Keywords: pseudo-gen

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Subsections

- 2.1.1 Exchange-correlation functional
- 2.1.2 Valence-core partition
- 2.1.3 Electronic reference configuration
- 2.1.4 Nonlinear core correction

## 2.1 Choosing the generation parameters

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2.1.1 Exchange-correlation functional

PP's must be generated with the same exchange-correlation (XC) functional that will be later used in calculations. The use of, for instance, a GGA (Generalized Gradient Approximation) functional tegether with PP's generated with Local-Density Approximation (LDA) is inconsistent. This is why the PP file contains information on the DFT level used in their generation: if you or your code ignore it, you do it at your own risk.

The `atomic` package allows PP generation for a large number of functionals, both LDA and GGA. Most of them have been extensively tested, but beware: some exotic or seldom-used functionals might contain bugs. Currently, `atomic` does not allow PP generation with meta-GGA (TPSS) or hybrid functionals. For the former, an old version of `atomic`, modified by Xiaofei Wang, is available. Work is in progress for the latter.

Some functionals may present numerical problems when the charge density goes to zero. For instance, the Becke gradient correction to the exchange may diverge for ρ→ 0. This does not happen in a free atom if the charge density behaves as it should, that is, as ρ(r)→exp(- αr) for r→∞. In a pseudoatom, however, a weird behavior may arise around the core region, r→ 0, because the pseudocharge in that region is very small or sometimes vanishing (if there are no filled s states). As a consequence, nasty-looking ``spikes'' appear in the unscreened pseudopotential very close to the nucleus. This is not nice at all but it is usually harmless, because the interested region is really very small. However in some unfortunate cases there can be convergence problems. If you do not want to see those horrible spikes, or if you experience problems, you have the following choices:

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  Use a better-behaved GGA, such as PBE
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  Use the nonlinear core correction, which ensures the presence of some charge close to the nucleus.

A further possibility would be to cut the gradient correction for small r (it used to be implemented, but it isn't any longer).

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2.1.2 Valence-core partition

This seems to be a trivial step, and often it is: valence states are those that contribute to bonding, core states are those that do not contribute. Things may sometimes be more complicated than this. For instance:

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  in transition metals, whose typical outer electronic configuration is something like (n = main quantum number) ndi(n + 1)sj(n + 1)pk, it is not always evident that the ns and np states (``semicore states'') can be safely put into the core. The problem is that nd states are localized in the same spatial region as ns and np states, deeper than (n + 1)s and (n + 1)p states. This may lead to poor transferability. Typically, PP's with semicore states in the core work well in solids with weak or metallic bonding, but perform poorly in compounds with a stronger (chemical) type of bonding.
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  Heavy alkali metals (Rb, Cs, maybe also K) have a large polarizable core. PP's with just one electron may not always give satisfactory results.
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  In some II-VI and III-V semiconductors, such as ZnSe and GaN, the contribution of the d states of the cation to the bonding is not negligible and may require explicit inclusion of those d states into the valence.

In all these cases, promoting the highest core states ns and np, or nd, into valence may be a computationally expensive but obliged way to improve poor transferability. .

You should include semicore states into valence only if really needed: their inclusion in fact makes your PP harder (unless you resort to US pseudization) and increases the number of electrons. In principle you should also use more than one projector per angular momentum, because the energy range to be covered by the PP with semicore electrons is much wider than without. For instance, it may happen that the error on the lattice parameter of a simple metal is larger with a semicore PP than with a valence-only PP.

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2.1.3 Electronic reference configuration

This may be any reasonable configuration not too far away from the expected configuration in solids or molecules. As a first choice, use the atomic ground state, unless you have a reason to do otherwise, such as for instance:

