Strong-correlated behavior of 4f electrons and 4f5d hybridization in PrO₂

Bringing oxygen atoms from infinite, passing equilibrium until short enough distances, we aim to reveal the 4f5d electron bonding property and its relevance to the peculiar physical properties within PrO₂ based on both accounting for electron Coulomb repulsion and spin-orbit coupling effects in combination with Wannier function methods. The microscopic mechanism of static Janh-Teller distortions and the physical insight into the dynamic Jahn-Teller effects are clarified. Peculiarly, the magnetic coupling is suggested to be via 4f-5d-O2p-5d-4f pathway in PrO₂, and the coupling between spin and orbital ordering of 4f electrons is for the first time disclosed. The 5d orbitals, hybridized with 4f electrons, are found to play important roles in these processes.