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Research spanning nearly a century has found that math plays an important role in the learning of chemistry. Here, we use a large dataset of student interactions with online courseware to investigate the details of this link between math and chemistry. The activities in the courseware are labeled against a list of knowledge components (KCs) covered by the content, and student interactions are tracked over a full semester of general chemistry at a range of institutions. Logistic regression is used to model student performance as a function of the number of opportunities a student has taken to engage with a particular KC. This regression analysis generates estimates of both the initial knowledge and the learning rate for each student and each KC. Consistent with results from other domains, the initial knowledge varies substantially across students, but the learning rate is nearly the same for all students. The role of math is investigated by labeling each KC with the level of math involved. The overwhelming result from regressions based on these labels is that only the initial knowledge varies strongly across students and across the level of math involved in a particular topic. The student learning rate is nearly independent of both the level of math involved in a KC and the prior mathematical preparation of an individual student. The observation that the primary challenge for students lies in initial knowledge, rather than learning rate, may have implications for course and curriculum design. 

Current neural networks for predictions of molecular properties use quantum chemistry only as a source of training data. This paper explores models that use quantum chemistry as an integral part of the prediction process. This is done by implementing self-consistent-charge Density-Functional-Tight-Binding (DFTB) theory as a layer for use in deep learning models. The DFTB layer takes, as input, Hamiltonian matrix elements generated from earlier layers and produces, as output, electronic properties from self-consistent field solutions of the corresponding DFTB Hamiltonian. Backpropagation enables efficient training of the model to target electronic properties. Two types of input to the DFTB layer are explored, splines and feed-forward neural networks. Because overfitting can cause models trained on smaller molecules to perform poorly on larger molecules, regularizations are applied that penalize nonmonotonic behavior and deviation of the Hamiltonian matrix elements from those of the published DFTB model used to initialize the model. The approach is evaluated on 15,700 hydrocarbons by comparing the root-mean-square error in energy and dipole moment, on test molecules with eight heavy atoms, to the error from the initial DFTB model. When trained on molecules with up to seven heavy atoms, the spline model reduces the test error in energy by 60% and in dipole moments by 42%. The neural network model performs somewhat better, with error reductions of 67% and 59%, respectively. Training on molecules with up to four heavy atoms reduces performance, with both the spline and neural net models reducing the test error in energy by about 53% and in dipole by about 25%.