# Comparing calculations in plane wave and atomic orbital bases for the same functional

Let's say I have two codes. One is plane wave, and the other is using atomic orbitals as basis sets. How can I compare these two codes with the same functional? And let's say I want to optimise the same structure with these two codes (PBE). Principally they should give the same result, right?

Not necessarily. The LAPW method is often considered as the golden standard when it comes to correctness of DFT calculations. This happens because the atom-like orbitals within the muffin tin sphere closely resemble the physical condition hence you can have a greater accuracy when it comes to DFT predictions.

But if you're refering to the case of using codes with atomic orbitals like GAUSSIAN over plane wave dft codes like Quantum ESPRESSO then the output accuracy depends on your system. It's better to use atomic basis set for aperiodic systems like molecules and a plane wave basis set for periodic basis sets like crystals.

• No. I mean codes that use numeric atomic orbitals instead of plane waves. So it's expected not to have the same result, right? In this matter, how can I know that one of this code not giving wrong results due to some basis set issues? Is there way to confirm that they both work correctly? – user1263 Oct 19 '20 at 15:02
• The correctness of your results depends on whether or not, the values youre inspecting have converged for that specific basis set size. There is no way to tell if a code is good apriori. – Anoop A Nair Oct 19 '20 at 15:09

Like other atomic basis sets, numerical atomic orbitals are excellent for getting qualitative results with few basis functions, so you can expect e.g. good geometries for cheap. However, since they're ideally exact for isolated atoms, numerical atomic orbitals are also much less susceptible to basis set superposition error than Gaussian or Slater type orbitals. They also allow for all-electron a.k.a. full-potential calculations to be performed routinely.

While a minimal basis of atomic orbitals is accurate for a single atom, you need polarization functions (e.g. D and F functions on oxygen) to be able to describe the breaking of the atomic symmetry in a molecule or crystal. Often, a triple-zeta basis (contains two polarization shells) yields results that are sufficiently converged.

Plane wave calculations, on the other hand, have been dominating solid-state calculations for a long time. Although results can be obtained easily with plane-wave codes, it is important to go up high enough in the kinetic energy cutoff $$E_{\rm cut}$$ so that the calculations are converged. (Several plane wave calculations in the literature are limited to STO-3G-like i.e. minimal-basis accuracy due to insufficient cutoffs!) You are also limited to using pseudopotentials or PAWs, since all-electron calculations are pretty much infeasible even for a single atom due to the uniform spatial resolution of plane waves.

Comparing different computational approaches for the same calculation is quite tricky: a proper comparison requires cranking the approaches to the limit! In an atomic-orbital calculation, you need to include a complete set of S functions, P functions, D functions, etc, until your observable has converged; in a plane wave calculation you need to increase $$E_{\rm cut}$$ until convergence has been achieved.

The laboriousness of these comparisons is a reason why a paper showing that different computational approaches yield the same answer for the PBE functional got published in Science just a few years ago, see Science 351, 6280 (2016).

It all depends on what you're interested in. There's probably no way to get the absolute energies to match between the codes, since even if both codes use pseudopotentials, they might be different. Geometries should be easier. Excitation spectra? Hard to say.

• I guess this is your 100th answer! – Thomas Oct 27 '20 at 15:06
• A small clarification, regarding "You are also limited to using pseudopotentials or PAWs, since all-electron calculations are pretty much infeasible". You can easily generate "all-electron" pseudopotentials which are much softer than bare Coulomb potentials, but of course they cannot be as soft as potentials where some of the electrons have been "pseudised". – Phil Hasnip Feb 8 at 17:05
• @PhilHasnip but then again those potentials are probably not available in the atomic orbital cod, which again means you're not comparing apples to apples... – Susi Lehtola Feb 8 at 22:34
• Yes, that's true, but they should give "the same" answers for all observables. – Phil Hasnip Feb 9 at 15:47