In contrast to wavefunction approaches, Density Functional Theory cannot be systematically improved. This has led to distrust due to the lack of uncertainty estimates within the DFT framework. However, due to its favorable scaling, DFT is still the method of choice for the simulation of big systems. In this contribution, we introduce an a priori estimation of uncertainty associated with a DFT calculation based on the identification of physical limiting behaviors, Hartree-Fock (HF) and Local Density Approximation (LDA).
The exact functional in DFT should behave like a piece-wise linear E(N) function, with derivative discontinuities at integer N. However, approximate methods deviate from this behavior. Hartree-Fock mean field approximation shows a concave E(N) behavior in between integer electron numbers whereas approximate local and semi-local functionals (e.g. LDAs, GGAs) are known to show a convex behavior. Among the different functionals, LDA provides the greatest curvatures.
Hence, LDA and HF are two limiting extreme cases that enable to define an error bar for DFT calculations. This error estimation has two advantages: 1) it can be carried out independently of the final simulation, 2) it can be carried out in the absence of benchmark data. Moreover, we will show that experimental results typically fall within this error bar.
This error being size dependent, we will also introduce a normalization that enables to approximate the error for similar structures directly from their volume (no simulation needed).