The size of metallic nanoparticles significantly affects their optical properties due to a phenomenon known as localized surface plasmon resonance LSPR . LSPR occurs when the conduction electrons in the nanoparticles collectively oscillate in response to an incident electromagnetic field, such as light. This oscillation leads to strong absorption and scattering of light, which results in unique optical properties that are highly dependent on the size, shape, and composition of the nanoparticles.As the size of metallic nanoparticles changes, their optical properties are affected in the following ways:1. Absorption and scattering: Smaller nanoparticles tend to have stronger absorption and weaker scattering, while larger nanoparticles exhibit stronger scattering and weaker absorption.2. Plasmon resonance wavelength: The LSPR wavelength shifts as the size of the nanoparticles changes. For smaller nanoparticles, the LSPR wavelength is shorter blue-shift , while for larger nanoparticles, the LSPR wavelength is longer red-shift .3. Bandwidth: The bandwidth of the LSPR peak broadens as the size of the nanoparticles increases, which can lead to a decrease in the intensity of the peak.To calculate the optical properties of metallic nanoparticles using quantum chemical methods, one can employ the Time-Dependent Density Functional Theory TD-DFT or the Discrete Dipole Approximation DDA . These methods allow for the calculation of the absorption and scattering cross-sections, as well as the LSPR wavelength, by considering the interaction between the incident electromagnetic field and the conduction electrons in the nanoparticles.TD-DFT is a widely used quantum chemical method that can accurately describe the electronic structure and optical properties of metallic nanoparticles. It involves solving the time-dependent Kohn-Sham equations to obtain the excited-state electronic structure and then calculating the oscillator strengths and transition energies to determine the absorption and scattering cross-sections.DDA is a numerical method that approximates the nanoparticle as an array of discrete dipoles, which interact with the incident electromagnetic field. By solving the resulting system of linear equations, one can obtain the induced dipole moments and calculate the absorption and scattering cross-sections.Both TD-DFT and DDA can be used to study the size-dependent optical properties of metallic nanoparticles, providing valuable insights into their potential applications in areas such as sensing, imaging, and photothermal therapy.