Impact of electron--spin coupling on exchange coupling parameters: a nonperturbative approach

Exchange coupling parameters $J_{ij}$ in the Heisenberg model are crucial for describing magnetic behavior at the atomic level. In magnetic materials, spin fluctuations can be accompanied by a self-consistent electronic response -- including charge and magnetization redistribution and changes in orbital occupations -- reflecting electron--spin coupling in the sense of electronic feedback to finite spin rotations. However, the quantitative importance of this coupling in extracting reliable $J_{ij}$ has not been fully clarified. Here, using fully self-consistent, nonperturbative evaluations, we show that finite-angle spin rotations induce such electronic feedback and quantify how strongly it renormalizes the extracted $J_{ij}$. We examine systems of both fundamental and practical interest, including perovskite SrMnO$_3$, Nd-based permanent-magnet compounds (Nd$_2$Fe$_{14}$B and Nd$_2$Co$_{14}$B), and elemental $3d$ transition metals.The nonperturbative approach yields exchange couplings that remain consistent over a wide range of rotation angles. Moreover, spin models parameterized in this way give reasonable agreement with experimental magnetic phase-transition temperatures, underscoring the quantitative role of electron--spin coupling. Overall, our results provide a practical route to constructing quantitatively reliable spin models for predictive finite-temperature simulations and magnetic-materials design.