I have a general question about planewave DFT codes. Many of the codes require a user to specify both a energy based cut-off for the wave function and the density of the electrons. Typically it would make sense that the energy of the cut-off for density would be 4x greater than that of the energy of a wave function cut-off based on the relation that the density is the wave function squared and the relation that Ecut =h(bar)^2/2m (Gcut^2). However, in certain calculations, it is required to have the density cut-off to be more than 4x higher than the wave function cut-off (e. g. 8x or even 12x higher.) and in another case the density cut-off is only about 2 to 3x higher. In the first case it seems like the wave function basis set would be limited in the first case and overly detailed in the second case. I am having trouble understanding why the ratio of the energy cut-off of the wave function to density would be anything different than 1:4 ?
Because implementations based on ultra-soft pseudo potentials or PAW require the use of so-called "augmentation charges" whose hardness is related to the overlap between the valence and core wave-functions.
And if you want more info try reading this:
Article From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method
To answer your question differently (the above answers all being correct), norm-conserving pseudopotentials are what you are implicitly assuming. For them, the pseudo wave function is literally the sum over plane waves weighted by coefficients. So when you square it to get the density, you double the length of the wave vectors and as you say this means you quadruple the cutoff.
But in non-norm-conserving schemes (PAW and/or ultrasoft for example), the wave function is *not* just sum over plane waves but has an added part around each atom (inside the "core radius") that requires higher resolution to describe. You only need the resolution in a compact region around each atom but you need it otherwise you don't accurately represent the added part of the wave function.
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