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_N2, 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.

中文翻译：

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迈向有效的化学反应直接动力学研究：一种新的矩阵完成算法

本文描述了基于多项式变体的矩阵补全 (PVMC) 算法的开发和测试。我们的目标是使用具有多维隧穿 (VTST-MT) 的变分过渡态理论来减少与反应速率系数计算相关的计算工作量。该算法通过利用这些特征值的基本属性，仅使用一小部分信息样本来恢复构成反应的最小能量路径 (MEP) 的量子力学 Hessians 的特征值。除了构成大多数矩阵完成（MC）算法基础的低秩属性之外，这项工作还在目标函数中引入了多项式约束。这使我们能够对矩阵列进行采样，这与大多数只能采样元素的传统 MC 方法不同，这使得 PVMC 很容易与量子化学计算兼容，因为对单个柱进行采样需要一次 Hessian 计算。适用于各种反应─S_N 2、氢原子转移、金属-配体协同催化和酶化学——我们证明 PVMC 平均只需要 6 到 7 次 Hessian 计算即可准确预测量子效应和变分效应。