There is same question on the research gate maybe that will help you out.
In Au, this is the 6s,p band, with one electron per atom. The total bandwidth is probably of the order of 20 eV or so. Divide this by the number of atoms in the particle to get a zero-order (!!!) estimate of the energetic distance between electronic states around the Fermi level.
[This estimate is not quite accurate since it neglects degeneracies/symmetry-related features of all kinds].
In a particle of 5nm diameter, you end up with approx. 4000 atoms, which directs you towards a 'gap' in the single-particle spectrum of about 5meV and accordingly smaller for larger particles.
But according to my expertise, I think there is no band in gold nanoparticles but they are affected by deposition properties and their rate of deposition.
Normally metals do not have any band gap. There is a continuous band. Band gaps are present in semi-conductors. So I expect that there should be no band gap in Au nanoparticles.
One of the most valuable article about gold nanoparticle synthesis, structure and optical properties is, “Synthesis, structural and optical properties of gold nanoparticle-graphene-selenocysteine composite bismuth ultrathin film electrode and its application to Pb(II) and Cd(II) determination”.
https://doi.org/10.1016/j.arabjc.2015.06.020
In addition to your queries, you might also be able to find more knowledge which may further be helpful in your research.
There is no band gap in gold nanoparticles, since gold is a metal. I think that you are misled by the optical properties of those nanoparticles, which are due to the excitation of localized surface plasmon resonances (LSPR). Such LSPR depend on the geometry and the shape of the particle. I strongly suggest this publication:
Article Gold nanorods: Synthesis, characterization and applications