  The dielectric function is one of the most important quantities that
describes the electrical and optical properties of solids. Accurate modeling of
the frequency-dependent dielectric function has great significance in the study
of the long-range van der Waals (vdW) interaction for solids and adsorption. In
this work, we calculate the frequency-dependent dielectric functions of
semiconductors and insulators using the $GW$ method with and without exciton
effects, as well as efficient semilocal density functional theory (DFT), and
compare these calculations with a model frequency-dependent dielectric
function. We find that for semiconductors with moderate band gaps, the model
dielectric functions, $GW$ values, and DFT calculations all agree well with
each other. However, for insulators with strong exciton effects, the model
dielectric functions have a better agreement with accurate $GW$ values than the
DFT calculations, particularly in high-frequency region. To understand this, we
repeat the DFT calculations with scissors correction, by shifting DFT Kohn-Sham
energy gap to match the experimental band gap. We find that scissors correction
only moderately improves the DFT dielectric function in low-frequency region.
Based on the dielectric functions calculated with different methods, we make a
comparative study by applying these dielectric functions to calculate the vdW
coefficients ($C_3$ and $C_5$) for adsorption of rare-gas atoms on a variety of
surfaces. We find that the vdW coefficients obtained with the nearly-free
electron gas-based model dielectric function agree quite well with those
obtained from the $GW$ dielectric function, in particular for adsorption on
semiconductors, leading to an overall error of less than 7% for $C_3$ and 5%
for $C_5$. This demonstrates the reliability of the model dielectric function
for the study of physisorption.
