Mathematical State Function Structure Matters in Metal Oxide Reduction Thermodynamic Modeling

Thermodynamic modeling investigations commonly focus on data sufficiency, regularization strategies, and optimizer choice, implicitly assuming that mathematically equivalent state function formulations yield equivalent physical insight. We show that this assumption fails. The structural form of the state function can dominate solution robustness, parameter identifiability, and physical interpretability. By systematically comparing alternative formulations under identical data, constraints, and optimization frameworks, this investigation identifies formulation-dependent failure modes and establishes criteria for constructing thermodynamically meaningful and numerically robust models. The compound energy formalism (CEF) achieves this rigor by explicitly constructing the Gibbs free energy as a function of sublattice occupancies and excess interaction terms. Implementing CEF is challenging because of the high dimensionality of the parameter space and the associated difficulty in finding unique, physically meaningful, global minima. In this study, we evaluate how mathematically equivalent but structurally distinct state function formulations influence robustness and the quality of the extracted thermodynamics in conjunction with our previously developed CrossFit CEF (CF-CEF) method, which integrates both experimental and first-principles data to reduce parameter degeneracy. Using the (Ba,Sr)FeO3-δ (BSF) perovskite family as a model system, we compare two common free energy expansions across multiple optimizers, parameter initializations, and normalization strategies. We demonstrate that state function form selection strongly affects enthalpy and entropy stability. Between the two equivalent functional forms, we find that the simpler three-parameter free energy expansion, ΔG = G° + BT - ATln(T), is highly susceptible to parameter compensation (i.e., trade-off), giving enthalpy differences of 10s of kJ/mol O and entropy differences of 10s of J/mol/K depending on the parameter fitting methodology. Conversely, using the thermodynamically equivalent three-parameter expression ΔG = H° + (A - S°)T - ATln(T) consistently decreases sensitivity to selected optimization techniques and yields physically intuitive results regardless of parameter optimizer or initialization. This work demonstrates that careful mathematical construction of state functions is as critical as data set quality, and that the ΔG = H° + (A - S°)T - ATln(T) construction should be used for future thermodynamic CF-CEF models.