Title: Exchange Interactions in Rare-earth Magnets $A_2$PrO$_3$ ($A$= alkali metals): Revisited

Abstract: Rare-earth materials hold promise to realize exotic magnetic states owing to synergy between electron correlations and spin-orbit coupling. Recently, quasi-two-dimensional honeycomb magnets $A_2$PrO$_3$ ($A$ = alkali metals) were predicted to be good candidates for Kitaev quantum spin liquids, as the Kitaev-type bond-dependent anisotropic interactions dominate over bond-independent isotropic Heisenberg ones. However, experimental observations are negative, questioning the energy hierarchy in Pr$^{4+}$ ions assumed in the conjecture on the basis of the conventional Russell-Saunders coupling scheme. We here revisit the exchange interactions in these Pr compounds, by explicitly taking into account the ionic states beyond the assumption. We show that, while increasing the octahedral crystal field splitting, which was assumed to be negligibly small, relative to the spin-orbit coupling, the Kitaev-type interactions are suppressed to be subdominant compared to prevailing Heisenberg ones. Our finding compromises the contradiction as arising from the peculiar ionic state of the high valence Pr$^{4+}$.