  The evolution of differential ages of passive galaxies at different redshifts
(cosmic chronometers) has been proved to be a method potentially able to
constrain the Hubble parameter in a cosmology-independent way, but the
systematic uncertainties must be carefully evaluated. In this paper, we compute
the contribution to the full covariance matrix of systematic uncertainties due
to the choice of initial mass function, stellar library, and metallicity,
exploring a variety of stellar population synthesis models. Through simulations
in the redshift range 0<z<1.5 we find that the choice of the stellar population
synthesis model dominates the total error budget on $H(z)$, with contributions
at a level of ~4.5%, discarding the most discordant model. The contribution due
to the choice of initial mass function is <0.5%, while that due to the stellar
library is ~6.6% on average. We also assess the impact of an uncertainty in the
stellar metallicity determination, finding that an error of ~10% (5%) on the
stellar metallicity propagates to a 9% (4%) error on $H(z)$. These results are
used to provide the combined contribution of these systematic effects on the
error budget. For current $H(z)$ measurements, where the uncertainties due to
metallicity and star formation history were already included, we show that,
using the more modern stellar libraries, the additional systematic uncertainty
is between 5.4% (at z=0.2) and 2.3% (at z=1.5). To reach the goal of keeping
the systematic error budget below the 1% level we discuss the efforts needed to
obtain higher resolution and signal-to-noise spectra and improvements in the
modeling of stellar population synthesis.
