A quantum chemical model for a series of self-assembled nanocages: the origin of stability behind the coordination-driven formation of transition metal complexes up to [M12L24]24+

Herein, we present a systematic computational model to study the electronic states and free energies of a self-assembled multi-metal complex series. By combining the previously developed model Hamiltonian approach for transition-metal complexes and the generalized Born model, the thermodynamics, optimized geometries, and electronic states of the [Pd12L24]24+ nanocage are revealed, together with [PdnLm]2n+ complex series. The effective model Hamiltonian is a theoretical method to obtain the d-electron wavefunction and potential energy including interaction energy between the transition-metal and ligands. In the present improvement, the electronic state on each transition-metal center is focused as a building unit and solved under the whole electronic field of the assembling system. We realize a reliable and systematic treatment of multi-transition-metal complexes with different sizes and charges. Consequently, our model could reproduce binding energies of the [PdnLm]2n+ complex series quantitatively as compared to density functional theory (DFT). Regarding free energy, we revealed that the assembling solute becomes unstable due to the electrostatic interaction, and effects of the solvent and counter anions mainly compensated it. Optimized geometries were also analysed. The local square-planar coordination structures around the palladium centres were characterized in the complex series. The relationships between the entire symmetrical geometries and the local coordination structures are also discussed. Finally, electronic structures of the [Pd12L24]24+ nanocage were well characterized as a single-determinant, where only dx2−y2 is unoccupied due to the ligand-field effect. We also found that the solvent polarized the electronic states of the Pd ions, whereas the counter anion suppressed the polarization. The present method realizes size-independent reliable and rapid computations, and therefore can be expected to further application studies on self-assembly dynamics.