Since their introduction in 1929, Pauling’s five rules have been used by scientists from many disciplines to rationalize and predict stable arrangements of atoms and coordination polyhedra in crystalline solids; amorphous materials such as silicate glasses and melts; nanomaterials, poorly crystalline solids; aqueous cation and anion complexes; and sorption complexes at mineral-aqueous solution interfaces. The predictive power of these simple yet powerful rules was challenged recently by George et al. (2020), who performed a statistical analysis of the performance of Pauling’s five rules for about 5000 oxide crystal structures. They concluded that only 13% of the oxides satisfy the last four rules simultaneously and that the second rule has the most exceptions. They also found that Pauling’s first rule is satisfied for only 66% of the coordination environments tested and concluded that no simple rule linking ionic radius to coordination environment will be predictive due to the variable quality of univalent radii.We address these concerns and discuss quantum mechanical calculations that complement Pauling’s rules, particularly his first (radius sum and radius ratio rule) and second (electrostatic valence rule) rules. We also present a more realistic view of the bonded radii of atoms, derived by determining the local minimum in the electron density distribution measured along trajectories between bonded atoms known as bond paths, i.e., the bond critical point (rc). Electron density at the bond critical point is a quantum mechanical observable that correlates well with Pauling bond strength. Moreover, a metal atom in a polyhedron has as many bonded radii as it has bonded interactions, resulting in metal and O atoms that may not be spherical. O atoms, for example, are not spherical in many oxide-based crystal structures. Instead, the electron density of a bonded oxygen is often highly distorted or polarized, with its bonded radius decreasing systematically from ~1.38 Å when bonded to highly electropositive atoms like sodium to 0.64 Å when bonded to highly electronegative atoms like nitrogen. Bonded radii determined for metal atoms match the Shannon (1976) radii for more electropositive atoms, but the match decreases systematically as the electronegativities of the M atoms increase. As a result, significant departures from the radius ratio rule in the analysis by George et al. (2020) is not surprising. We offer a modified, more fundamental version of Pauling’s first rule and demonstrate that the second rule has a one-to-one connection between the electron density accumulated between the bonded atoms at the bond critical point and the Pauling bond strength of the bonded interaction.Pauling’s second rule implicitly assumes that bond strength is invariant with bond length for a given pair of bonded atoms. Many studies have since shown that this is not the case, and Brown and Shannon (1973) developed an equation and a set of parameters to describe the relation between bond length and bond strength, now redefined as bond valence to avoid confusion with Pauling bond-strength. Brown (1980) used the valence-sum rule, together with the path rule and the valence-matching principle, as the three axioms of bond-valence theory (BVT), a powerful method for understanding many otherwise elusive aspects of crystals and also their participation in dynamic processes. We show how a priori bond-valence calculations can predict unstrained bond-lengths and how bond-valence mapping can locate low-Z atoms in a crystal structure (e.g., Li) or examine possible diffusion pathways for atoms through crystal structures.In addition, we briefly discuss Pauling’s third, fourth, and fifth rules, the first two of which concern the sharing of polyhedron elements (edges and faces) and the common instability associated with structures in which a polyhedron shares an edge or face with another polyhedron and contains high-valence cations. The olivine [α-(MgxFe1–x)2SiO4] crystal structure is used to illustrate the distortions from hexagonal close-packing of O atoms caused by metal-metal repulsion across shared polyhedron edges.We conclude by discussing several applications of BVT to Earth materials, including the use of BVT to: (1) locate H+ ions in crystal structures, including the location of protons in the crystal structures of nominally anhydrous minerals in Earth’s mantle; (2) determine how strongly bonded (usually anionic) structural units interact with weakly bonded (usually cationic) interstitial complexes in complex uranyl-oxide and uranyl-oxysalt minerals using the valence-matching principle; (3) calculate Lewis acid strengths of cations and Lewis base strengths of anions; (4) determine how (H2O) groups can function as bond-valence transformers by dividing one bond into two bonds of half the bond valence; (5) help characterize products of sorption reactions of aqueous cations (e.g., Co2+ and Pb2+) and oxyanions [e.g., selenate (Se6+O4)2− and selenite (Se4+O3)2−] at mineral-aqueous solution interfaces and the important role of protons in these reactions; and (6) help characterize the local coordination environments of highly charged cations (e.g., Zr4+, Ti4+, U4+, U5+, and U6+) in silicate glasses and melts.

