Chemical Similarity Enables a Unified Free-Energy Framework for f -Block Solvation Thermodynamics

Abstract

f-Block elements underpin many essential technologies, yet their efficient mutual separation remains a foundational challenge in chemistry. The lanthanide series provide a stringent test case: the trivalent lanthanide ions (Ln3+) exhibit near-identical solvation chemistry across the series, distinguished only by the lanthanide contraction. Here, we turn this chemical similarity into a computational advantage. Treating each solvated Ln3+ as a distinct alchemical state, we exploit configurational overlap between neighboring ions by applying the multistate Bennett acceptance ratio (MBAR) to estimate relative hydration free energies from pooled ensembles. We extend this framework to biased simulations by coupling coordination-number-based well-tempered metadynamics with MBAR, enabling enhanced sampling of rare coordination fluctuations within a unified free-energy estimator. Benchmarked in water as a controlled, well-characterized solvent, the workflow yields convergence, overlap, and effective sample size (ESS) diagnostics alongside free-energy trends in good agreement with experiment. MBAR-enabled energy–entropy decomposition reveals discrepancies in the underlying energetic and entropic contributions despite good agreement in total free energy, motivating a multifidelity test with a machine-learned polarizable interatomic potential. MBAR-based reweighting reveals insufficient Hamiltonian overlap, exposing a transferability bottleneck; the pooled classical ensemble does not support reliable post hoc reweighting to the target Hamiltonians. Domain-specific fine-tuning, renewed sampling, and explicit overlap reassessment are therefore required before MACE-polar can be incorporated as a higher-fidelity layer. This framework establishes “UfS”─a unified free-energy framework for f-block solvation─as a scalable, uncertainty-aware approach to lanthanide solvation thermodynamics and a foundation for future applications to related actinides, speciation-rich environments, and data-driven design of new separations chemistries.