Machine learning optimization of the collocation point set for solving the electronic Schrödinger equation

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Ku, Jonas
Kamath, Aditya
Carrington, Tucker Jr
Manzhos, Sergei
Wave Function , Energy , Chemical Calculations , Approximation , Mathematical Methods
The rectangular collocation approach makes it possible to solve the Schrödinger equation with basis functions that do not have amplitude in all regions in which wavefunctions have significant amplitude. Collocation points can be restricted to a small region of space. As no integrals are computed, there are no problems due to discontinuities in the potential, and there is no need to use integrable basis functions. In this paper, we show, for the electronic Schrödinger equation, that machine learning can be used to drastically reduce the size of the collocation point set. This is demonstrated by solving the Kohn-Sham equations for CO and H2O. We use a combination of Gaussian process regression and a genetic algorithm to reduce the collocation point set size by more than an order of magnitude (from about 51,000 points to 2,000 points) while retaining mHartree accuracy.