This paper presents a new theoretical and computational method to describe how light interacts with extended materials when the usual electric dipole approximation (EDA) is not enough. The EDA assumes the electric field of light is the same across the material. That works when the sample is much smaller than the light’s wavelength, but it can fail for larger samples or for light with a spatially structured field. The authors develop a way to keep the full spatial structure of the light without truncating the usual multipole expansion.

The core idea is to start from the Power–Zienau–Woolley (PZW) multipolar Hamiltonian, a version of the light–matter interaction that naturally includes the full spatial dependence of the electric field. To make this practical for realistic materials, they represent the material’s electronic Hamiltonian in a basis of maximally localized Wannier functions (MLWFs). MLWFs are localized electronic orbitals that can be obtained from standard first-principles calculations. In that basis the position operator becomes simple, and the spatial integral that couples the material polarization to the electric field can be evaluated efficiently. They demonstrate the method on a one-dimensional model (a trans‑polyacetylene chain) and also test nonuniform fields obtained from electrodynamics simulations using the finite-difference time-domain (FDTD) method.

Using this framework, the authors map out when the EDA is reliable and when it fails. They find that the EDA is surprisingly robust for uniformly illuminated one- or two-dimensional materials when the light travels perpendicular to the material. In contrast, for three-dimensional materials, or when light strikes lower-dimensional materials at oblique angles, the EDA breaks down once the wavelength becomes comparable to the system size. The EDA also fails when only part of the material is illuminated or when the field intensity changes a lot across the sample. For gently varying fields, a few multipole correction terms can fix problems. But for fields that vary strongly over the material, adding many multipole terms becomes impractical; the present approach captures those effects without that cost and at roughly the same computational effort as a standard dipole calculation.