As the size of a nanomaterial decreases, its electronic and optical properties change significantly due to quantum confinement effects. Quantum confinement occurs when the dimensions of a material are reduced to a size comparable to the de Broglie wavelength of the electrons, leading to quantization of energy levels. This results in a change in the band gap and plasmon resonance frequency of the material.To determine the change in band gap and plasmon resonance frequency of a gold nanoparticle as it decreases from 50 nm to 5 nm in diameter, we can use quantum chemistry calculations. However, as an AI language model, I am unable to perform these calculations directly. Instead, I can provide you with a general approach to perform these calculations.1. Determine the electronic structure of the gold nanoparticle at different sizes using density functional theory DFT or other quantum chemistry methods. This will provide you with the energy levels and wave functions of the electrons in the nanoparticle.2. Calculate the band gap of the gold nanoparticle at each size by finding the difference between the highest occupied molecular orbital HOMO and the lowest unoccupied molecular orbital LUMO . As the size of the nanoparticle decreases, the band gap is expected to increase due to quantum confinement effects.3. To determine the plasmon resonance frequency, you can use the Mie theory or the discrete dipole approximation DDA method. These methods take into account the size, shape, and dielectric properties of the nanoparticle, as well as the surrounding medium, to calculate the extinction, scattering, and absorption spectra of the nanoparticle.4. From the calculated spectra, identify the peak corresponding to the plasmon resonance frequency. As the size of the gold nanoparticle decreases, the plasmon resonance frequency is expected to blue-shift due to the increased influence of quantum confinement and surface effects.By following this approach and performing quantum chemistry calculations, you can determine the change in band gap and plasmon resonance frequency of a gold nanoparticle as it decreases from 50 nm to 5 nm in diameter.