A Numerical Method for the Efficient Calculation of Scattering Form Factors

Scintillating molecular crystals have emerged as prime candidates for directional dark matter detector targets. This anisotropy makes them exquisitely sensitive due to the daily modulation induced by the directional dark matter wind. However, predicting the interaction rate for arbitrary molecules requires accurate modeling of the many-body ground as well as excited states, a task that has been historically computationally expensive. Here, we present a theory and computational framework for efficiently computing dark matter scattering form factors for molecules. We introduce SCarFFF, a GPU-accelerated code to compute the fully three-dimensional anisotropic molecular form factor for arbitrary molecules. We use a full time-dependent density functional theory framework to compute the lowest-lying singlet excited states, adopting the B3YLP exchange functional and a double-zeta Gaussian basis set. Once the many-body electronic structure is computed, the form factors are computed in a small fraction of the time from the transition density matrix. We show that ScarFFF can compute the first 12 form factors for a molecule of 10 heavy atoms in approximately 5 seconds, opening the door to accurate, high-throughput material screening for optimal directional dark matter detector targets. Our code can perform the calculation in three independent ways, two semi-analytical and one fully numeric, providing optimised methods for every precision goal.

💡 Research Summary

The paper introduces SCarFFF, a GPU‑accelerated Julia package designed to compute dark‑matter–electron scattering form factors for arbitrary molecules with unprecedented speed and flexibility. The authors motivate the work by highlighting the need for anisotropic scintillating molecular crystals as directional dark‑matter detectors; the daily modulation of the dark‑matter wind makes such targets highly sensitive, but accurate predictions of interaction rates require detailed many‑body electronic structure calculations for both ground and excited states, which have traditionally been computationally prohibitive.

The theoretical framework starts from the standard expression for the DM–electron scattering rate, separating the particle‑physics matrix element from a material‑specific form factor f_S(q). This form factor is the Fourier transform of the transition charge density between the many‑electron ground state |Ψ_g⟩ and an excited state |Ψ_s⟩. To obtain the transition density, the authors employ time‑dependent density‑functional theory (TD‑DFT) within the Casida formalism, using the B3YLP exchange‑correlation functional and a double‑ζ Gaussian basis (e.g., 6‑31G*, cc‑pVDZ, cc‑pVTZ). The Casida eigenvectors X and Y encode single‑excitation character and correlation effects, respectively, and are combined into a transition density matrix (TDM) T_{pq}. Because the molecular orbitals are expanded in Gaussian atomic orbitals, the matrix elements ⟨φ_a|e^{i q·r}|φ_i⟩ can be evaluated analytically or numerically without reconstructing full wavefunctions.

SCarFFF implements three distinct computational pathways for the Fourier transform of the transition density:

1. FFT Method – The transition density Φ_{g→s}(r) is sampled on a three‑dimensional rectangular grid and transformed using a fast Fourier transform. This approach is extremely simple and fast but requires a sufficiently large simulation box and fine grid spacing to avoid aliasing; it is best suited for low‑precision, high‑throughput screening.

2. Semi‑Analytical Gaussian Method – By exploiting the known Fourier transforms of Gaussian basis functions, the code evaluates the required integrals analytically, dramatically reducing memory usage while retaining medium precision (≈1 % error). This method bridges the gap between the crude FFT and the most accurate approach.

3. Fully Numerical Integration – Direct numerical integration of the transition density without an FFT yields the highest precision (sub‑percent errors) at the cost of increased CPU/GPU time. This is the preferred route when exact rates are needed, for example in benchmark studies or for light‑mediator models where the form factor at very low momentum transfer is critical.

Performance benchmarks demonstrate that for a molecule containing ten heavy atoms (e.g., p‑xylene), SCarFFF can compute the first twelve singlet transition form factors in roughly five seconds on a modern GPU. This represents a speed‑up of two to three orders of magnitude compared with conventional quantum‑chemistry packages. Convergence studies show that a simulation box of ~24 Å (scale = 2) is sufficient for most dark‑matter models, while larger boxes only marginally improve the result.

The code is open‑source (GitHub link provided) and modular: any transition density matrix and any Gaussian basis set can be supplied, making the framework adaptable to a wide range of chemical systems. Currently only spin‑independent form factors are implemented, but the authors outline a roadmap to include spin‑dependent response functions, triplet excitations, temperature effects, and extensions to periodic crystals.

In summary, SCarFFF offers a highly efficient, accurate, and versatile tool for calculating anisotropic molecular form factors, enabling rapid high‑throughput screening of candidate scintillating crystals for directional dark‑matter detection. Its combination of TD‑DFT accuracy, GPU acceleration, and multiple precision modes positions it as a critical resource for both the particle‑physics and materials‑science communities seeking to design next‑generation dark‑matter detectors.