The doping phase diagram of V$_2$O$_3$ revisited

V$_2$O$_3$ is a prototypical Mott-Hubbard insulator, with hundreds theoretical papers published in course of the last 30+ years. Intriguingly, doping with Ti rapidly supresses the low-$T$ antiferromagnetic insulator (AFI) phase so that the material becomes a paramagnetic metal (PM) at all $T$s, while doping with Cr slightly stabilizes the AFI phase, and rapidly transforms the high-$T$ PM phase into a paramagnetic insulator (PI). It's been often interpreted as a chemical pressure effect, with Ti allegedly exerting a positive, and Cr a negative pressure. However, the documented ionic radius of Ti$^{3+}$ ($R_{Ti}=0.81$ \AA) is slightly $larger$ than that $R_V=0.78$ \AA, and $R_{Cr}=0.755$ \AA is slightly $smaller$. Both experiments and our $ab$ $initio$ calculations suggest that either substitution expands the lattice, and anyway the effect is too small. In order to address this controversy, we have performed structural optimization of a 16 formula unit supercell with a Cr or Ti impurity, and found that the former $enhances$, and the latter $reduces$ the tendency to insulating behavior. We have observed this effect in both zero-$T$ LDA+U calculations and finite-$T$ DFT+DMFT calculations. Thus, we conclude that the pecularities of the V$_2$O$_3$+(Cr,Ti) phase diagram are not a simple chemical pressure effect, but indicate high sensitivity to the local geometry. This conclusion is consistent with the (theoretical) observation that the local crystal field plays a critical role in the Mott physics in V$_2$O$_3$, and with (experimental) fact that the insulating state in V$_2$O$_3$ is easily destroyed by ionic irradiation (J.G. Ramirez et al, PRB 91, 205123, 2015).