Thin films of rare earth metal oxides are interesting materials for many technology applications, which requires a method for controlled growth of such films. If suitable precursors are available, atomic layer deposition (ALD) is the method of choice for nanoscale thin film deposition. Previous studies have identified promising cyclopentadienyl-containing (C5H5, Cp) precursors for rare earth oxide ALD, but little is known about the growth reactions. In this paper, We use first principles periodic density functional theory (DFT) computations to study key reactions in ALD growth of La2O3 and Er2O3. We start from the hydroxylated (001) surface of the hexagonal phase as a model for each oxide. To predict the most stable adsorbate once the metal precursor pulse is finished, we analyze the interaction of the precursor molecule with the oxide surface, the energetics of successive ligand eliminations and the resulting surface structures. For La2O3 we find (i) transfer of hydrogen from the surface to a Cp ligand has a barrier of 0.8 eV, (ii) non-ALD desorption of precursor fragments is favored, (iii) the final adsorption fragment is predicted to be La(Cp)(2). In contrast, at the Er2O3 surface, (i) hydrogen transfers spontaneously from the surface to the adsorbing precursor, (ii) reactive adsorption is thermodynamically favored over desorption, and (iii) the final adsorbate is predicted to be Er(Cp). We predict that ligand elimination is significantly more favorable on surfaces of Er2O3 relative to La2O3, and so that Er2O3 ALD is a better process. We rationalize this as due to stronger Er-O bonding, but also due to the restoration of a less distorted surface. These studies provide new insights into the key reactions occurring during ALD of rare earth oxides and new understanding of experimental findings.