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8

You're right that this part is key: My assumption so far is that this satisfies the condition $\langle \tilde{p}_i \vert \tilde{\phi}_j \rangle = \delta_{ij}$. To make the notation a bit simpler, let's define $S$ as the matrix with $$S_{ij} = \langle f_i \vert \tilde{\phi}_j \rangle\tag{1}.$$ Its inverse matrix is $S^{-1}$. The expression for the ...

8

Short answer: no. The idea of the GAPW method described in Theor. Chem. Acc. 103, 124 (1999) is simply to speed up the evaluation of the Coulomb and exchange-correlation contributions. Quoting from the conclusions: Starting from the GPW approach we substituted the PW auxiliary basis for the electron density by an APW auxiliary basis which besides plane ...

5

When you're using a norm-conserving pseudopotential, the charge density $\rho^\mathrm{NC}(r)$ is calculated directly from the modulus-squared of the wavefunction: \begin{align}\tag{1} \rho^\mathrm{NC}(r) = \sum_{bk}\left\vert\psi_{bk}(r)\right\vert^2, \end{align} where $b$ and $k$ are band and k-point indices, respectively, and I've restricted the summation ...

4

Velocity is not really used in quantum mechanics, since it is the momentum that is the canonical variable. Leave velocity to classical physics. The momentum operator ${\bf p}$ makes sense for whatever Hamiltonian. It will just only share eigenstates with the Hamiltonian in cases where $\hat{H}=\hat{H}({\bf p})$, e.g. the free particle $\hat{H}={\bf p}^2/2m$. ...

3

This isn't a perfect comparison of what you asked, but I think it can help. I'll refer to PAW datasets as "pseudopotentials" here since that is how they are used in practice. According to this relatively recent paper, when compared to all-electron calculations, the Standard Solid State Pseudopotential library (which has many PSLibrary ...

2

PAW is proposed to deal with the interaction between electrons and ions. However, PBE or GGA (PBE just one kind of GGA) is just the exchange-correlation functional. You can think PAW and PBE are related to the two different terms in the Kohn-Sham equation. So your question is totally meaningless because you are comparing two completely different concepts.

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