Discretization error cancellation in electronic structure calculation: a quantitative study

TL;DR

This paper quantitatively investigates how discretization errors cancel out in quantum chemistry calculations, showing that energy differences are more accurate than absolute energies, supported by numerical simulations and analytical models.

Contribution

It provides a detailed numerical and analytical analysis of discretization error cancellation in electronic structure calculations, clarifying the phenomenon with quantitative insights.

Findings

Errors in energy differences are significantly smaller than in energies.

Errors on energies and energy differences converge at the same rate asymptotically.

Analytical models explain the error cancellation phenomenon quantitatively.

Abstract

It is often claimed that error cancellation plays an essential role in quantum chemistry and first-principle simulation for condensed matter physics and materials science. Indeed, while the energy of a large, or even medium-size, molecular system cannot be estimated numerically within chemical accuracy (typically 1 kcal/mol or 1 mHa), it is considered that the energy difference between two configurations of the same system can be computed in practice within the desired accuracy. The purpose of this paper is to provide a quantitative study of discretization error cancellation. The latter is the error component due to the fact that the model used in the calculation (e.g. Kohn-Sham LDA) must be discretized in a finite basis set to be solved by a computer. We first report comprehensive numerical simulations performed with Abinit on two simple chemical systems, the hydrogen molecule on the one hand, and a system consisting of two oxygen atoms and four hydrogen atoms on the other hand. We observe that errors on energy differences are indeed significantly smaller than errors on energies, but that these two quantities asymptotically converge at the same rate when the energy cut-off goes to infinity. We then analyze a simple one-dimensional periodic Schr\"odinger equation with Dirac potentials, for which analytic solutions are available. This allows us to explain the discretization error cancellation phenomenon on this test case with quantitative mathematical arguments.