A Matrix Completion Algorithm to Recover Modes Orthogonal to the Minimum Energy Reaction Path

Variational Transition State Theory (VTST) is a way to overcome several of the shortcomings of the potential energy-based Transition State Theory (TST) in describing reaction mechanisms and rates. However, because of its sheer computational cost resulting from prohibitive quantum mechanical hessians and entropy estimations, VTST is yet to be widely accepted compared to conventional TST in computational kinetics studies. In this work, we aim to reduce such costs by demonstrating that matrix completion algorithms used in signal processing approaches, which exploit some underlying matrix structure in order to find missing values, can be used to recover the frequencies of modes orthogonal to the minimum energy path (MEP). In particular, we develop a Harmonic Variety-Based Matrix Completion (HVMC) algorithm in order to make the computation of the potential energy terms (which includes the diagonalized hessian) of the Reaction Path Hamiltonian (RPH) more efficient. Manipulation of these terms leads to a matrix of where each column represents a point along the MEP, and each row represents the square of an eigenvalue frequency corresponding to a specific vibrational mode. We demonstrate proof-of-concept of HVMC with model SN2 reactions. At a given sampling density (that is, the proportion of matrix elements available to us), elements are randomly hidden and the HVMC algorithm is used to recover the missing data. The performance is quantified in two error measurements. First, relative column-wise and matrix errors between the recovered and true matrices are computed using the Frobenius norm. Second, free energies of vibration and zero-point energies computed from each column are compared between the true and recovered matrices. We run 100 HVMC trials at a given sampling density in order to get a more accurate understanding of the average performance. Furthermore, we define the minimum sampling density as the sampling density it takes for each system to reach a certain degree of accuracy. We find that HVMC recovers the missing frequencies to high fidelity even with up to 70% of elements missing and that the minimum sampling density decreases as system size increases. With this success, our future work will therefore focus on practical implementations of HVMC which include utilization of strategic sampling schemes, gradient-based methods to compute initial approximate eigenvalues, and cheap eigenmodes from molecular mechanics hessians.