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.


Supplementary Note 1: Calculation details
So far, the first-principles calculations have proved central in predicting new advance materials and/or explaining their behavior mechanism, many times leading to subsequent experimental observation 1,2 . However, it is hard to describe the strongly correlated systems containing transition metals with d electrons, lanthanide and actinide elements with f electrons based on standard density function theory (DFT) method. Hence, many groups are tending to promote the algorithm to remedy the shortcomings in theoretical predictions, such as the self-interaction correction local spin density (SIC-LSD) approach 3 , DFT + U (onsite Coulomb interaction) 4,5 , the hybrid DFT functional 6,7 , and DFT + DMFT (dynamic meanfield theory) 8,9 . Based on these newly developed methods, numerous meaningful theoretical research on f orbitals concerned materials have been performed. For instance, based on the SIC-LSD approach, A. Svane's group studied the valency property of REs clearly 10 , and in their latter work, they perfectly described the electronic structures of PuO 2+x 11 and the magnetism of heavy RE elements 12 . F. Tran and co-workers simulated the Jahn-Teller effect in PrO 2 with DFT + U with spin-orbit coupling (SOC) and their results were in line with experimental observations. 13 Lately, using the hybrid DFT functional, T. Duchon etc. 7 showed that the f contribution in CeO 2 was of covalent nature with experimental evidence.
All in all, we keep our minds on that all calculating methods must serve for the firstprinciples researches and all researches must be faithful to the scientific facts. Our calculations were carried out by the WIEN2k code which was based on spin-polarized DFT with the full-potential (linearized) augmented plane-wave and local orbitals [FP -(L) APW + lo] method. [14][15][16] During structural optimization, to obtain the equilibrium PrO 2 structure, which is approximate to experimental result, we carried out kinds of the exchange correlation potential a k-point grid of size 7 × 7 × 7, and all structural data are estimated by the Murnaghan-Birch equation of states fitting, 17 as the example shown in Fig. S1. Compared 3 the fitted lattice constant (a) and bulk modulus (B) with experimental data, gathered in Table. SI, we found that the result computed by the generalized gradient approximation (GGA) 18 where E %&% ' ()*+ is the total energy of PrO 2 with difference d Pr

Supplementary Note 2: Test for optimization method
As we know, the equilibrium properties, such as bulk modulus (B), phonon frequencies, magnetism and ferroelectricity, are sensitive to the lattice constant a. PrO 2 possess a cubicfluorite crystallographic structure. In order to be agreement with the experimental lattice constant of PrO 2 (a = 5.386 Å) 24 , we have preformed DFT calculations with many exchangecorrelation functionals for the structural optimization. Here U is not taken into consideration.
Each of structure has reached the minimized ground state energy with the fitting of Birch-Murnaghan equation of state (BM-EOS), as the example displayed in Fig. S1. Their results are summarized in Table. SI. The volume is underestimated and bulk modulus is correspondingly overestimated obtained by LDA functional owing to its description to over To avoid this problem, a widely accepted method is DFT + U, which introduces the Coulomb potential in a screen Hartree-Fock like manner. Considering the spin-orbital coupling (SOC) based on the second variational approach, the DFT + U approach opens a gap.
In order to find a suitable value of U Pr , we have tested the U with the equilibrium structure.
As shown in Fig. S3a, with the U Pr increasing, the occupied 4f states move into the low energy level while the empty 4f move toward the high energy level, which opens the band gap. Fig. S3b and 3c

Supplementary Note 4: Analysis of electronic evolutions
According to the changing trend of the magnetic moment ( Fig. 2a right column), we divided the dependence d Pr-O system for PrO 2 into four parts labeled by i, ii, iii and iv. In S.I.