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Our previously defined Sparkle model (Inorg. Chem. 2004, 43, 2346) has been reparameterized for Eu(III) as well as newly parameterized for Gd(III) and Tb(III). The parameterizations have been carried out in a much more extensive manner, aimed at producing a new, more accurate model called Sparkle/AM1, mainly for the vast majority of all Eu(III), Gd(III), and Tb(III) complexes, which possess oxygen or nitrogen as coordinating atoms. All such complexes, which comprise 80% of all geometries present in the Cambridge Structural Database for each of the three ions, were classified into seven groups. These were regarded as a "basis" of chemical ambiance around a lanthanide, which could span the various types of ligand environments the lanthanide ion could be subjected to in any arbitrary complex where the lanthanide ion is coordinated to nitrogen or oxygen atoms. From these seven groups, 15 complexes were selected, which were defined as the parameterization set and then were used with a numerical multidimensional nonlinear optimization to find the best parameter set for reproducing chemical properties. The new parameterizations yielded an unsigned mean error for all interatomic distances between the Eu(III) ion and the ligand atoms of the first sphere of coordination (for the 96 complexes considered in the present paper) of 0.09 A, an improvement over the value of 0.28 A for the previous model and the value of 0.68 A for the first model (Chem. Phys. Lett. 1994, 227, 349). Similar accuracies have been achieved for Gd(III) (0.07 A, 70 complexes) and Tb(III) (0.07 A, 42 complexes). Qualitative improvements have been obtained as well; nitrates now coordinate correctly as bidentate ligands. The results, therefore, indicate that Eu(III), Gd(III), and Tb(III) Sparkle/AM1 calculations possess geometry prediction accuracies for lanthanide complexes with oxygen or nitrogen atoms in the coordination polyhedron that are competitive with present day ab initio/effective core potential calculations, while being hundreds of times faster.