Values of molecular bond lengths, bond angles and (less frequently) bond torsion angles are readily available from databases, from crystallographic software, and/or from interactive molecular and crystal visualization programs such as Jmol. However, the methods used to calculate these values are less well known. In this paper, the computational methods are described in detail, and live Excel implementations, which permit readers to readily perform the calculations for their own molecular systems, are provided. The methods described apply to both fractional coordinates in crystal space and Cartesian coordinates in Euclidean space (space in which the geometric postulates of Euclid are valid) and are vector/matrix based. In their simplest computational form, they are applied as algebraic expansions which are summed. They are also available in matrix formulations, which are readily manipulated and calculated using the matrix functions of Excel. In particular, their general formulation as metric matrices is introduced. The methods in use are illustrated by a detailed example of the calculations. This contribution provides a significant practical application which can also act as motivation for the study of matrix mathematics with respect to its many uses in chemistry.

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从原子到键、角度和扭转：来自晶体空间的分子度量，以及两个 Excel 实现

分子键长、键角和（不太常见的）键扭转角的值很容易从数据库、晶体学软件和/或交互式分子和晶体可视化程序（如 Jmol）中获得。然而，用于计算这些值的方法鲜为人知。在本文中，详细描述了计算方法，并提供了实时 Excel 实现，使读者可以轻松地对自己的分子系统进行计算。所描述的方法适用于晶体空间中的分数坐标和欧几里得空间中的笛卡尔坐标（欧几里得几何假设有效的空间）并且是基于向量/矩阵的。在最简单的计算形式中，它们被用作求和的代数展开式。它们也可用于矩阵公式，可以使用 Excel 的矩阵函数轻松操作和计算。特别是，介绍了它们作为度量矩阵的一般公式。使用的方法通过计算的详细示例进行说明。这一贡献提供了一个重要的实际应用，也可以作为研究矩阵数学在化学中的许多用途的动力。