## Abstract

The isotopic fractionation factor and element partition coefficient can be calculated only after the geometric optimization of the molecular clusters is completed. Optimization directly affects the accuracy of some parameters, such as the average bond length, molecular volume, harmonic vibrational frequency, and other thermodynamic parameters. Here, we used the improved volume variable cluster model (VVCM) method to optimize the molecular clusters of a typical oxide, quartz. We documented the average bond length and relative volume change. Finally, we extracted the harmonic vibrational frequencies and calculated the equilibrium fractionation factor of the silicon and oxygen isotopes. Given its performance in geometrical optimization and isotope fractionation factor calculation, we further applied the improved VVCM method to calculate isotope equilibrium fractionation factors of Cd and Zn between the hydroxide (Zn–Al layered double hydroxide), carbonate (cadmium-containing calcite) and their aqueous solutions under superficial conditions. We summarized a detailed procedure and used it to re-evaluate published theoretical results for cadmium-containing hydroxyapatite, emphasizing the relative volume change for all clusters and confirming the optimal point charge arrangement (PCA). The results showed that the average bond length and isotope fractionation factor are consistent with those published in previous studies, and the relative volume changes are considerably lower than the results calculated using the periodic boundary method. Specifically, the average Si–O bond length of quartz was 1.63 Å, and the relative volume change of quartz centered on silicon atoms was − 0.39%. The average Zn–O bond length in the Zn–Al-layered double hydroxide was 2.10 Å, with a relative volume change of 1.96%. Cadmium-containing calcite had an average Cd–O bond length of 2.28 Å, with a relative volume change of 0.45%. At 298 K, the equilibrium fractionation factors between quartz, Zn–Al-layered double hydroxide, cadmium-containing calcite, and their corresponding aqueous solutions were \(\Delta ^{30/28} {\text{Si}}_{{{\text{Qtz-H}}_{4} {\text{SiO}}_{4} }} = 2.20{\permil} \), \(\Delta^{18/16} {\text{O}}_{ {\text{Qtz}}{-} ( {\text{H}}_{2} {\text{O}} )_{\text{n}}} = 36.05{\permil}\), \(\Delta^{66/64} {\text{Zn}}_{ {\text{Zn}} {-} {\text{Al LDH-Zn}} ( {\text{H}}_{2} {\text{O}} )_{\text{n}}^{2+}} = 1.12{\permil}\) and \(\Delta^{114/110} {\text{Cd}}_{ {\text{(Cd--Cal)-Cd}} ( {\text{H}}_{2} {\text{O}} )_ {\text{n}}^{2 +} } = - 0.26{\permil}\) respectively. These results strongly support the reliability of the improved VVCM method for geometric optimization of molecular clusters.