# Procedure to classify errors in Kohn-Sham DFT

I was reading this paper which basically outlines the two main types of errors that one encounters in Kohn-Sham DFT : Density-based errors and Functional-based errors. I understand the practical differences between these two: The former refers to the case where the electron density generated through the self-iterative process, is one of poor quality. The latter is just a reflection of the fact that in practice, we sort to approximate functionals and this can lead to errors. My doubt is regarding how one could pin down the nature of error in a DFT calculation. Let us confine ourselves to analyzing the functional error. From what I gather, this seems to be the general procedure:

1. Perform an SCF calculation for a system with a given functional (say for example, LDA)
2. Fix the charge density from the LDA calculation, but now use a different functional (say, PBE) and do another SCF calculation.

If one observes qualitatively different results between these two steps, then it should imply that the error is inherently functional-based. This is because in one case we perform LDA calculations with the LDA density and in the other, PBE calculations with the LDA density.

My doubt is as follows: How are these two cases different putting into context the H-K theorems? One of the H-K theorems state that the ground state energy is a unique functional of the density. Doesn't this mean that fixing a density is equivalent to fixing the potential?

I'm looking forward to your inputs on this.

From what I gather, this seems to be the general procedure:

1. Perform an SCF calculation for a system with a given functional (say for example, LDA)
2. Fix the charge density from the LDA calculation, but now use a different functional (say, PBE) and do another SCF calculation.

This wouldn't be density-corrected DFT; this is just using the LDA orbitals as a guess for a self-consistent PBE calculation. What you want is to do a single-shot calculation i.e. evaluate the PBE functional with the LDA density, and contrast it to the value you get for PBE with an optimal electron density.

Also, LDA and PBE are quite close, since they are both pure density functionals and thereby are both extremely prone to self-interaction errors. Instead, one typically compares self-consistent DFT to DFT evaluated with Hartree-Fock orbitals. The whole point of density-corrected DFT is that in some cases one can get much better results from density functional approximations if one just uses a better density.

Hartree-Fock is often used as reference since it is free of self interaction. Alternatively, one can also use electron densities from higher-level ab initio methods than Hartree-Fock in case the Hartree-Fock density is far from correct.

There are several more recenty papers from Burke and coworkers on the topic where some of your unclarities may be explained, e.g. J. Phys. Chem. Lett. 9, 6385 (2018) and J. Chem. Theory Comput. 15, 6636 (2019). A benchmark on the topic has also just been published by Jan Martin: J. Chem. Theory Comput., in press (2021).

• 'What you want is to do a single-shot calculation i.e. evaluate the PBE functional with the LDA density, and contrast it to the value you get for PBE with an optimal electron density.' --How would one do this? Feb 25, 2021 at 19:08
• @Iivars98 that depends on the code. The general answer is you'd use the optimized LDA density as the initial guess for the PBE calculation, and take the results from the first iteration, since the energy will be the one evaluated from the LDA density. The self-consistent PBE energy will be the final, converged value of the same calculation. Feb 26, 2021 at 1:37
• I use Quantum ESPRESSO for the most part. But I get the idea. Circling back to one of the concerns in the question.. how does the H-K theorem relate here? In its bare form, the ground state energy is a unique functional of the ground state charge density. The LDA density from the first step in your comment should correspond to an energy (say, E1). Now, if you use a PBE functional to calculate the total energy, are you calculating a completely different quantity (say, E2)? I believe this is where my confusion lies. Thanks for the help. Feb 26, 2021 at 3:59
• Yes, they are different functionals. We don't know what the right functional is! That's where the density correction scheme comes in. If you use a density from another method that has other types of failures, your hybrid method may have a better accuracy than either of the two you started with. Feb 26, 2021 at 7:26
• Interesting discussion... Makes me wonder.. you have density-corrected DFT and hybrid DFT, where you generate density from Hartree-Fock self-consistent orbitals in the former and Hartree-Fock density but with a Kohn-Sham Scheme in the latter. What about the self interaction error (SIC) in both these cases? In the former, it can be made zero if you take purely HF-orbitals. But does the latter still suffer from SIC? Mar 4, 2021 at 7:58

Let's assume you calculated the density, $$\rho,$$ 100% correctly.

Now, you have the task to "correlate" it with the energy. This correlation is the functional $$F$$.

Let's suppose that the unknown best functional is $$F[\rho]=a \times \rho^\alpha + b \times \rho^\beta$$.

If you are using $$E[\rho] = a \times \rho^\alpha + b \times \rho^\gamma$$ your calculations has an error due to the difference between the parameter $$\beta$$ and $$\gamma$$.

So, the challenge is not only to calculate the density correctly but also known the correct functional that relate both quantities. Only in this paper1, they used 83 functionals!

1. A look at the density functional theory zoo with the advanced GMTKN55 database for general main group thermochemistry, kinetics and noncovalent interactions. Goerigk, L., et al. Phys. Chem. Chem. Phys., 2017,19, 32184-32215 (doi: 10.1039/C7CP04913G)