# Different between ENCUT and ENAUG in the VASP setting?

I am trying to figure out the difference between ENCUT and ENAUG in VASP settings, I found the following description on the wiki of VASP.

• ENAUG specifies the cut-off energy of the plane wave representation of the augmentation charges in eV.
• ENCUT specifies the cutoff energy for the plane-wave-basis set in eV.

I also found the ENMAX and EAUG values from the pseudopotential file of the oxygen element

   ENMAX  =  400.000; ENMIN  =  300.000 eV
ICORE  =        2    local potential
LCOR   =        T    correct aug charges
LPAW   =        T    paw PP
EAUG   =  605.392


Apparently, they are different. I understand the ENCUT and ENMAX are used to determine how many plane-wave-basis will be included in the basis set. My question is:

Regarding the ENAUG, what does it mean by the augmentation charges?

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 to over occupied bands, for simplicity (in general, we'd also need to include the band occupation). By convention, we also omit the fundamental charge of an electron, $$e$$, since it has unit value in atomic units.

In order to fully represent this charge density, we need to have a Fourier grid which is twice as big as the wavefunction cut-off wavevector in each direction. This factor of two comes directly from the fact that the density is the wavefunction squared.

In an ultrasoft (or PAW) method, the charge density is no longer simply the modulus-squared of the Bloch functions, but gains an "augmentation" charge term,

\begin{align}\tag{2} \rho^\mathrm{USP}(r) = \sum_{bk}\left(\left\vert\psi_{bk}(r)\right\vert^2 + \sum_{nmI}Q^I_{nm}(r)\langle\psi_{bk}\vert\beta_{n}^I\rangle\langle\beta_{m}^I\vert\psi_{bk}\rangle\right), \end{align} where $$I$$ is an atom index, and $$n$$ and $$m$$ are projector indices. $$Q^I_{nm}(r)$$ are the augmentation charges themselves.

If we have a Fourier grid twice as large as the cut-off wavevector, then we can represent the first term exactly; however, the second term depends on $$Q^I_{nm}(r)$$, which is not related to $$\psi$$. Can $$Q^I_{nm}(r)$$ be represented exactly on this Fourier grid, or does it need a larger grid?

$$Q(r)$$ is actually designed to have larger Fourier components, this is in fact why the ultrasoft/PAW method can give lower cut-off energies for $$d-$$ and $$f-$$block elements. This means that it cannot usually be represented well on the same Fourier grid we used for the soft charge density, and we need a larger grid for the full augmented charge density. It is the size of this "augmentation charge grid" which is controlled by EAUG.