Normalizing flows rank molecular crystal polymorphs without alchemical sampling, matching Einstein crystal free energies at about 40% of the cost
Synopsis
Ranking polymorphs of molecular crystals by thermodynamic stability has relied on the Einstein crystal method (ECM), which must sample a separate chain of overlapping alchemical states for every polymorph, a cost that grows unmanageable across large crystal landscapes. This work hands the mapping in targeted free-energy perturbation to a normalizing flow: the model reversibly transforms molecular-dynamics supercell configurations onto an analytical reference distribution, and targeted Bennett acceptance ratio (tBAR) reweighting then yields lattice free energies. On succinic acid, Veliparib and Mivebresib the estimates agree with ECM, handle supercells of up to about 3×10³ degrees of freedom, and take on average roughly one over 2.5 of the ECM time.
Interpretation
Lattice free energies of molecular crystal polymorphs can now be reweighted directly from the physical molecular-dynamics trajectory already available for each polymorph, without building and sampling a chain of alchemical intermediate states per form. The benchmark ECM is accurate but must ergodically sample a separate chain of overlapping alchemical Boltzmann distributions for each polymorph, which is prohibitive on the large landscapes produced by crystal structure prediction; harmonic free-energy methods are efficient but lose accuracy in flexible, anharmonic systems. At 300 K, supercell free energies for the α, β and γ forms of succinic acid, forms I and II of Veliparib, and packings I, II, IIIs and IIIb of Mivebresib agree with the ECM benchmark, with free-energy differences falling within each other's error bars; across all systems, ECM required on average about 2.5 times more computational time than a reweighted model estimate.
The method learns a reversible coordinate transformation that carries a polymorph's Boltzmann distribution onto a common analytical reference distribution, then uses statistical reweighting to correct for the learned map being approximate. The hard part of targeted free-energy perturbation is constructing sufficiently accurate maps, which are too complex to write from physical or chemical intuition, so a normalizing flow neural network learns them; a normalizing flow is a generative model that provides exact analytical likelihoods for samples. Models were trained on finite-temperature supercells of succinic acid, Veliparib and Mivebresib containing up to 32 molecules and about 3×10³ degrees of freedom, with a 4:1 training-validation split, free energies reweighted only from validation samples, and each BAR evaluation using 5000 model samples and 5000 physical samples.
Perspective
The demonstrated scope is three single-component molecular crystals (succinic acid, Veliparib, Mivebresib) near 300 K and 1 atm, with supercells of up to 32 molecules and about 3×10³ degrees of freedom, and one ergodically sampled physical trajectory per polymorph; the authors describe the method as best suited to smaller supercells of medium-to-large molecules and recommend it for high-throughput polymorph ranking.
The authors report that in supercells containing 16 Mivebresib molecules the model approaches its representational limit, with free energies rising more slowly during training in a way characteristic of underfitting; for Veliparib with the tailor-made force field no ECM ground truth was computed because some perturbed ECM states were hard to sample ergodically, and GAFF and the tailor-made force field disagree on polymorph stability ranking, which the authors stress makes force-field choice important.