Most periodic density functional theory (DFT) codes use plane-wave basis sets in conjunction with three-dimensional periodic boundary conditions. In contrast, for molecular systems of finite size, Gaussian basis sets are often used. The former are quite efficient for periodic systems, but more expensive methods such as hybrid functionals become intractable for large system sizes. The latter is well-suited for finite clusters but rarely is appropriate for periodic systems.

CP2K uses a mixed Gaussian and plane-wave approach (GAPW) for periodic systems. Crystal models periodic systems using atom-centered Gaussian functions. PARSEC expresses wavefunctions in real space, avoiding explicit basis sets.

What are some positives and negatives of these alternate approaches, when compared to more conventional periodic DFT packages?

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    $\begingroup$ Some initial thoughts: Negatives could be the cost of using Gaussians, especially for the evaluation of integrals and the slow convergence to the complete basis set limit. Positives could be the description of surfaces with adsorbants, as pointed out in this question: materials.stackexchange.com/questions/68/…. $\endgroup$ – Nike Dattani May 1 '20 at 20:15
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    $\begingroup$ I am using [SIESTA][1] for a while now. From the SIESTA site: SIESTA's efficiency stems from the use of a basis set of strictly-localized atomic orbitals. When I began to work with DFT, I tested [ABINIT][2] and [Quantum ESPRESSO][3] (QE), both using plane-wave approach. SIESTA has the advantaged to be, by far, the fastest and lowest memory consumer. One of the disadvantage is the low number of properties it can calculate compare to ABINIT and QE. [1]: departments.icmab.es/leem/siesta [2]: abinit.org [3]: quantum-espresso.org $\endgroup$ – Camps May 1 '20 at 20:20
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    $\begingroup$ Another nice example is ADF/BAND, which also uses purely localized (Slater, fewer integrals to perform) basis sets for periodic DFT calculations. $\endgroup$ – Felipe S. S. Schneider May 1 '20 at 20:27
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    $\begingroup$ @Fabian, that's great! But what's the downside? Why, for instance, is everyone not using CP2K instead of Quantum Espresso or VASP? $\endgroup$ – Andrew Rosen May 5 '20 at 11:39
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    $\begingroup$ They require fitted basis sets, something you might not want in a first-principle calculation. Then there is the completeness of the basis set, as @NikeDattani already pointed out. Nothing guarantees an increase in accuracy when using a larger basis set. In my experience these codes are also more difficult to work with. It is easier to produce good results with PW based codes. $\endgroup$ – Fabian May 5 '20 at 13:26

Pure plane-wave basis sets have the following advantages when used in periodic DFT (or HF) simulations:

  • Orthogonal
  • Computationally simple (operators with derivatives are particularly straightforward)
  • Low-scaling methods allow easy transformations between real- and reciprocal-space
  • Basis set size does not scale with electron count
  • Independent of atomic positions
  • Their accuracy is controlled with a single parameter, and it is systematically improvable
  • Model all space with equal accuracy

However there are some disadvantages:

  • Basis set size scales with simulation volume - vacuum is not "free"
  • Basis sets are usually large "per atom" - it is not usually practical to construct the full Hamiltonian explicitly (or any other operator) and you must solve the eigenequations iteratively
  • Model all space with equal accuracy - no scope for focusing effort on "interesting" regions
  • Extend throughout space (no simple real-space truncation possible in integrals - e.g. the Fock operator is computationally expensive)

In contrast, (periodic) local basis sets generally have the following advantages:

  • Basis set size does not scale with simulation volume
  • Basis set is typically compact, with few basis states "per atom"
  • Model space with variable accuracy - basis can be tuned to improve representation in regions of interest, and reduce accuracy in uninteresting regions
  • Basis functions are local, and real-space truncations are straightforward in multi-basis set integrals
  • Some basis choices (e.g. Gaussians) allow analytic integration of some energy terms

and the following disadvantages:

  • Non-orthogonal
  • Computationally complicated (often)
  • They depend on the atomic positions (leading to Pulay forces)
  • Basis set size scales with electron count
  • Model space with variable accuracy - need to decide a priori where to spend the computational effort, i.e. which regions are "interesting"
  • No single parameter to control their accuracy; not always systematically improvable
  • Some basis set choices are not easy to transform between real- and reciprocal-space

Roughly speaking, plane-wave methods are efficient when computing and applying the terms of the Hamiltonian, but lead to a much larger dimensionality in the eigenvalue problem and must compute a subset of states; local basis sets often take more time constructing the eigenvalue problem, but it is quite compact and can be solved directly (e.g. with LAPACK) to generate the full eigenspectrum.

