A Numerical Method for the Efficient Calculation of Scattering Form Factors

TL;DR

The paper tackles the computational bottleneck in predicting dark-matter–electron scattering rates in anisotropic molecular detectors by introducing SCarFFF, a GPU-accelerated framework that computes fully three-dimensional molecular form factors from TD-DFT transition density matrices. It develops three complementary analytic/numeric pathways—the Cartesian analytic method, a spherical grid method, and a fast Fourier transform (FFT) approach—to evaluate the molecular form factor $f_S(\mathbf q) = \langle \Psi_s | \tilde{n}_e(-\mathbf q) | \Psi_g \rangle$ efficiently. The methods achieve rapid, high-precision results (e.g., the first 12 form factors for ~10 heavy-atom molecules in about 5 s) and enable high-throughput material screening for directional DM detectors, while including verification steps such as Parseval’s theorem and oscillator-strength reconstruction. This framework, with its modular pipeline from geometry optimization to TD-DFT and then to form-factor evaluation, significantly accelerates design iterations for detector media and paves the way for broader DM–SM coupling studies in complex molecular systems.

Abstract

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.

Figures (6)

• Figure 1: A weakly interacting particle $\chi$ scatters with an electron in a multiparticle ground state, transferring to it some momentum and energy, and leaving it in a higher energy eigenstate.
• Figure 2: The accuracy of the FFT method is controlled in part by the volume of the sampled region. The "scale = 1" volume encloses the atomic coordinates in a rectangular box with a $5 a_0$ margin in all directions. Its qualitative accuracy is poor, although it correctly identifies the order of magnitude. In the "scale = 2" and "scale = 3" examples, we extend the integration volume by a factor of two or three before applying the FFT, yielding more accurate and finely grained pictures of $f_S(\mathbf q)$, shown here in the $q_z = 0$ plane.
• Figure 3: We show the isotropic average form factor $(4\pi)^{-1} \int d\Omega \, |f_S|^2$ for the first excited states of p-xylene and anthracene, calculated with the spherical analytic method of Section \ref{['sec:SphericalMethod']}. This $f_S$ is found by a sum over spherical harmonic modes $f_S(\mathbf q) = \sum_{\ell m} Y_l^m(\hat{q}) \mathcal{R}_{lm}(q)$, which we truncate at finite $\ell$. We find that the first excited state of p-xylene is very well described by the $\ell \leq 12$ angular modes, while an accurate representation of anthracene above $q > 7$ keV requires the $\ell > 12$ modes.
• Figure 4: The molecule set used to test SCarFFF. This test set is composed of 100 molecules composed of hydrogen, carbon, nitrogen, sulphur, and fluorine, with an exactly uniformly distributed heavy (non-hydrogen) atom count from 1 to 20. These are chosen to be a combination of long chain and aromatic compounds, so as to represent a wide spectrum of molecules.
• Figure 5: Workflow of SCarFFF. Given a SMILES string and a basis set set, we first optimise the geometry and perform the DFT and TD-DFT computations. We then build data structures defining the molecule, and pass it through one of the three methods. See the text for more details. Boxes highlighted in blue represent outputs of the code.