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.

本文报道了利用量子力学计算，对水溶液中金属离子与单个氨配体形成的配合物的稳定常数（log K1）开展预测的相关研究。本研究针对34种金属离子，采用密度泛函理论（DFT）计算了式1在298 K气相条件下的吉布斯自由能变（ΔG(DFT)），并预期该计算结果将与水溶液中配合物形成过程的吉布斯自由能变（ΔG(aq)）存在相关性。式1为：[M(H2O)6]n+(g) + NH3(g) = [M(H2O)5NH3]n+(g) + H2O(g)。ΔG(aq)涵盖了溶剂化过程变化对配合物形成的影响，而该部分效应并未纳入式1的计算范畴。此前研究推测，对于不同金属离子，这类溶剂化变化的程度将保持恒定或随log K1值呈系统性变化，因此本研究尝试建立ΔG(DFT)与ΔG(aq)之间的简单线性关联。用于计算ΔG(aq)的大部分log K1(NH3)数值并非实验实测值，而是先前通过多种经验关联方法估算得到（Hancock 1978, 1980）。针对不同电荷类型的金属离子（M2+、M3+和M4+），本研究分别建立了ΔG(DFT)与ΔG(aq)的线性关联模型。在ΔG(DFT)对ΔG(aq)的散点图中，关联斜率从M2+离子的2.201逐步降至M4+离子的1.076，而截距则随电荷从M2+到M4+逐步升高。M3+离子存在两组独立的关联模型，分别对应配位数（CN）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。所有这些独立关联模型的相关系数（R）均处于0.97−0.99的高区间内，表明本研究采用的方法可作为未来预测难以通过实验直接获取的水相化学行为的基础。据此，我们预测最稳定同位素半衰期仅为3.6小时的超铀元素铹的三价离子Lr3+的log K1(NH3)值为1.46。本研究的计算结果还有助于深入理解调控水溶液中配合物形成过程的各类影响因素。上述所有DFT计算均对标量相对论效应（RE）进行了校正。金元素曾被描述为‘相对论性元素’（Koltsoyannis 1997）。对于第11族离子，相对论效应的主要作用是促进线性配位几何结构的形成，并显著增强M−L键的共价性。针对M+离子（H+、Cu+、Ag+、Au+）的关联模型涉及[M(H2O)2]+配合物的优选线性配位模式，因此针对M=H+、Cu+、Ag+和Au+的情况，我们计算了式2所示气相反应的ΔG。式2为：[M(H2O)2]+(g) + NH3(g) = [M(H2O)NH3]+(g) + H2O(g)。本研究还针对式2开展了未进行相对论效应校正的额外DFT计算。结果显示，在未校正相对论效应的情况下，H+的log K1(NH3)值几乎无变化，Cu+的该值小幅下降，Ag+的下降幅度更大；而Au(I)的log K1(NH3)值则会从校正相对论效应后的9.8大幅降至未校正时的1.6。上述结果表明，在皮尔逊提出的硬软酸碱分类体系中，水溶液中多数软酸的行为特征，可能源于周期表中靠近金的元素所受的相对论效应。