  Optical conductivity measurements are combined with density functional theory
calculations in order to understand the electrodynamic response of the
frustrated Mott insulators Herbertsmithite $\mathrm{ZnCu_{3}(OH)_{6}Cl_{2}}$
and the closely-related kagome-lattice compound
$\mathrm{Y_{3}Cu_{9}(OH)_{19}Cl_{8}}$. We identify these materials as
charge-transfer rather than Mott-Hubbard insulators, similar to the high-$T_c$
cuprate parent compounds. The band edge is at 3.3 and 3.6 eV, respectively,
establishing the insulating nature of these compounds. Inside the gap, we
observe dipole-forbidden local electronic transitions between the Cu $3d$
orbitals in the range 1--2 eV. With the help of \textit{ab initio} calculations
we demonstrate that the electrodynamic response in these systems is directly
related to the role of on-site Coulomb repulsion: while charge-transfer
processes have their origin on transitions between the ligand band and the Cu
$3d$ upper Hubbard band, \textit{local} $d$-$d$ excitations remain rather
unaffected by correlations.
