Computational studies of nonlinear optical responses in quantum-confined systems have become essential for understanding and predicting phenomena such as second-harmonic generation (SHG) and third-harmonic generation (THG). These effects are strongly influenced by quantum confinement, which modifies electronic structures and enhances nonlinear susceptibilities. Time-dependent density functional theory (TDDFT) and nonlinear susceptibility calculations provide powerful tools for investigating these effects in 2D materials and quantum dots under intense electromagnetic fields.

Quantum confinement alters the electronic and optical properties of nanostructures by restricting electron motion to discrete energy levels. In 2D materials like transition metal dichalcogenides (TMDCs), reduced dimensionality leads to strong excitonic effects and large nonlinear optical coefficients. For quantum dots, the discretization of energy levels results in resonant enhancements of nonlinear responses when the incident photon energy matches interband transitions. Computational methods can systematically explore these effects by solving the many-body Schrödinger equation under confinement potentials.

TDDFT is widely used for simulating nonlinear optical responses due to its balance between accuracy and computational efficiency. The approach extends standard density functional theory (DFT) to time-dependent perturbations, allowing the calculation of dynamic polarizabilities and hyperpolarizabilities. For SHG and THG, the nonlinear susceptibilities χ(2) and χ(3) are derived from the induced polarization under an intense laser field. In confined systems, the electronic transitions between quantized states contribute significantly to these susceptibilities. For example, monolayer MoS2 exhibits a large χ(2) value on the order of 10⁻⁷ m/V due to broken inversion symmetry and strong exciton binding.

Nonlinear susceptibility calculations require careful treatment of electron-electron interactions and local field effects. The Sternheimer approach within TDDFT allows direct computation of frequency-dependent hyperpolarizabilities by solving perturbed Kohn-Sham equations. For quantum dots, the nonlinear response is highly sensitive to size and shape. CdSe quantum dots with diameters around 3 nm show THG enhancements when the incident photon energy is near half the bandgap, a consequence of resonant two-photon absorption. Computational studies confirm that the magnitude of χ(3) in such systems can exceed 10⁻¹⁹ m²/V² under optimal conditions.

The role of intense fields in nonlinear response cannot be overlooked. High-intensity laser pulses induce nonperturbative effects such as multiphoton absorption and high-harmonic generation. TDDFT simulations under strong fields reveal that quantum-confined systems exhibit saturation behavior at high intensities due to Pauli blocking. For 2D materials, the SHG intensity initially scales quadratically with the incident field but deviates as the field strength approaches the exciton binding energy. In graphene quantum dots, THG signals show a cubic dependence on field strength up to a threshold, beyond which nonlinear absorption dominates.

Comparative studies between different materials highlight the impact of dimensionality on nonlinear responses. While 2D materials benefit from in-plane confinement and reduced dielectric screening, quantum dots offer tunability through size-dependent bandgaps. For instance, WS2 monolayers exhibit stronger SHG than their bulk counterparts due to enhanced excitonic effects, whereas PbS quantum dots show size-tunable THG peaks in the near-infrared range. Computational models accurately reproduce these trends by incorporating excitonic binding energies and Coulomb interactions.

Challenges remain in accurately simulating high-order nonlinearities and many-body effects. The adiabatic approximation in TDDFT may underestimate excitonic contributions, necessitating hybrid functionals or GW corrections. Additionally, electron-phonon coupling can modulate nonlinear responses in room-temperature conditions, requiring nonadiabatic molecular dynamics approaches. Recent advances in machine learning potentials offer promise for accelerating these calculations while maintaining accuracy.

In summary, computational predictions of nonlinear optical responses in quantum-confined systems rely on advanced TDDFT and susceptibility calculations. These methods reveal how confinement enhances SHG and THG through quantized electronic states and excitonic effects. Both 2D materials and quantum dots exhibit tunable nonlinearities under intense fields, with applications in optoelectronics and photonics. Future developments in computational techniques will further refine predictions and guide the design of nanomaterials with tailored nonlinear properties.