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  You do not want to deal with unbound states. Very often states with highest angular momentum l are not bound in the atom (an example: the 3d state in Si is not bound on the ground state 3s23p2, at least with LDA or GGA). In such a case one has the choice between
  –
    using one configuration for s and p, another, more ionic one, for d, as in Refs.[4,5];
  –
    choosing a single, more ionic configuration for which all desired states are bound;
  –
    generate PP's on unbound states: requires to choose a suitable reference energy.
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  The results of your PP are very sensitive to the chosen configuration. This is something that in principle should not happen, but I am aware of at least one case in which it does. In III-V zincblende semiconductors, the equilibrium lattice parameter is rather sensitive to the form of the d potential of the cation (due to the presence of p - d coupling between anion p states and cation d states [12]). By varying the reference configuration, one can change the equilibrium lattice parameter by as much as 1 - 2%. The problem arises if you want to calculate accurate dynamical properties of GaAs/AlAs alloys and superlattices: you need to get a good theoretical lattice matching between GaAs and AlAs, or otherwise unpleasant spurious effects may arise. When I was confronted with this problem, I didn't find any better solution than to tweak the 4d reference configuration for Ga until I got the observed lattice-matching.
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  You know that for the system you are interested in, the atom will be in a given configuration and you try to stay close to it. This is not very elegant but sometimes it is needed. For instance, in transition metals described by a PP with semicore states in the core, it is probably wise to chose an electronic configuration for d states that is close to what you expect in your system (as a hand-waiving argument, consider that the (n + 1)s and (n + 1)p PP have a hard time in reproducing the true potential if the nd state, which is much more localized, changes a lot with respect to the starting configuration). In Rare-Earth compounds, leaving the 4f electrons in the core with the correct occupancy (if known) may be a quick and dirty way to avoid the well-known problems of DFT yielding the wrong occupancy in highly correlated materials.
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  You don't manage to build a decent PP with the ground state configuration, for whatever reason.

NOTE 1: you can calculate PP for a l as high as you want, but you are not obliged to use all of them in PW calculations. The general rule is that if your atom has states up to l = lc in the core, you need a PP with angular momenta up to l = lc + 1. Angular momenta l > lc + 1 will feel the same potential as l = lc + 1, because for all of them there is no orthogonalization to core states. As a consequence a PP should have projectors on angular momenta up to lc; l = lc + 1 should be the local reference state for PW calculations. This rule is not very strict and may be relaxed: high angular momenta are seldom important (but be careful if they are). Moreover separable PP pose serious constraints on local reference l (see below) and the choice is sometimes obliged. Note also that the highest the l in the PP, the more expensive the PW calculation will be.

NOTE 2: a completely empty configuration (s0p0d0) or a configuration with fractional occupation numbers are both acceptable. Even if fractional occupation numbers do not correspond to a physical atomic state, they correspond to a well-defined mathematical object.

NOTE 3: PP could in principle be generated on a spin-polarized configuration, but a spin-unpolarized one is typically used. Since PP are constructed to be transferrable, they can describe spin-polarized configurations as well. The nonlinear core correction is needed if you plan to use PP in spin-polarized (magnetic) systems.

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2.1.4 Nonlinear core correction

The nonlinear core correction[11] accounts at least partially for the nonlinearity in the XC potential. During PP generation one first produces a potential yielding the desired pseudo-orbitals and pseudoenergies. In order to extract a ``bare'' PP that can be used in a self-consistent DFT calculation, one subtracts out the screening (Hartree and XC) potential generated by the valence charge only. This introduces a trasferability error because the XC potential is not linear in the charge density. With the nonlinear core correction one keeps a pseudized core charge to be added to the valence charge both at the unscreening step and when using the PP.

The nonlinear core correction must be present in one-electron PP's for alkali atoms (especially in ionic compounds) and for PP's to be used in spin-polarized (magnetic) systems. It is recommended whenever there is a large overlap between valence and core charge: for instance, in transition metals if the semicore states are kept into the core. Since it is never harmful, one can take the point of view that it should always be included, even in cases where it will not be very useful.

The pseudized core charge used in practice is equal to the true core charge for r≥rcc, differs from it for r < rcc in such a way as to be much smoother. The parameter rcc is typically chosen as the point at which the core charge ρc(rcc) is twice as big as the valence charge ρv(rcc). In fact the effect of nonlinearity is important only in regions where ρc(r)∼ρv(r). Alternatively, rcc can be provided in input, Note that the smaller rcc, the more accurate the core correction, but also the harder the pseudized core charge, and vice versa.

Next: 2.2 Type of pseudization Up: 2 Step-by-step Pseudopotential generation Previous: 2 Step-by-step Pseudopotential generation Contents