Molecules in the $z_{\rm abs}$~=~2.8112 damped system toward PKS~0528--250

We present a detailed analysis of a high resolution spectrum of the damped Ly$\alpha$ system at $z_{\rm abs}$~=~2.8112toward PKS~0528-250. The absorption redshift is slightly larger than the emission redshift of the quasar. We estimate the column density of H$_2$ molecules $N$(H$_2$)~$\sim$~6$\times$10$^{16}$~cm$^{-2}$ and the fractional abundance of H$_2$, $f$~=~5.4$\times$10$^{-5}$. The excitation temperature derived for different transitions suggests that the kinetic temperature of the cloud is $\sim$200~K and the density $n$~$\sim$~1000~cm$^{-3}$. The cloud therefore has a dimension of $\sim$1~pc along the line of sight. Since it obscures the broad-line emission region, its transverse dimension should be larger than 10~pc.\par We obtain upper limits on the column densities of C~{\sc i} ($<$~10$^{12.7}$~cm$^{-2}$) and CO ($<$~10$^{13.2}$~cm$^{-2}$; $N$(CO)/$N$(H~{\sc i})~$<$~7$\times$10$^{-9}$).We suggest that the ratio $N$(H$_2$)/$N$(C~{\sc i}) is a useful indicator of the physical conditions in the absorber. Simple photo-ionization models assuming solar relative abundances show that radiation fields with spectra similar to typical AGNs or starbursts are unable to reproduce all the constraints and in particular the surprisingly small $N$(C~{\sc i})/ $N$(H$_2$) and $N$(Mg~{\sc i})/$N$(H$_2$) ratios. In view of the models we explored, the most likely ionizing spectrum is a composite of a UV- "big bump" possibly produced by a local starburst and a power-law spectrum from the QSO that provides the X-rays. Dust is needed to explain the production of molecules in the cloud. The amount of dust is broadly consistent with the [Cr/Zn] abundance determination.