If you consider the analytic expression for the dielectric function in the random-phase approximation (RPA), you will notice that its frequency dependence (that is, its ω dependence) is determined by a sum over a function consisting of a numerator that involves some Fermi functions and a denominator comprised of ω and a difference of two single-particle energies. The Fermi functions in the numerator restrict the energy difference in the denominator to one involving a single-particle energy below the chemical potential and a single-particle energy above the chemical potential. This leads to the fact that for ω = 0 the frequency-dependent part of the dielectric function scales inversely with the fundamental energy gap of the system under investigation. A large static (ω=0) dielectric function therefore signifies a small energy gap. Nanoparticles have small energy gaps (because of the confinement). For frequencies below the plasmon frequency (frequencies) the behaviour of the dielectric function as a function of ω is rather non-trivial, however for the chemical potential located inside a relatively small energy gap, the trend is a rapid decrease in the dielectric function for increasing values of ω. This is essentially what one observes in the case of nano-particles, as indicated by you (for gold nano-spheres, one may wish to consult the attached paper by Derkachova et al., 2016).

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4875142/

Dear Zaid Hamid 

Please consult the following articles for the details answer of your question:

https://www.hindawi.com/journals/isrn/2012/706398/

http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1516-14392016000200478

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4409379/ 

Dear Zaid,

I do not really know whether I understands your question, but I assume that you mean "dielectric constant" ?

In any dispersive material system, the dielectric response function is time/frequency dependent. This means that it is not a constant, although in some frequency ranges it can become frequency independent. For linear and causal systems, there is a general relation - Kramers-Kronig relationship -  that relates the real part of the dielectric response function(non-dissipative) to its imaginary(dissipative) part. 

Now, starting from frequency w->infinity, the real part is constant and equal one - dielectric constant of vacuum. As you pass through various "absorption bands (X-rays, UV, VIS, IR, FIR and various dielectric relaxational absorption process), the dissipation is finite there and because the real part is kind of integral of the imaginary part, It must increase as the frequency decreases. Finally for w->0, you obtain the dc limit - the dc dielectric constant of the material. 

There exist various semi-empirical relations that relate this real part of the dielectric response function to the electron band structure of the material (be that glasses, solids, liquids, etc...) and the band gap of the material in question, but to my knowledge, there is not a general expression resulting from "first principle calculations". I might be wrong here though. The other effect that has been pointed out by Behman above , are size quantum effects.

So, I am not really surprised that in also in your system the real part of the dielectric response function increases with decreasing frequency of measurement.

With best regards

Petr

P.S.: You might like to consult the following :

1. J.D.Jackson :Classical Electrodynamics (Wiley, 1999)

2. J.C.Phillips and G.Lucovsky: Bonds and Bands in Semiconductors (Momentum Press 1973).

3. F.Han: Modern Course in the Quantum Theory of Solids (World Scientific 2013)