Extending Multi-Input Linear Correction to Energy Representation Theory: Accurate Solvation Free Energy Prediction Independent of Volume Information

We extend the multi-input linear correction (MILC) framework to the energy representation (ER) theory to improve the accuracy and robustness of solvation free energy (SFE) predictions. While ER theory offers a computationally efficient route to SFE from solute-solvent pair interaction distributions, its standard functionals often suffer from systematic biases. Although the truncated hypernetted chain (tHNC) functional was developed to reduce these errors, it relies on an empirical truncation parameter that is temperature-dependent and difficult to define for mixed solvents. To overcome these limitations, we propose the MILC-ER scheme, which utilizes a combination of descriptors─approximate SFEs for fully charged and zero-charge states, and the average interaction energy─to capture both electrostatic and excluded-volume effects without empirical truncation. Our model achieves exceptional accuracy on the FreeSolv database (excluding carboxylic acids), yielding a mean absolute deviation (MAD) of 0.35 kcal/mol relative to Bennett acceptance ratio (BAR) values. Critically, we employ a nested cross-validation protocol to provide an unbiased assessment of the model's generalization capability. The results demonstrate that simple linear models consistently outperform complex nonlinear machine learning algorithms, such as Random Forests, confirming that the systematic errors in ER functionals scale linearly with the chosen physical descriptors. This linear framework is not only statistically robust and well-conditioned for small-data regimes but also physically interpretable as a generalized extension of conventional correction schemes. Furthermore, the MILC-ER coefficients remain transferable across different temperatures and successfully generalize to mixed-solvent environments, as demonstrated for ethanol-water systems. These findings establish MILC-ER as a high-precision, computationally efficient, and transferable tool for large-scale solvation studies.