  Transition-metal heterostructures offer the fascinating possibility of
controlling orbital degrees of freedom via strain. Here, we investigate
theoretically the degree of orbital polarization that can be induced by
epitaxial strain in LaNiO$_3$ films. Using combined electronic structure and
dynamical mean-field theory methods we take into account both structural
distortions and electron correlations and discuss their relative influence. We
confirm that Hund's rule coupling tends to decrease the polarization and point
out that this applies to both the $d^8\underline{L}$ and $d^7$ local
configurations of the Ni ions. Our calculations are in good agreement with
recent experiments, which revealed sizable orbital polarization under tensile
strain. We discuss why full orbital polarization is hard to achieve in this
specific system and emphasize the general limitations that must be overcome to
achieve this goal.
