Emergence of Atomic Energy States: Predicting Ionization Energies of One-Electron Systems via Nucleus-Driven Force Propagation in Real Space

Although probabilistic electron wave-function approaches provide solid mechanistic insights for molecules, their inherent reliance on static, boundary-dependent approximations limits the direct description of real-space molecular dynamics. This study demonstrates the emergence of highly accurate atomic energy states by a single nucleus-driven force-propagating framework within an open system.When compared withNIST ionization energy dataforone-electron systems,our model yields a 0.0000% relative error for the hydrogen atom (Z = 1), where the energy spectrum naturally emerges from the nucleus as wave maximum peaks in real space. Furthermore, by relying solely on the atomic number (Z) as an independent variable alongside fixed global parameters, the model autonomously generates macroscopic energy values across various elements,maintaining relative errors of less than 1% up to Pr (Z = 59) and remaining even for massive nuclei such as Au (1.88%) and Rn (2.59%), with relative errors quite low at 1.88% and 2.59%, respectively. These highly accurate predictions are achieved through the Causal Recording (CR) framework, which tracks the irreversible and continuous propagation of classical nuclear forces. Comparisons with the emergentLagrangianpathconfirmedthatmacroscopictrajectoriesandwavebehaviors autonomously emerge from the collective accumulation oflocalforce events.Ultimately,these results successfully confirm that intra-atomic electron behavior can be described by the classical force of the nucleus in real space.