For systems with periodic boundary condition , plane wave basis is usually adopted. While for molecular systems, gaussian basis set is normally adopted. For periodic systems, the gaussian basis is transformed from real lattice space to momentum space (or k space) through Bloch function. My question is that what is the advantage of the plane wave basis? And a relevant question is that how to determine the number of k points required to converge the energy and density matrix for a periodic system?

  • $\begingroup$ "How to determine the number of k-points" should probably be a separate question. It might have already been asked here (perhaps do a search). $\endgroup$ – Nike Dattani Jun 2 at 4:56

In my opinion a solid, brief overview of DFT is given here.


The pro/cons of plane waves and other basis sets are discussed and I will list them here in case the link goes dead.


  • Fourier coefficients stored in regular grid.
  • Efficient FFT algorithms between r- and G-space representation.
  • O(N^2) scaling on CPU
  • Complete and orthonormal basis set.
  • Not atom-centered -> unbiased.
  • Systematically improvable by increasing the cut-off of the Fourier coefficients.


  • Large set of basis coefficients. Hamiltonian cannot be stored.
  • Sharp nodes of wave functions of core electrons are very expensive. Need pseudo-potential.
  • Vacuum as expensive as atoms.
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  • $\begingroup$ +1 nice answer! $\endgroup$ – Nike Dattani Jun 2 at 11:56

@Susi Lehtola Thanks for sharing the link. I agree that the linked question is more or less the same as the question here. However, I think most of the answers is limited to DFT. I know there are electronic packages that could do calculation for periodic systems with coupled cluster method.

Like PySCF: https://pubs.acs.org/doi/abs/10.1021/acs.jctc.7b00049?casa_token=okpTgl35nWYAAAAA:OcmfSpa8IpK_P8PCgfmCbDfVbLAZ-ILEuZoTdkMvPTUzhqhr7yvBrVejAg998vDoyHf-zzTmm95qZRaq)

For simplified model Hamiltonian, conventional solid state physicist also developed second quantization techniques in momentum space.

Like classical spin wave approach for ferromagnetism


Also more recently, quantum monte carlo approach is developed to study Hubbard model


All these method adopt a plane-wave basis. I think in the real lattice space, effect of delocalization might not be well captured. But these effect might only consequential for same special system (correspond to long wave length limit in k space). For normal systems, the interaction might be decay within first few nearest neighbours (especially for ground state).

My question is that how is the performance of the plane-wave basis for these post-Hartree-Fock approaches?

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    $\begingroup$ The PySCF coupled-cluster study you linked uses a Gaussian basis set. This question is still a duplicate, and any further discussion should happen in the original post $\endgroup$ – Susi Lehtola Jun 3 at 8:03
  • $\begingroup$ Ok,I agree to bind this question to the linked question. But I just don’t want to limit the discussion to DFT anyway. $\endgroup$ – Paulie Bao Jun 3 at 17:11

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