中文翻译：

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鲍林对氧化物矿物的规则：基于量子力学约束和键价理论在地球材料中的现代应用的重新检验

自 1929 年推出以来，鲍林的五项规则已被许多学科的科学家用于合理化和预测晶体固体中原子和配位多面体的稳定排列；无定形材料，例如硅酸盐玻璃和熔体；纳米材料，结晶性差的固体；水性阳离子和阴离子络合物；和矿物水溶液界面处的吸附复合物。这些简单而强大的规则的预测能力最近受到 George 等人的挑战。(2020)，他对大约 5000 个氧化物晶体结构的鲍林五项规则的性能进行了统计分析。他们得出的结论是，只有 13% 的氧化物同时满足最后四个规则，而第二个规则有最多的例外。他们还发现，只有 66% 的测试配位环境满足鲍林的第一条规则，并得出结论，由于单价半径的可变质量，没有简单的规则将离子半径与配位环境联系起来。我们解决了这些问题并讨论了量子力学补充鲍林规则的计算，特别是他的第一条（半径和半径比规则）和第二条（静电价规则）规则。我们还提出了原子键合半径的更现实视图，该视图是通过确定沿键合原子之间的轨迹（称为键合路径，即键合临界点 (rc)）测量的电子密度分布中的局部最小值得出的。键临界点处的电子密度是与鲍林键强度密切相关的量子力学可观测值。而且，多面体中的金属原子具有与键合相互作用一样多的键合半径，导致金属和 O 原子可能不是球形的。例如，O 原子在许多基于氧化物的晶体结构中不是球形的。相反，键合氧的电子密度通常是高度扭曲或极化的，其键合半径从与钠等高正电原子键合时的约 1.38 Å 到与氮等高电负性原子键合时的 0.64 埃有系统地减小。金属原子的键合半径与更多正电原子的 Shannon (1976) 半径相匹配，但随着 M 原子的电负性增加，匹配会系统地降低。因此，George 等人的分析中明显偏离了半径比规则。（2020）并不奇怪。我们提供修改后的，鲍林第一条规则的更基本版本，并证明第二条规则在键临界点处键合原子之间累积的电子密度与键合相互作用的鲍林键强度之间存在一对一的联系。鲍林的第二条规则隐含地假设对于给定的键合原子对，键强度与键长无关。许多研究表明情况并非如此，Brown 和 Shannon (1973) 开发了一个方程和一组参数来描述键长和键强度之间的关系，现在重新定义为键价以避免与鲍林键混淆-力量。Brown (1980) 使用价和规则，连同路径规则和价匹配原则，作为键价理论 (BVT) 的三个公理，了解晶体的许多其他难以捉摸的方面以及它们参与动态过程的有力方法。我们展示了先验键价计算如何预测无应变键长以及键价映射如何定位晶体结构（例如 Li）中的低 Z 原子或检查原子通过晶体结构的可能扩散路径。此外，我们简要讨论鲍林的第三、第四和第五条规则，其中前两条涉及多面体元素（边和面）的共享以及与多面体与另一个多面体共享边或面并包含高-价阳离子。橄榄石 [α-(MgxFe1–x)2SiO4] 晶体结构用于说明由共享多面体边缘的金属-金属排斥引起的 O 原子六方密堆积引起的扭曲。我们通过讨论 BVT 在地球材料中的几种应用得出结论，包括使用 BVT 来： (1) 在晶体结构中定位 H+ 离子，包括质子在地幔中名义上无水矿物晶体结构中的位置；(2) 使用化合价匹配原理确定复合铀氧化物和铀酰含氧盐矿物中键合（通常是阴离子）结构单元与弱键合（通常是阳离子）间隙配合物相互作用的强度；(3)计算阳离子的路易斯酸强度和阴离子的路易斯碱强度；(4) 通过将一个键分成两个键合价一半的键，确定 (H2O) 基团如何充当键合价转换器；(5) 帮助表征水性阳离子（例如，Co2+ 和 Pb2+）和氧阴离子 [例如，亚硒酸盐 (Se6+O4)2- 和亚硒酸盐 (Se4+O3)2-] 在矿物-水溶液界面和质子在这些反应中的重要作用；(6) 帮助表征硅酸盐玻璃和熔体中高电荷阳离子（例如 Zr4+、Ti4+、U4+、U5+ 和 U6+）的局部配位环境。