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.