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Register a new userA Bayesian approach to NMR crystal structure determination
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Nuclear Magnetic Resonance (NMR) spectroscopy is particularly well-suited to determine the structure of molecules and materials in powdered form. Structure determination usually proceeds by finding the best match between experimentally observed NMR chemical shifts and those of candidate structures. Chemical shifts for the candidate configurations have traditionally been computed by electronic-structure methods, and more recently predicted by machine learning. However, the reliability of the determination depends on the errors in the predicted shifts. Here we propose a Bayesian framework for determining the confidence in the identification of the experimental crystal structure, based on knowledge of the typical error in the electronic structure methods. We also extend the recently-developed ShiftML machine-learning model, including the evaluation of the uncertainty of its predictions. We demonstrate the approach on the determination of the structures of six organic molecular crystals. We critically assess the reliability of the structure determinations, facilitated by the introduction of a visualization of the of similarity between candidate configurations in terms of their chemical shifts and their structures. We also show that the commonly used values for the errors in calculated $^{13}$C shifts are underestimated, and that more accurate, self-consistently determined uncertainties make it possible to use $^{13}$C shifts to improve the accuracy of structure determinations.
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The calculation of optimal structures in reaction-diffusion models is of great importance in many physicochemical systems. We propose here a simple method to monitor the number of interphases for long times by using a boundary flux condition as a control. We consider as an illustration a 1-D Allen-Cahn equation with Neumann boundary conditions. Numerical examples are given and perspectives for the application of this approach to electrochemical systems are discussed.
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We introduce and evaluate a set of feature vector representations of crystal structures for machine learning (ML) models of formation energies of solids. ML models of atomization energies of organic molecules have been successful using a Coulomb matrix representation of the molecule. We consider three ways to generalize such representations to periodic systems: (i) a matrix where each element is related to the Ewald sum of the electrostatic interaction between two different atoms in the unit cell repeated over the lattice; (ii) an extended Coulomb-like matrix that takes into account a number of neighboring unit cells; and (iii) an Ansatz that mimics the periodicity and the basic features of the elements in the Ewald sum matrix by using a sine function of the crystal coordinates of the atoms. The representations are compared for a Laplacian kernel with Manhattan norm, trained to reproduce formation energies using a data set of 3938 crystal structures obtained from the Materials Project. For training sets consisting of 3000 crystals, the generalization error in predicting formation energies of new structures corresponds to (i) 0.49, (ii) 0.64, and (iii) 0.37 eV/atom for the respective representations.
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