The little-known non-orthogonality of hydrogenic radial wavefunctions: an ℓ -dependent operator perspective

Abstract

Abstract It is often assumed that hydrogenic radial wavefunctions R nℓ ( r ) inherit universal orthogonality from the total wavefunctions ψ nℓm . This leads to the inference that all R nℓ are eigenfunctions of a single universal radial operator and thus form an orthonormal basis for L 2 ([0, ∞), r 2 d r ). We demonstrate this is false: orthogonality holds only within fixed- ℓ subspaces, where all R nℓ share the same self-adjoint operator h ˆ ℓ . For ℓ ≠ ℓ ′ , radial functions belong to distinct Sturm–Liouville families with operators differing in their centrifugal terms ∝ ℓ ( ℓ + 1)/ r 2 . This structural difference—not numerical artifact—explains the non-zero radial overlap (e.g. the 2 s –2 p overlap equals − 3 / 2 ). We classify all radial overlap integrals into four cases based on ( n , n ′ , ℓ , ℓ ′ ) , demonstrate non-orthogonality with explicit counterexamples, and explain the operator-family origin of this behavior. This perspective prevents subtle errors in basis expansions and perturbation theory for hydrogenic systems.