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.

本研究报道了采用量子力学计算方法，对水溶液中金属离子与单个氨（NH3）配体形成配合物的生成常数（log K1）开展预测的相关工作。针对34种金属离子，本研究采用密度泛函理论（DFT）计算了298 K下气相反应式(1)的吉布斯自由能（ΔG）值，记为ΔG(DFT)，并假设该值可与水溶液中配合物生成的吉布斯自由能（ΔG(aq)）建立关联。式(1)：[M(H₂O)₆]ⁿ⁺(g) + NH₃(g) = [M(H₂O)₅NH₃]ⁿ⁺(g) + H₂O(g)。ΔG(aq)值包含了溶剂化作用的复杂变化对配合物生成的影响，而该影响并未被纳入式(1)的计算范畴。研究假设，针对不同金属离子，这类溶剂化变化的程度为恒定值，或随log K1值呈系统性变化，因此本研究旨在建立ΔG(DFT)与ΔG(aq)之间的简单关联关系。用于计算ΔG(aq)的多数log K1(NH3)值并非实验实测值，而是此前由Hancock等人于1978年、1980年通过多种经验关联方法估算得到的。研究针对不同电荷态的金属离子（M²⁺、M³⁺、M⁴⁺）分别建立了ΔG(DFT)与ΔG(aq)之间的线性关联。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，截距则随电荷从M²⁺到M⁴⁺逐渐升高。针对M³⁺离子则存在两组独立的关联关系，分别对应配位数（CN）为6的小尺寸金属离子，以及配位数在7~9区间的大尺寸金属离子。所有这些独立关联的相关系数（R）均处于0.97~0.99的较高区间，表明本研究采用的方法可作为未来预测水相化学行为的基础——这类水相化学行为往往难以通过实验手段获取。据此，本研究预测超铀元素铹的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更深入地理解调控水溶液中配合物生成的各类因素。上述所有DFT计算均纳入了标量相对论效应（RE）的校正项。Koltsoyannis于1997年曾将金（Au）描述为‘相对论性元素’。对于第11族离子而言，标量相对论效应的主要作用是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]⁺配合物的优选线性配位模式，因此本研究针对M=H⁺、Cu⁺、Ag⁺、Au⁺开展了气相反应式(2)的ΔG值DFT计算。式(2)：[M(H₂O)₂]⁺(g) + NH₃(g) = [M(H₂O)NH₃]⁺(g) + H₂O(g)。本研究还开展了不包含RE校正项的式(2)DFT计算。结果显示，在不考虑RE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)值会从纳入RE时的9.8大幅降至未纳入RE时的1.6。上述结果表明，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，可能源于周期表中紧邻金的元素所具有的相对论效应。