Density Functional Theory-Based Prediction of the Formation Constants of Complexes of Ammonia in Aqueous Solution: Indications of the Role of Relativistic Effects in the Solution Chemistry of Gold(I)

A prediction of the formation constants (log K1) for complexes of metal ions with a single NH3 ligand in aqueous solution, using
quantum mechanical calculations, is reported. ΔG values at 298 K in the gas phase for eq 1 (ΔG(DFT)) were calculated for
34 metal ions using density functional theory (DFT), with the expectation that these would correlate with the free energy of
complex formation in aqueous solution (ΔG(aq)). [M(H2O)6]n+(g) + NH3(g) = [M(H2O)5NH3]n+(g) + H2O(g) (eq 1). The ΔG(aq)
values include the effects of complex changes in solvation on complex formation, which are not included in eq 1. It was
anticipated that such changes in solvation would be constant or vary systematically with changes in the log K1 value for different
metal ions; therefore, simple correlations between ΔG(DFT) and ΔG(aq) were sought. The bulk of the log K1(NH3) values used
to calculate ΔG(aq) were not experimental, but estimated previously (Hancock 1978, 1980) from a variety of empirical correlations.
Separate linear correlations between ΔG(DFT) and ΔG(aq) for metal ions of different charges (M2+, M3+, and M4+) were found.
In plots of ΔG(DFT) versus ΔG(aq), the slopes ranged from 2.201 for M2+ ions down to 1.076 for M4+ ions, with intercepts
increasing from M2+ to M4+ ions. Two separate correlations occurred for the M3+ ions, which appeared to correspond to small
metal ions with a coordination number (CN) of 6 and to large metal ions with a higher CN in the vicinity of 7−9. The good
correlation coefficients (R) in the range of 0.97−0.99 for all these separate correlations suggest that the approach used here
may be the basis for future predictions of aqueous phase chemistry that would otherwise be experimentally inaccessible. Thus,
the log K1(NH3) value for the transuranic Lr3+, which has a half-life of 3.6 h in its most stable isotope, is predicted to be 1.46.
These calculations should also lead to a greater insight into the factors governing complex formation in aqueous solution. All
of the above DFT calculations involved corrections for scalar relativistic effects (RE). Au has been described (Koltsoyannis
1997) as a “relativistic element”. The chief effect of RE for group 11 ions is to favor linear coordination geometry and greatly
increase covalence in the M−L bond. The correlation for M+ ions (H+, Cu+, Ag+, Au+) involved the preferred linear coordination
of the [M(H2O)2]+ complexes, so that the DFT calculations of ΔG for the gas-phase reaction in eq 2 were carried out for M =
H+, Cu+, Ag+, and Au+. [M(H2O)2]+(g) + NH3(g) = [M(H2O)NH3]+(g) + H2O(g) (eq 2). Additional DFT calculations for eq 2 were
carried out omitting corrections for RE. These indicated, in the absence of RE, virtually no change in the log K1(NH3) value for
H+, a small decrease for Cu+, and a larger decrease for Ag+. There would, however, be a very large decrease in the log
K1(NH3) value for Au(I) from 9.8 (RE included) to 1.6 (RE omitted). These results suggest that much of “soft” acid behavior in
aqueous solution in the hard and soft acid−base classification of Pearson may be the result of RE in the elements close to Au
in the periodic table.

本研究报道了利用量子力学计算，对水溶液中金属离子与单个氨（NH3）配体形成的配合物的一级稳定常数对数（log K1）进行预测的工作。针对34种金属离子，采用密度泛函理论（DFT, density functional theory）计算了298 K下气相中方程式1的吉布斯自由能变（ΔG(DFT)），预期其可与水溶液中配合物形成的吉布斯自由能变（ΔG(aq)）建立关联。方程式1为：[M(H2O)6]ⁿ⁺(g) + NH3(g) = [M(H2O)5NH3]ⁿ⁺(g) + H2O(g)。ΔG(aq)涵盖了溶剂化作用变化对配合物形成的影响，而该影响未包含在方程式1中。我们推测，对于不同金属离子，此类溶剂化变化的程度或为恒定，或随log K1值呈系统性变化，因此尝试建立ΔG(DFT)与ΔG(aq)之间的简单关联。用于计算ΔG(aq)的log K1(NH3)数据，多数并非实验实测值，而是此前通过多种经验关联方法估算得到（Hancock，1978、1980）。研究发现，针对不同电荷的金属离子（M²+、M³+和M⁴+），ΔG(DFT)与ΔG(aq)分别存在独立的线性相关性。在ΔG(DFT)对ΔG(aq)的散点图中，斜率范围从M²+离子的2.201降至M⁴+离子的1.076，截距则随金属离子电荷从+2升高至+4而递增。对于M³+离子，存在两组独立的相关性，其分别对应配位数（CN, coordination number）为6的小体积金属离子，以及配位数约为7~9的大体积金属离子。所有这些独立关联的相关系数（R）均处于0.97~0.99之间，表明本研究采用的方法可作为未来预测水相化学的基础，而这类预测往往难以通过实验实现。据此，我们预测了超铀元素铹(III)离子（Lr³+，其最稳定同位素的半衰期为3.6 h）的log K1(NH3)值为1.46。本计算还有助于深入理解水溶液中配合物形成的控制因素。上述所有DFT计算均对标量相对论效应（RE, relativistic effects）进行了校正。金（Au）曾被描述为"相对论性元素"（Koltsoyannis，1997）。相对论效应对第11族离子的主要影响是使其倾向于形成线性配位几何构型，并显著增强M-L键的共价性。针对M+离子（H+、Cu+、Ag+、Au+）的关联，需考虑[M(H2O)2]+配合物的优先线性配位模式，因此针对M=H+、Cu+、Ag+和Au+，我们计算了方程式2的气相反应吉布斯自由能变ΔG。方程式2为：[M(H2O)2]+(g) + NH3(g) = [M(H2O)NH3]+(g) + H2O(g)。我们还针对方程式2开展了未校正相对论效应的额外DFT计算。结果显示，在未考虑相对论效应的情况下，H+的log K1(NH3)值几乎无变化，Cu+的该值小幅降低，Ag+的该值下降幅度更大；但Au(I)的log K1(NH3)值会从考虑相对论效应时的9.8大幅降至未校正时的1.6。上述结果表明，在皮尔逊提出的"软硬酸碱（HSAB, hard and soft acid−base）分类体系"中，水溶液中多数软酸的特性，可能源于周期表中紧邻金的元素所受的标量相对论效应。