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The Lever Equation
Ligand electrochemical parameters, commonly known as Lever parameters, comprise a set of values assigned to ligands describing the incremental change in a metal's reduction potential as a result of attaching the ligand. Lever parameters enable the estimation of the reduction potential of any octahedral transition metal complex by the Lever equation, shown here. In this equation, Sm and Im are characteristic constants of the metal, and EL(i) is the Lever parameter of the ligand attached to coordination site i.
Lever parameters have played a major role in understanding redox processes involved in inorganic electrochemistry, enzymatic reactions, catalysis, solar cells, biochemistry, and materials science. Despite thier broad usefulness, Lever parameters are not well understood at a first-principles level. One of our ongoing theoretical projects is to obtain a fundamental understanding of the relationships between EL parameters and the electronic structures of ligands and their metal complexes.
Through the course of our research, we have demonstrated that a ligand's EL parameter is related to pi-type orbital interactions between the ligand's MOs and the t2g d-orbitals of the metal, and to electron repulsions caused by the ligand's sigma-donation. Thus, by calculating sigma repulsions and pi orbital interactions using density functional theory (DFT), we can predict a ligand's EL parameter with reasonable accuracy, as shown in the correlation plot.
The good general correlation demonstrates that we are indeed acquiring an understanding of the fundamental nature of Lever parameters. However, the plot also shows that there is still a lot we don't know. The light blue triangles are anionic ligands, for which our theoretical arguments are clearly incomplete. The two red dots in the plot are N2 and CO. Here again, we see considerable error in the prediction. We are currently investigating the effects of including solvation, hardness factors, and relativistic contributions in the calculations.
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