Global Plane Waves from Local Gaussians: Periodic Charge Densities in a Blink

We introduce ELECTRAFI, a fast, end-to-end differentiable model for predicting periodic charge densities in crystalline materials. ELECTRAFI constructs anisotropic Gaussians in real space and exploits their closed-form Fourier transforms to analytically evaluate plane-wave coefficients via the Poisson summation formula. This formulation delegates non-local and periodic behavior to analytic transforms, enabling reconstruction of the full periodic charge density with a single inverse FFT. By avoiding explicit real-space grid probing, periodic image summation, and spherical harmonic expansions, ELECTRAFI matches or exceeds state-of-the-art accuracy across periodic benchmarks while being up to $633\times$ faster than the strongest competing method, reconstructing crystal charge densities in a fraction of a second. When used to initialize DFT calculations, ELECTRAFI reduces total DFT compute cost by up to $\sim$20 \%, whereas slower charge density models negate savings due to high inference times. Our results show that accuracy and inference cost jointly determine end-to-end DFT speedups, and motivate our focus on efficiency.

Many important materials are first studied on computers before they are made in the lab. A common method for doing this is called density-functional theory, or DFT, which predicts how electrons are arranged inside a material. DFT is very useful, but it can be slow because the computer has to repeatedly refine the electron density until it finds a stable answer. In this paper, we use machine learning to provide a much better starting guess, which helps the DFT calculations finish faster by starting closer to the true solution. The challenge is that the machine-learning model itself also has to be fast. If it is not, the time saved in DFT is lost when making the prediction. We built ELECTRAFI, a model that predicts electron densities in crystals using a large set of simple 3D “blobs" called Gaussians. Instead of checking every point in space one by one, ELECTRAFI converts these blobs directly into the mathematical format used by many DFT programs. This makes the prediction both naturally periodic, like a crystal, and very fast. We found that ELECTRAFI is as accurate as the best existing methods while being hundreds of times faster. When used inside real DFT calculations, it reduces the total compute time by up to about 20%. This is significant because billions of compute hours are spent globally on DFT calculations each year, and methods like ELECTRAFI can help make large-scale materials simulation faster, cheaper, and less energy-intensive.