Pseudopotentials are widely used throughout physics (and in some parts of quantum chemistry); in brief, they use the screening effects of the core electrons on the valence electrons to replace the nuclear potential with a softer[1] potential, and to remove the core electrons from the problem that is being solved. For a detailed discussion of pseudopotentials, I recommend Richard Martin’s book[2].

The ideas in pseudopotential theory date back at least fifty years, and come from a solid-state background. The difficulty with choosing a basis set for solid-state problems comes from the two different behaviours of the wavefunctions: near the nuclei, they are strongly bound, rapidly varying and rather close to atomic wavefunctions; in the interstitial space, they are much smoother, and slowly varying.

There are several methods which try to address this problem head-on, by using a basis set with both characters: atomic-like functions near the nuclei, supplemented with plane waves (e.g. the FLAPW method). An early approach, which is helpful in understanding the whole problem, is the orthogonalised plane wave approach (OPW) where the basis functions are written as plane waves, orthogonalised to local, atom-centred functions:

$$\phi_q(\mathbf{r}) = e^{i\mathbf{q}\cdot\mathbf{r}} - \sum_j c_j u_j(\mathbf{r})$$

The functions \(u_j(\mathbf{r})\) are the atom-centred functions, often taken to be the core wavefunctions from the atoms. This formalism links directly to the pseudopotential formalism, and gives a clear idea of the process being undertaken. With a pseudopotential, we only solve the Schrodinger equation for the valence electrons, and we substitute the full potential with a pseudopotential which is shallower and smoother than the full nuclear potential. This potential is normally only applied up to a certain cut-off radius, beyond which the pseudopotential is exactly equal to the full potential.

A good reproduction of the scattering of the nuclear potential often requires that different potentials are found for each angular momentum value. You will find the terms semi-local (where a potential is non-local in angular variables but local in radius, r: \(\hat{V} = \sum_{lm} \vert Y_{lm}\rangle V_(r) \langle Y_{lm}\vert\) and non-local (where the pseudopotential is written in a separable form, such as \(\sum_j f_j(\mathbf{r}) g_j(\mathbf{r^\prime})\) which is fully non-local in all variables; this becomes an integral operator) though a detailed discussion is beyond the scope of this blog.

Types of pseudopotential

There are three commonly used types: norm-conserving; ultrasoft; and projector-augmented waves (PAWs)[3]. It is important to understand the basic ideas behind each of these types. When considering pseudopotentials, it is important to be aware of the accuracy of the potential and its transferability: that is, how accurate it is in different environments, in particular going from the simple, atomic calculation where it is generated to a more complex environment.

Norm-conserving pseudopotentials require the norm of the pseudofunctions to be the same as the norm of the all-electron wavefunctions:

$$\int_0^{r_c} r^2 \vert \psi^{PS}_i(\mathbf{r})\vert^2 dr = \int_0^{r_c} r^2 \vert \psi^{AE}_i(\mathbf{r})\vert^2 dr$$

where \(r_c\) is the core radius of the pseudopotential. This requirement was found to make more accurate, transferrable pseudopotentials (it can be shown that it enforces a further condition, that the energy derivative of the logarithmic derivative of the pseudofunctions also match those of the all-electron wavefunctions).

These pseudopotentials are still commonly used, but are challenging particularly for first-row elements and transition metals, where there are no core electrons to screen the potential. Accuracy and transferability often lead to small core radii, which in turn makes the potential harder.

Ultrasoft pseudopotentials relax the norm-conservation constraint, and often introduce multiple potentials for each angular momentum channel. This enables a softer potential, and matching the all-electron eigenvalues at more energies, often giving a larger core radius than is possible for a norm-conserving pseudopotential with the same accuracy.

Projector-augmented waves (PAWs) give what is effectively an all-electron method with the cost of a pseudopotential method (though there are some subtleties here). There is a very strong link between PAWs and ultrasoft potentials[4]. The method defines a transformation between the pseudofunctions and the all-electron wavefunctions, that changes the pseudofunctions only within spheres centred on atoms.

Within these spheres, the wavefunctions are expanded as partial waves, with coefficients that are found using projector functions (which are very similar to projector functions found in pseudopotentials). The PAW method allows for the easy reconstruction of all-electron wavefunctions, but does not allow the core wavefunctions to respond during the simulation.

A caution

I want to end with a caution: there are now many libraries of pseudopotentials that are available with electronic structure codes, and there is a strong temptation to simply use the potential without any testing. This is a very dangerous thing to do, and every potential should always be tested carefully before embarking on production calculations. You should be clear about how they were generated (in particular about the core radii) and how they respond in situations of differing density or valence. A pseudopotential is an approximation, though a well understood and well-defined one, and that requires testing and understanding.

[1] Soft when applied to potentials means smooth and shallow, while hard means rapidly varying and generally deep. The number of bound states in the potential will correlate with its depth, and because higher states are orthogonal to lower, they must have more nodes. More nodes in a function generally means a higher second derivative (i.e. kinetic energy) which in turn makes representation either on a real-space grid or with Fourier components (plane waves) more costly.

[2] Electronic Structure, Richard M. Martin, Cambridge 2004. Book website

[3] There are some people who do not class PAWs as pseudopotentials, though this distinction is largely academic.

[4] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999) DOI:10.1103/PhysRevB.59.1758