Multipolar fluctuations in localized $4f^2$-electron systems from dynamical mean-field theory: application to ${\mathrm{P}\mathrm{r}\mathrm{C}\mathrm{d}\mathrm{N}\mathrm{i}}_4$

Multipolar-fluctuation analysis based on density functional theory combined with dynamical mean-field theory is extended from $4f^1$- to $4f^2$-electron systems. Two methodological developments are presented. First, many-particle states are constructed within the $j$-$j$ coupling scheme in the $j=5\mathrm{/}2$ subspace, so as to reproduce the crystalline-electric-field splitting of the $4f^2$ Hund’s-rule ground-state multiplet. Second, multipolar susceptibilities and interactions between $4f^2$ multiplets are derived through a mapping from single-particle to many-particle bases. We apply this framework to PrCdNi${}_4$, which has a non-Kramers ${\mathrm{\Gamma }}_3$ doublet ground state, and propose a $(3z^2-r^2)$-type antiferroquadrupolar order at $\mathit{q}=(2\pi ,\pi ,0)$ driven by the competition between nearest- and fourth-nearest-neighbor interactions. The formalism developed here is applicable to general $4f^n$-electron systems.