The in-plane anomalous Hall effect occurs when the magnetization lies within the same plane as the electric field and the Hall current, and requires magnetic point groups lacking rotational or mirror symmetries. While it is observed in both Weyl semimetals and elemental ferromagnets, the microscopic role of higher-order multipoles remains unclear. Here, we develop a microscopic framework that combines time-reversal-symmetric Wannier functions with a symmetry-adapted multipole basis to decompose the first-principles Wannier Hamiltonian into electric, magnetic, magnetic-toroidal, and electric-toroidal multipoles. This approach allows us to selectively analyze multipole components of the Wannier Hamiltonian and identify how symmetry-classified terms affect the angular dependence of the conductivity. Applying this framework to body-centered cubic iron, we find that high-rank magnetic and magnetic-toroidal terms produce angular variations comparable to those from magnetic-dipole terms, and that magnetic-toroidal 16-poles, in particular, give opposite-sign features. Guided by this multipole analysis, we further examine uniaxial strain along the [103] direction as an example of externally tuning the angular response. The strain substantially modifies the angular dependence, illustrating that multipole-resolved Hamiltonian analysis can provide microscopic guidance for understanding and controlling the in-plane anomalous Hall effect.

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• Anomalous Hall effect
• Magnetic order
• Magnetism
• Multipole order
• Density functional theory
• Wannier function methods