Applying R-Matrix Theory to Atom-Molecule Inelastic Collisions: the case study of H$_2$O + H

TL;DR

This work applies the calculable $R$-Matrix framework to inelastic atom–molecule collisions, using H + H$_2$O as a benchmark to test accuracy against conventional CC theory. The inner-region eigenproblem is solved with a Bloch operator and a Lagrange-mesh expansion to obtain the boundary $R$-matrix, from which scattering information is extracted without extensive outer-region propagation. The study achieves near-CC accuracy for rotationally inelastic cross sections while delivering substantial computational speedups, particularly when leveraging GPU-accelerated diagonalization with MAGMA. The results demonstrate the method’s scalability and potential for studying larger polyatomic systems relevant to astrophysical and atmospheric environments, with direct $S$-matrix extraction at the inner boundary for neutral-neutral collisions. Overall, the paper establishes $R$-Matrix theory as a viable, efficient alternative to CC for complex inelastic scattering, enabling systematic exploration of molecule–molecule interactions in high-performance computing contexts.

Abstract

The present study presents a comprehensive theoretical investigation of atom and asymmetric top molecule inelastic scattering based on the R-matrix formalism. The proposed methodology establishes a rigorous framework for treating inelastic collisions in the space-fixed coordinate system. The excellent numerical performance of the method is demonstrated through the comparison of state-to-state rotationally inelastic R-matrix cross sections for the H + H$_2$O system with those obtained using conventional close-coupling (CC) theory. The R-matrix approach is shown to deliver results of comparable accuracy while achieving substantially reduced computation times. The method is furthermore shown to achieve more than one order-of-magnitude speedup by exploiting GPU-accelerated diagonalisation through the MAGMA library. This combination of accuracy and computational efficiency positions the R--matrix approach as a powerful and scalable tool for investigating inelastic scattering involving complex polyatomic systems, thereby paving the way for systematic studies of molecule-molecule interactions in astrophysical, atmospheric, and cold-matter environments.

Figures (4)

• Figure 1: State-to-state cross sections for the rotational relaxation of H2O by collision with H from several para/ortho initial states. Solid lines represent the CC cross sections and dotted lines the R--Matrix cross-sections. Solid and dotted lines are fully overlapping, except for fine resonances structures.
• Figure 2: Elastic cross section for the Na$^{+}$ + He collision. Black solid lines represent the results obtained using the R--Matrix method in a $[0.2,50]\,a_0$ interval, dashed blue lines represent the R--Matrix calculations restricted to the shorter interval $[0.2,30]\,a_0$ without propagation while dotted red lines represent calculations over $[0.2,30]\,a_0$ supplemented by propagation from $30$ to $50\,a_0$. The red dotted and black solid lines are fully overlapping
• Figure 3: Comparison of the R--Matrix ( in blue ) and CC (in red) computation time. We consider three grids of respectively 100, 250, and 500 energies in the [0.1,1000] cm$^{-1}$ interval. The computation time required is specified over each bar.
• Figure 4: Comparison of the R--Matrix time performance using CPU ( in blue ) and GPUs (in green) to carry out the diagonalizations. We have divided the diagonalizations in intervals of the total angular momentum $J$. For each interval we have solve the diagonalization of all the values included in the interval and the two total parities.