# The Elephant in the Room of Density Functional Theory Calculations

## TL;DR

This study uses multiwavelet methods to generate highly accurate reference energies for molecules, enabling a detailed assessment of the accuracy of various basis sets in density functional theory calculations.

## Contribution

It provides the first systematic quantification of basis set errors in DFT using quasi-exact reference energies, highlighting the accuracy limits of common basis sets.

## Key findings

- NAOs achieve chemical accuracy with typical basis sizes
- GTOs require larger basis sets for similar accuracy
- Systematic error cancellations improve atomization energy accuracy

## Abstract

Using multiwavelets, we have obtained total energies and corresponding atomization energies for the GGA-PBE and hybrid-PBE0 density functionals for a test set of 211 molecules with an unprecedented and guaranteed $\mu$Hartree accuracy. These quasi-exact references allow us to quantify the accuracy of standard all-electron basis sets that are believed to be highly accurate for molecules, such as Gaussian-type orbitals (GTOs), all-electron numeric atom-centered orbitals (NAOs) and full-potential augmented plane wave (APW) methods. We show that NAOs are able to achieve the so-called chemical accuracy (1 kcal/mol) for the typical basis set sizes used in applications, for both total and atomization energies. For GTOs, a triple-zeta quality basis has mean errors of ~10kcal/mol in total energies, while chemical accuracy is almost reached for a quintuple-zeta basis. Due to systematic error cancellations, atomization energy errors are reduced by almost an order of magnitude, placing chemical accuracy within reach also for medium to large GTO bases, albeit with significant outliers. In order to check the accuracy of the computed densities, we have also investigated the dipole moments, where in general, only the largest NAO and GTO bases are able to yield errors below 0.01 Debye. The observed errors are similar across the different functionals considered here.