Exact finite-spectrum vibrational thermodynamics and radial information structure of diatomic hydrides in the newing potential

Analytical molecular potentials acquire their greatest physical value when the bound spectrum, thermal response, and eigenstate structure are examined within a single framework. We determine the complete \(J=0\) vibrational branches of \(H_2\) , LiH, and ScH in the Newing potential and connect their spectral organization to finite-level thermodynamics and reduced radial information measures. The three parameter sets support 15, 26, and 22 bound states. Independent finite-difference diagonalization reproduces the analytical levels with maximum relative deviations below \(2× 10⁻⁴\) on the finest grid. The resulting excitation ladders exhibit markedly different activation scales: the first gap is 0.646791 eV for \(H_2\) , compared with 0.198669 eV for LiH and 0.201394 eV for ScH. These differences govern the onset of vibrational population, the heat-capacity structure, and the finite-state entropy ceilings. Dimensionless half-line Shannon measures obtained from the reduced radial wavefunctions and their Fourier-sine transforms further resolve the interplay between spatial confinement, nodal growth, and conjugate-space spreading. The analysis establishes a direct connection from potential geometry to state counting, thermodynamic activation, and radial information structure in finite molecular spectra.