There is no reason in principle why you cannot use a hybrid approach (e.g. like CP2K) whereby you transform to a different basis set to perform certain parts of the calculation. You can gain some of the advantages of both, but unfortunately you may suffer from some of the disadvantages of both as well -- for example, when switching from plane-waves to Gaussians the Fock operator becomes much more compact and computationally tractable, but you need to ensure that there are Gaussians in all the "interesting" regions of space. The computational cost of the transformation can also be problematic.

Two final comments:

  • "Muffin tin" programs use mixed basis sets, using localised basis functions to represent the regions of space near nuclei, and plane-waves in the interstitial regions. This is efficient in both regions, but matching the descriptions at the boundaries can be tricky

  • Wannier transformations allow a "lossless" transformation of the occupied Kohn-Sham states from a plane-wave representation to a local representation. However, the transformation scales cubically and is not well-conditioned, usually relying on a "guess" transformation which would be generated from a local basis set (typically LCAO)


The main positives:

  • you can do all-electron calculations
  • you don't need to set up pseudopotentials / PAWs
  • you can study core properties
  • you can use hybrid functionals cheaper / run post-HF calculations

The negatives:

  • basis set is geometry dependent, so you get superposition error
  • it's harder to get results close to the complete basis set limit

Either approach is bad for empty space: plane-waves have uniform precision everywhere, whereas atomic orbitals are localized. Other approaches like finite elements and multiresolution grids fare a lot better here; they can easily represent both core orbitals and empty space.

edit: see also the answer with references here https://mattermodeling.stackexchange.com/a/1944/142

  • $\begingroup$ Could you clarify what you mean with "run post-HF calculations cheaper"? Post HF methods are distinctly not density functional methods. I would disagree that plane waves are bad at describing empty space. In fact, they are very good at describing a slowly oscillating electron density, hence you need pseodopotentials for the core regions. $\endgroup$ – Fuzzy May 17 '20 at 18:20
  • $\begingroup$ I clarified the statement: I meant run hybrid functionals for cheaper, and run post-HF calculations since the orbital space is unambigously defined. Yes, the problem is exactly that plane waves have the same precision everywhere: you usually don't need to describe small-length features in regions where there are no atoms! $\endgroup$ – Susi Lehtola May 17 '20 at 18:38
  • $\begingroup$ I see what you mean now, thanks for clarifying. $\endgroup$ – Fuzzy May 17 '20 at 18:55
  • $\begingroup$ Can you list a reference for what you have told? Thanks. $\endgroup$ – Jack Nov 10 '20 at 2:32
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    $\begingroup$ Plane waves are not geometry independent, since they depend on the cell. $\endgroup$ – Susi Lehtola Nov 13 '20 at 0:11

One important property of atom-centered basis sets is that electrons can only be localized on atoms. This is a problematic property when modeling solid systems with defects.

For instance, at a color center, an electron is localized at a vacancy site. How can you model this with atom-centered basis sets? You have place a ghost atom at the vacancy site, which means you place an empty basis set without nucleus at the vacancy, and only then an electron could localize there.

This is an easy example, but I hope it illustrates that you have to put additional information into the calculation to get the correct result, while you do not need to provide this information in plane-wave DFT. And I cannot imagine how many details one could possibly miss this way when modeling a more complex material.

  • $\begingroup$ This is more-or-less just discussing why plane-wave periodic DFT is more useful for periodic systems than periodic DFT with Gaussian basis sets. It's not really getting at the heart of my question. $\endgroup$ – Andrew Rosen May 17 '20 at 18:28

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