Accurate Lattice Free Energies of Packing Polymorphs from Probabilistic Generative Models

Finite-temperature lattice free energy differences between polymorphs of molecular crystals are fundamental to understanding and predicting the relative stability relationships underpinning polymorphism, yet are computationally expensive to obtain. Here, we implement and critically assess machine-learning-enabled targeted free energy calculations derived from flow-based generative models to compute the free energy difference between two ice crystal polymorphs (Ice XI and Ic), modelled with a fully flexible empirical classical force field. We demonstrate that even when remapping through an analytical reference distribution, such methods enable a cost-effective and accurate calculation of free energy differences between disconnected metastable ensembles when trained on locally ergodic data sampled exclusively from the ensembles of interest. Unlike classical free energy perturbation methods, such as the Einstein crystal method, the targeted approach analysed in this work requires no additional sampling of intermediate perturbed Hamiltonians, offering significant computational savings in the system sizes compared in this work. To systematically assess the accuracy of the method, we monitored the convergence of free energy estimates during training by implementing an overfitting-aware weighted averaging strategy. By comparing our results with ground-truth free energy differences computed with the Einstein crystal method, we assess the accuracy and efficiency of two different model architectures, employing two different representations of the supercells degrees of freedom (Cartesian vs. quaternion-based). We conduct our assessment by comparing free energy differences between crystal supercells of different sizes and temperatures and assessing the accuracy in extrapolating lattice free energies to the thermodynamic limit. While at low temperatures and in small system sizes, the models perform with similar accuracy, we note that for larger systems and high temperatures, the choice of representation is key to obtaining generalisable results of quality comparable to that obtained from the Einstein crystal method. We hope the current work to be a useful stepping stone towards efficient free energy calculations in larger, more conformationally flexible systems.