Evolution of the Coulomb interactions in correlated transition-metal perovskite oxides from the constrained random phase approximation

Determining the strength of electronic correlations of correlated electrons plays important roles in accurately describing the electronic structures and physical properties of transition-metal (TM) perovskite oxides. Here, we study the evolution of electronic interaction parameters as a function of $d$-electron occupancy in an extended class of TM perovskite oxides $AB$O$_3$ ($A$=Sr, Ca, and $B$=3$d$-5$d$ TM elements) using the constrained random-phase-approximation method adopting two distinct models: $t_{2g}$-$t_{2g}$ and $d$-$dp$. For Sr$B$O$_3$ with $B$=Fe, Ru, and Ir, the $t_{2g}$-$t_{2g}$ model faces critical challenges, as the low-energy Hamiltonian spanning $t_{2g}$ manifolds is ill-defined. The $t_{2g}$-$t_{2g}$ model suggests that, for early $AB$O$_3$ series ($B$=$d^1$-$d^3$), the bare Coulomb interaction parameters $V$ remain nearly constant due to the competition between extended $t_{2g}$ Wannier orbitals and bandwidth reduction. As the $d$-electron filling increases, both partially screened Coulomb interaction parameters $U$ and fully screened Coulomb interaction parameters $W$ decrease, which are attributed to enhanced $e_g$-$t_{2g}$ and $e_g$-$p$ screenings. In contrast to the $t_{2g}$-$t_{2g}$ model, the $d$-$dp$ model effectively handles both early and late $AB$O$_3$ perovskites and reveals different trends. Specifically, $V$ varies inversely with the spreads of $d$-orbitals. $W$ reaches its minimum at the $d^3$ occupancy due to an interplay between increasing $d$-orbital localization and increasing screening effects. An unusual trend is observed for $U$, with local maxima at both $d^1$ and $d^4$ occupations. This can be understood from two aspects: (1) the increasing full screening effects from $d^1$ to $d^3$ and (2) the strongest $d$-$d$ and the weakest $d$-$p$ screening effects near $d^4$ for Sr$B$O$_3$.