Core cutoff radius and energy cutoff radius in pseudo potential DFT

Question

From what I understand the core cutoff radius is the distance where the pseudo wavefunction and the full electron wavefunction start to resemble each other. In other words, the pseudo wavefunction and full electron wavefunction have the same energy and electron densities for distances larger than the core cutoff radius. The energy cutoff radius is the energy that determines the size of the plane wave basis set.

My question is how they are related. Most computational software control the energy cutoff radius, and the energy cutoff radius is lower for ultrasoft pseudopotentials. My question is why is this so? I understand conceptually that ultrasoft pseudopotentials are smoother. Hence the Fourier series is less complicated. Hence the basis set does not need to be large. However, I have trouble putting this vague (if correct) concept into concrete reasoning. What are relevant equations that directly relate the energy cutoff and core cutoff?

Answer 1

The reason pseudopotentials are use is because (at low pressures), core electrons do not participate in chemical bonding or reactions, and more over have high kinetic energies. If you expand and electronic wave function in a plane wave basis, high energy wave functions require high frequency basis functions, therefore in order to capture the wave function of a core electron accurately, you would need a high plane wave cutoff, which is computationally expensive.

To avoid this, the core electrons are replaced by a pseudopotential, which is by construction smooth. The core radius is to some extent a measure of the 'hardness' of the pseudopotential. A soft pseudopotential will generally have a larger core radius, and will be smooth. A hard pseudopotential will have a lower core radius, and in order to capture the rapidly varying (more jagged) potential near the core, will be less smooth, and therefore require higher frequencies in its Fourier transform to recover the correct electronic wave functions. Harder pseudopotential are necessary, for example, at high pressures, when the nuclei are squeezed together --- if the core parts of the pseudopotential overlap, then the calculation will fail catastrophically.

Answer 2

For simplicity consider an atom. If the atom has m energy levels (wavefunction), you can convert this system into n energy levels problem where m > n by replacing the innermost (core) electrons with an effective potential. For e.g. Ca has 20 electrons 1s 2e 2s 2e 2p 6e 3s 2e 3p 6e 4s 2e In pseudopotential you can replace the 1s by an effective potential. So, in this two approach AE (all electron) and PSP (pseudopotential) both describe the same atom where the AE will be used as reference. Hence the energy (eigenvalue) of the AE wavefunction and pseudo wavefunction are same. There are two types of PSP generally used the Norm Conserving and ultrasoft. In both the approaches, a cut off radius (Rcut) value is set beyond which it is imposed that the AE and PSP wavefunction exactly fall on top of each other. Below this cut off radius you can have the pseudo wavefunction in any possible shape you want. However the shape of the wavefunction below Rcut is optimized such that it is smooth (i.e. you need small basis set to represent the wavefunction). If the norm of the charge within this Rcut coming from pseudo wavefunction is constrained to be equal to charge of AE wvafeunction then it is norm conserving PSP. In case of ultrasoft this constraint is not enforced. The loss in charge due to removing this constraint called augmentation charges is complemented through additional complictated calculations. Consider 3d orbital. It is localised. Hence, in case of norm conserving psp you will need a large basis set and hence higher cutoff compared to ultrasoft where the wavefunction can be made much smoother.

Generally in the DFT codes the cutoff radius is set/fixed in the psp as they can significantly effect the results. The energy cut off entirely depends on what system studied (atoms), what kind of potential used. If the Rcut is increased, one has more freedom to play with the shape of wavefunction and hence make smoth (small basis set -> cut off). In other words it can be correlated to the kinetic energy of the wavefunction. If the KE of the pseudo wavefunction is reduced you need small Ecut.