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

TL;DR

This paper introduces a new quadratic-scaling algorithm for quantum coupled-channel calculations that guarantees convergence, enabling more efficient and accurate simulations of complex molecular scattering phenomena.

Contribution

The authors develop the WISE algorithm, a robust and rigorous method that reduces the computational complexity of quantum scattering calculations from cubic to quadratic scaling.

Findings

Achieves exact quantum results with quadratic scaling on complex molecular collisions.

Successfully incorporates closed-channel effects and Feshbach resonances.

Demonstrates efficiency and accuracy on He + CO and CO + N₂ collision systems.

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.