bulk material has continuous spectrum and band gap, but nanoparticle has discrete spectrum and no band gap. It has highest occupied and lowest unoccupied orbitals.
You must specify your nanoparticle to proceed further.
your particle size reduction indirectly means the band gap energy increment. bulk band gap energy varies for materials to materials . in the same way nanoparticles band gap energy also varies from material to materials. so we cannot interpret the band gap energy with particles size without knowing the material name. because this interpretation varies from materials to materials. I hope this will help you to understand.
if quantum confinement takes place in the nanoparticles (it is required that the size of nanoparticles to be about 10 nm dependent on the material nature) so energy bands become discrete and the band gap would increased.
The bandgap (energy difference betwen valence band maximum and conduction band minimum) increases if the extension of the particle decreases. Furthermore, the increase depends on the potential barrier height with respect of the confined carrier.
My recommendation is to read the chapters concerning Quantum wells in a textbook. The difference between a Quantum well and a nanoparticle is the confinement in 3 dimensions. To calculate the change of the bandgap, it is necessary to know certain Parameters like the diameter of the particle, the barrier height and the effective mass in the material. For 1eV enhancement, the Diameter would be in the order of some Nanometer or lower. Furthermore, the density of states changes significiantly. I would expect almost deltalike states like in an atom.