A Unified Analytical Framework for Hydrogenic Probability Distributions: From Orthogonal Polynomials to 3D Nodal Topology.

We present an analytical framework for characterizing hydrogenic probability distributions across radial, angular, and three-dimensional geometries for arbitrary quantum numbers (n,ℓ,m) and nuclear charge Z. Using scaled Rodrigues representations, we derive: (i) a closed-form polynomial equation that directly determines all radial anti-nodes (most probable radii) for arbitrary (n,ℓ), with algebraic cancellation inherently filtering nodal minima to isolate physical maxima; (ii) a universal radial probability integral enabling exact evaluation over arbitrary intervals [m₁a₀, m₂a₀] for any (n,ℓ) without numerical quadrature; (iii) polynomial extremum conditions for angular maxima of |Y_ℓm|² expressed without derivatives of associated Legendre functions; and (iv) a complete three-dimensional nodal topology comprising (n−ℓ−1) spherical nodes, (ℓ−|m|) conical nodes, and an axial nodal line of multiplicity |m|, forming (n−ℓ)(ℓ−|m|+1) connected probability compartments. Accompanying C++ implementations compute radial/angular extrema and probability integrals for any (n,ℓ,m,Z) without quadrature, achieving 85× speedup over recursive schemes and enabling high-n calculations beyond numerical hardware limits. This framework bridges orthogonal-polynomial structure with atomic architecture for research and pedagogy.