Approximating Periodic Potential Energy Surfaces with Sparse Trigonometric Interpolation

The potential energy surface (PES) describes the energy of a chemical system as a function of its geometry and is a fundamental concept in computational chemistry. A PES provides much useful information about the system, including the structures and energies of various stationary points, such as local minima, maxima, and transition states. Construction of full-dimensional PESs for molecules with more than 10 atoms is computationally expensive and often not feasible. Previous work in our group used sparse interpolation with polynomial basis functions to construct a surrogate reduced-dimensional PESs along chemically significant reaction coordinates, such as bond lengths, bond angles, and torsion angles. However, polynomial interpolation does not preserve the periodicity of the PES gradient with respect to angular components of geometry, such as torsion angles, which can lead to nonphysical phenomena. In this work, we construct a surrogate PES using trigonometric basis functions, for a system where the selected reaction coordinates all correspond to the torsion angles, resulting in a periodically repeating PES. We find that a trigonometric interpolation basis not only guarantees periodicity of the gradient but also results in slightly lower approximation error than polynomial interpolation.