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.

本研究采用量子力学计算（quantum mechanical calculations）方法，针对水溶液中金属离子与单氨（NH3）配体形成的配合物的稳定常数对数（log K1）展开预测，并报道相关研究结果。研究针对34种金属离子，利用密度泛函理论（DFT）计算了反应式1在298 K气相下的吉布斯自由能变化值（ΔG(DFT)），预期其可与水溶液中配合物形成的吉布斯自由能变化（ΔG(aq)）建立关联。反应式1如下：[M(H2O)6]ⁿ⁺(g) + NH3(g) = [M(H2O)5NH3]ⁿ⁺(g) + H2O(g)。ΔG(aq)涵盖了溶剂化复杂变化对配合物形成的影响，而此类影响并未纳入式1的计算范畴。此前研究认为，对于不同金属离子，此类溶剂化变化的影响或为恒定值，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)之间的简单关联。本研究用于计算ΔG(aq)的log K1(NH3)数据大多并非实验实测值，而是此前通过各类经验关联方法估算所得（Hancock 1978, 1980）。研究发现，针对不同电荷价态的金属离子（M²⁺、M³⁺及M⁴⁺），ΔG(DFT)与ΔG(aq)之间分别存在线性关联关系。在ΔG(DFT)对ΔG(aq)的散点图中，其斜率范围从M²⁺离子的2.201降至M⁴⁺离子的1.076，而截距则随金属离子价态从M²⁺到M⁴⁺逐步升高。针对M³⁺离子则存在两组独立的关联关系，这两组关联分别对应配位数（coordination number, 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。所有上述独立关联的相关系数（R）均处于0.97~0.99的优异区间，表明本研究采用的方法可作为未来预测水溶液化学性质的可行基础——此类化学性质若仅通过传统实验手段往往难以获取。据此，本研究预测了超铀元素铹（Lr³⁺，其最稳定同位素的半衰期为3.6小时）的log K1(NH3)值为1.46。本研究的计算结果还有助于进一步阐明水溶液中配合物形成的关键影响因素。上述所有密度泛函理论计算均对标量相对论效应（RE）进行了校正。金（Au）被描述为"relativistic element"（Koltsoyannis 1997）。对于第11族金属离子而言，相对论效应的主要作用是促使其倾向于形成线性配位几何构型，并大幅增强M-L键的共价性。针对M⁺离子（H⁺、Cu⁺、Ag⁺、Au⁺）的关联分析涉及[M(H2O)2]⁺配合物的偏好线性配位特性，因此针对M=H⁺、Cu⁺、Ag⁺及Au⁺的气相反应式2的ΔG值的密度泛函理论计算均基于此展开。反应式2如下：[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密度泛函理论计算。计算结果表明，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。上述结果提示，在皮尔逊硬软酸碱理论中，水溶液中多数"soft acid"的行为特性，或源于周期表中靠近金的元素的标量相对论效应。