Curvature of the lanthanide contraction: an explanation

A number of studies have shown that for isostructural series of the lanthanides (elements La through Lu), a plot of equivalent metal-ligand bond lengths versus atomic number differs significantly from linearity and can be better fit as a quadratic equation. However, for hydrogen type wave functions, it is the inverse of the average distance of the electron from the nucleus (an estimate of size) that varies linearly with effective nuclear charge. This generates an apparent quadratic dependence of radius with atomic number. Plotting the inverse of lanthanide ion radii (the observed distance minus the ligand size) as a function of effective nuclear charge gives very good linear fits for a variety of lanthanide complexes and materials. Parameters obtained from this fit are in excellent agreement with the calculated Slater shielding constant, k.