This study focuses on conducting a comparative study of the extraction capacities of alizarin-oxalate (AR-Ox) ligands with La3+ and Nd3+ in acidic, neutral, and alkaline mediums. Density functional theory calculations at ωB97X-D/6-311++G(d,p)/SDD level have been performed for structural, thermochemical, frontier-orbital (highest occupied molecular orbitals and lowest unoccupied molecular orbitals), natural bond orbital, reduced density gradient (RDG), and density of state analysis for alizarin-oxalate-La(III) (AR-Ox-La) and alizarin-oxalate-Nd(III) (AR-Ox-Nd) complexes. The bonding characteristics of La3+ & Nd3+ ions with alizarin-oxalate ligand have been analysed using the quantum theory of atoms in molecules, revealing the presence of an intermediate type of bond between closed-shell and shared-shell electrons in (La/Nd)-O, (La/Nd)-C. The reduced density gradient (RDG) and iso-surface generated through the Multiwfn program shows mostly hydrogen-like and van der Waals interaction between La3+/Nd3+ and oxygen atoms of alizarin-oxalate ligand except for some of the complexes showing the presence of non-bonded/repulsive (La/Nd)-O interaction. Thermochemical, DOS, and natural bond orbital analysis reveals alizarin-oxalate-(La3+/Nd3+) complexes in the alkaline medium is more stable than in neutral and acidic medium, and the stability of AR-Ox-Nd complexes is more than AR-Ox-La complexes. It is observed that participation of oxygen atoms from both alizarin and oxalate in bond formation with lanthanides enhances the stability of alizarin-oxalate-lanthanide complexes, emphasizing the pivotal role of ligand coordination modes. This work illustrates the subtle differences in chelating properties of alizarin-oxalate ligands with La3+ and Nd3+ for designing new ligands for efficient selective lanthanide separation.