Electronic Structure and Dynamical Correlations in Antiferromagnetic BiFeO$_3$

We study the electronic structure and dynamical correlations in antiferromagnetic BiFeO$_3$, a prototypical room-temperature multiferroic, using a variety of static and dynamical first-principles methods. Conventional static Hubbard corrections (DFT+$U$, DFT+$U$+$V$) incorrectly predict a deep-valence Fe $3d$ peak (around $-7\,\text{eV}$) in antiferromagnetic BiFeO$_3$, in contradiction with hard-X-ray photoemission. We resolve this failure by using a recent generalization of DFT+$U$ to include a frequency-dependent screening -- DFT+$U(\omega)$ -- or using a dynamical Hubbard functional (dynH). The screened Coulomb interaction $U(\omega)$, computed with spin-polarized RPA and projected onto maximally localized Fe $3d$ Wannier orbitals, is expressed as a sum-over-poles, yielding a self-energy that augments the Kohn--Sham Hamiltonian. This DFT+$U(\omega)$ approach predicts a fundamental band gap of $1.53\,\text{eV}$, consistent with experiments, and completely eliminates the unphysical deep-valence peak. The resulting simulated HAXPES spectrum reproduces the experimental lineshape with an accuracy matching or exceeding that of far more demanding DFT+DMFT calculations. Our work demonstrates the critical nature of dynamical screening in complex oxides and establishes DFT+$U(\omega)$ as a predictive, computationally efficient method for correlated materials.