Machine learning classification can significantly reduce the cost of calculating the Hamiltonian matrix in CI calculations

Hamiltonian matrices in electronic and nuclear contexts are highly computation intensive to calculate, mainly due to the cost for the potential matrix. Typically, these matrices contain many off-diagonal elements that are orders of magnitude smaller than diagonal elements. We illustrate that here for vibrational H-matrices of H2O, C2H3 (vinyl), and C2H5NO2 (glycine) using full-dimensional ab initio-based potential surfaces. We then show that many of these small elements can be replaced by zero with small errors of the resulting full set of eigenvalues, depending on the threshold value for this replacement. As a result of this empirical evidence, we investigate three machine learning approaches to predict the zero elements. This is shown to be successful for these H-matrices after training on a small set of calculated elements. For H-matrices of vinyl and glycine, of order 15 552 and 8828, respectively, training on a percent or so of elements is sufficient to obtain all eigenvalues with a mean absolute error of roughly 2 cm(-1).