Toward Efficient Direct Dynamics Studies of Chemical Reactions: A Novel Matrix Completion Algorithm

Abstract

This paper describes the development and testing of a polynomial variety-based matrix completion (PVMC) algorithm. Our goal is to reduce computational effort associated with reaction rate coefficient calculations using variational transition state theory with multidimensional tunneling (VTST-MT). The algorithm recovers eigenvalues of quantum mechanical Hessians constituting the minimum energy path (MEP) of a reaction using only a small sample of the information, by leveraging underlying properties of these eigenvalues. In addition to the low-rank property that constitutes the basis for most matrix completion (MC) algorithms, this work introduces a polynomial constraint in the objective function. This enables us to sample matrix columns unlike most conventional MC methods that can only sample elements, which makes PVMC readily compatible with quantum chemistry calculations as sampling a single column requires one Hessian calculation. For various types of reactionsS N 2, hydrogen atom transfer, metal–ligand cooperative catalysis, and enzyme chemistrywe demonstrate that PVMC on average requires only six to seven Hessian calculations to accurately predict both quantum and variational effects.

Recent Advances toward Efficient Calculation of Higher Nuclear Derivatives in Quantum Chemistry

“…This is because the computational cost and therefore the success of CS (or low-rank MC), depends on the sparsity (or rank) of that particular problem. For instance, PVMC can be a few times cheaper than Hessian calculations across the entire MEP . However, the true cost depends on the complexity (rank) of the reaction which can only be approximately determined.…”

“…In our group, we are addressing (1) by developing matrix completion (MC) algorithms that leverage polynomial and low-rank structures of higher derivatives, specifically eigenvalues of Hessians that constitute the MEP. − The approach is grounded in the potential energy term of the reaction path Hamiltonian (RPH) in which translations, rotations, and the reaction coordinate are projected out. The potential energy is described in terms of normal mode displacements { q i }: , V ( s , q ) = V 0 ( s ) + 1 2 ∑ i = 1 F − 1 ω i 2 q i 2 where F = 3 N at – 6, V 0 ( s ) is the energy at a point on the MEP described by arc length s (=0 at transition state, ± ∞ at reactant/product), and { ω i 2 } are squares of vibrational frequencies obtained from Hessian eigenvalues.…”

“…To address this challenge, we recently proposed a new algorithm, polynomial VMC (PVMC). In addition to minimizing the problem rank, PVMC imposes a polynomial structure on eigenvalues across rows of C : min boldC , boldQ ( 1 − λ ) ∥ C ∥ S p p + 1 2 λ ∥ boldC − boldQ boldS ∥ F 2 nobreak0em0.25em⁡ subject to nobreak0em0.25em⁡ scriptP normalΩ ( C ) = scriptP normalΩ ( C true ) where ∥·∥ F is the Frobenius norm, λ is a hyper-parameter that describes relative weighting of the rank and polynomial terms, Q is the matrix of polynomial coefficients, and S is the matrix consisting of powers of s up to degree 4. In addition to the stationary points (reactant, product, transition state) that are assumed to always be sampled, we propose a Euclidean distance criterion to select the values of s and therefore columns (Hessians) that must be calculated for accurate recovery of C .…”

“…For instance, PVMC can be a few times cheaper than Hessian calculations across the entire MEP. 36 However, the true cost depends on the complexity (rank) of the reaction which can only be approximately determined. Since H-transfer reactions have at least one mode that is strongly coupled to the reaction coordinate and therefore result in higher problem ranks (than, say S N 2), the number of H-containing bonds being broken or formed may serve as an approximate indicator of reaction complexity.…”