Quantum-Chemical Calculation of Acidity Indices of Cu(II) Aqua Ions by the Density Functional Theory Method within the Framework of the Molecular-Continuum Model of Hydration

Quantum-chemical calculations of the acidity indices of Cu(II) aqua ions were performed using the B3LYP, PBE0, and ω-B97XD functionals in combination with the 6-311++G(d,p), aug-cc-pVDZ, and def2-TZVP atomic basis sets. In the molecular-continuum model, the Cu(H₂O)\(_{{18}}^{{2 + }}\) aqua complex (six H₂O molecules in the first and 12 molecules in the second hydration sphere) was taken to be the reference Cu(II) aqua ion. Its interaction with the dielectric environment in the solution was considered in the polarized continuum model. In the initial aqua complex, protons were sequentially removed, and the resulting Cu(II) aquahydroxo complexes were subjected to full geometry optimization. The Gibbs free energies of ionization processes were calculated as the difference between the total Gibbs energies of the reaction products and the initial reagents. Free energies for intermediate aquahydroxo complexes were obtained from thermochemical analysis, and for the proton in an aqueous solution \({\text{H}}_{{({\text{aq}})}}^{ + }\), the free energy was calculated by a special approach using the experimental Gibbs energy of hydration. Analysis of the calculation results showed that the B3LYP/aug-cc-pVDZ and ω-B97XD/aug-cc-pVDZ combinations provided a satisfactory agreement with the experimental values for all four stages of ionization (hydrolysis) of Cu(II) aqua ions. Therefore, it can be concluded that these combinations of the density functional theory level and the atomic basis set can be recommended for calculating the formation constants of Cu(II) complexes in aqueous solutions. Along with this, it is noted that for the first stage of ionization of the Cu(II) aqua ion, from which the primary reconstruction of the aqua complex begins, satisfactory pK_a1 values are achieved at almost all calculation levels. Therefore, other calculation levels used may also prove satisfactory for calculating the formation constants of Cu(II) complexes in aqueous solutions.