Here, we describe two sets of thermodynamic ionic radii ( r 1 and r 2 ) of lanthanide ions derived from the analysis of large set of the stability constants log β 1 and log β 2 of the ML and ML 2 complexes of lanthanide ions M (from Ce 3+ to Lu 3+ ) with organic ligands (L) in water. It has been demonstrated that the stability constants of two metals M i and M j with a given ligand L are related by simple equations log β 1 j = r 1 i / r 1 j. log β 1 i and log β 2 j = r 2 i / r 2 j. log β 2 i which formally correspond to purely electrostatic interactions between spherical cations with organic molecule. Predictive performance of these equations was assessed in fivefold cross-validation procedure. The standard deviation ( s ) of predictions varies from 0.3 to 1.0 for log β 1 and from 0.4 to 1.2 for log β 2 as a function of the difference of Shannon ionic radii (Δ r ij ) of M i and M j : s 1 = 0.27 + 4.25Δ r ij for log β 1 and s 2 = 0.39 + 5.16Δ r ij for log β 2 . The new radii r 1 and r 2 steadily decrease across the Ln series, i.e., they follow the same trend as Shannon effective radii of lanthanide ions. The calculations were performed using experimental data on 2854 log β 1 values for 445 organic ligands and 947 log β 2 values for 156 organic ligands in the complexes with 13 metals.

中文翻译：

---

源自金属-配体配合物稳定性常数的镧系元素离子的热力学半径

在这里，我们描述了两组镧系元素离子的热力学离子半径（r 1 和 r 2 ），这些半径来自对镧系元素离子 M 的 ML 和 ML 2 复合物的大组稳定性常数 log β 1 和 log β 2 的分析（从 Ce 3+ 到 Lu 3+ ) 与水中的有机配体 (L)。已经证明，具有给定配体 L 的两种金属 M i 和 M j 的稳定性常数通过简单的方程 log β 1 j = r 1 i / r 1 j 相关。log β 1 i 和 log β 2 j = r 2 i / r 2 j。log β 2 i 形式上对应于球形阳离子与有机分子之间的纯静电相互作用。在五重交叉验证程序中评估了这些方程的预测性能。对于 log β 1，预测的标准偏差 (s) 从 0.3 到 1.0，从 0.4 到 1。2 对于 log β 2 作为 Mi 和 M j 的香农离子半径 (Δ r ij ) 之差的函数： s 1 = 0.27 + 4.25Δ r ij 对于 log β 1 和 s 2 = 0.39 + 5.16Δ r ij对于 log β 2 。新的半径 r 1 和 r 2 在整个 Ln 系列中稳步减小，即它们遵循与镧系元素离子的香农有效半径相同的趋势。计算是使用 445 个有机配体的 2854 log β 1 值和 13 种金属的配合物中 156 个有机配体的 947 log β 2 值的实验数据进行的。