.

Nonlocal effects further complicate plasmonic behavior at atomic scales. The electron density near a metal surface exhibits spill-out beyond the classical boundary, altering the local optical response. Nonlocal models, such as the specular reflection model (SRM) or the semiclassical infinite barrier model (SCIB), introduce spatially dependent dielectric functions to account for this spill-out. These models show that nonlocality blueshifts plasmon resonances and reduces field confinement, effects that are experimentally observable in electron energy loss spectroscopy (EELS) and optical scattering measurements.

Quantum confinement also plays a role in very small nanoparticles, where discrete electronic states replace the continuum assumed in classical models. For clusters with diameters below 2 nm, the plasmon resonance splits into multiple peaks corresponding to transitions between quantized energy levels. TD-DFT and ab initio methods accurately predict these transitions, whereas classical models fail to resolve the discrete electronic structure.

Comparisons between quantum and classical descriptions reveal stark differences in predicted plasmonic properties. Classical models typically overestimate field enhancements in sub-nanometer gaps by factors of 10 to 100 due to ignoring tunneling and nonlocality. Quantum calculations, in contrast, show that field enhancement saturates or even decreases at atomic separations. Similarly, classical models predict a smooth scaling of plasmon resonance energy with particle size, while quantum methods capture size-dependent oscillations due to electronic shell effects.

The choice between quantum and classical approaches depends on the system size and the desired level of accuracy. For nanostructures larger than 10 nm, classical electrodynamics suffices for most applications. However, for sub-nanometer features or precise predictions of optical response, quantum mechanical methods are indispensable. Hybrid approaches, combining classical electromagnetics with quantum corrections, offer a practical compromise for intermediate systems.

In summary, quantum mechanical approaches like TD-DFT and ab initio methods are essential for modeling plasmonic excitations at sub-nanometer scales, where classical descriptions fail. These methods account for electron tunneling, nonlocal effects, and quantum confinement, providing accurate predictions of optical response and field enhancements. While computationally intensive, they are necessary for advancing applications in nanophotonics, sensing, and quantum optics, where atomic-scale precision is critical. Future developments in efficient algorithms and computational power will further bridge the gap between quantum accuracy and practical simulations of plasmonic systems.