A quadratic-scaling algorithm with guaranteed convergence for quantum coupled-channel calculations

Abstract

Rigorous quantum dynamics calculations provide essential insights into complex scattering phenomena across atomic and molecular physics, chemical reaction dynamics, and astrochemistry. However, the application of the gold-standard quantum coupled-channel (CC) method has been fundamentally constrained by a steep cubic scaling of computational cost [${O}(N^3)$]. Here, we develop a general, rigorous, and robust method for solving the time-independent Schr\"odinger equation for a single column of the scattering S-matrix with quadratic scaling [${O}(N^2)$] in the number of channels. The Weinberg-regularized Iterative Series Expansion (WISE) algorithm resolves the divergence issues affecting iterative techniques by applying a regularization procedure to the kernel of the multichannel Lippmann-Schwinger integral equation. The method also explicitly incorporates closed-channel effects, including those responsible for multichannel Feshbach resonances. We demonstrate the power of this approach by performing rigorous calculations on He + CO and CO + N$_2$ collisions, achieving exact quantum results with demonstrably quadratic scaling. Our results establish a new computational paradigm, enabling state-to-state quantum scattering computations for complex molecular systems and providing a novel window onto the intricate multichannel molecular collision dynamics.