Here we discuss an alternative avenue that leverages Targeted Free Energy Perturbation and normalizing flows to efficiently compute free energy differences between polymorphic crystal cells. In this context, normalising flows are employed to learn bijective mappings between the Boltzmann distribution of each polymorph and a common, simple reference distribution using limited molecular dynamics (MD) data.[2] This method bypasses the need for computationally expensive intermediate state sampling characteristic of classical free energy perturbation techniques.
We built confidence in this approach by initially studying conformational ensembles of isolated small molecules, where we rigorously assessed the accuracy and data efficiency of different free energy estimators, including the Bennett acceptance ratio (BAR) and multistate Bennett acceptance ratio (MBAR), when reweighting flow-based probabilistic generative models (PGMs). This work established that BAR and MBAR are robust and data-efficient for obtaining accurate free energy differences, even in the presence of model overfitting, providing a strong foundation for their application to solid-state systems.[3]
To ensure the applicability of our method to crystalline supercells of packing polymorphs, we investigated ice polymorphs (Ice XI and Ic). This study highlighted the importance of choosing appropriate representations for the molecular degrees of freedom, with a quaternion-based representation proving particularly effective for accurately modelling the rotational degrees of freedom in ice crystals, especially in larger systems and at higher temperatures. This work demonstrated the feasibility of cost-effective and accurate free energy difference calculations between distinct packing arrangements using normalizing flows.[4]
Building upon these foundations, we applied our normalizing flow-enhanced TFEP methodology to industrially relevant pharmaceutical compounds: succinic acid, Veliparib, and Mivebresib. By training PGMs on MD data from different polymorphs of these molecules, we achieved compelling agreement in absolute lattice free energies and relative stabilities with the ECM ground truth at a significantly lower computational cost. Furthermore, we observed that consistent free energy differences across different supercell sizes suggest that calculations on smaller systems may be sufficient for efficiently ranking large sets of computationally predicted crystal structures.[5]
Overall, our collective findings demonstrate that normalizing flow-enhanced TFEP offers a powerful and practical alternative to traditional methods for accurately determining lattice free energies and ranking molecular polymorphs. The robustness of BAR and MBAR for reweighting, combined with the ability of normalizing flows to efficiently map between complex thermodynamic states, establishes this approach as a promising tool for accelerating the prediction and understanding of polymorphism in the pharmaceutical industry, where efficient high throughput applications are desirable.
[1] Jarzynski, C. (2002). Targeted free energy perturbation. Physical Review E, 65(4), 046122.
[2] Noé, F., Olsson, S., Köhler, J., & Wu, H. (2019). Boltzmann generators: Sampling equilibrium states of many-body systems with deep learning. Science, 365(6457), eaaw1147.
[3] Olehnovics, E., Liu, Y. M., Mehio, N., Sheikh, A. Y., Shirts, M. R., & Salvalaglio, M. (2024). Assessing the accuracy and efficiency of free energy differences obtained from reweighted flow-based probabilistic generative models. Journal of Chemical Theory and Computation, 20(14), 5913-5922.
[4] Olehnovics, E., Liu, Y. M., Mehio, N., Sheikh, A. Y., Shirts, M. R., & Salvalaglio, M. (2025). Accurate Lattice Free Energies of Packing Polymorphs from Probabilistic Generative Models. Journal of Chemical Theory and Computation.
[5] Olehnovics, E., Liu, Y. M., Mehio, N., Sheikh, A. Y., Shirts, M. R., & Salvalaglio, M. (2025), Lattice free energies of molecular crystals using normalizing flow, Chemrxiv.