2.2 Type of pseudization
Keywords: pseudo-gen

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Subsections

- 2.2.1 Pseudization energies
- 2.2.2 Pseudization radii
- 2.2.3 Choosing the local potential

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2.2 Type of pseudization

The `atomic` package implements two different NC pseudization algorithms, both claiming to yield optimally smooth PP's:

- Troullier-Martins [7] (TM)
- Rappe-Rabe-Kaxiras-Joannopoulos [9] (RRKJ).

Both algorithms replace atomic orbitals in the core region with smooth nodeless pseudo-orbitals. The TM method uses an exponential of a polynomial (see Appendix B); the RRKJ method uses three or four Bessel functions for the pseudo-orbitals in the core region. The former is very robust. The latter may occasionally fail to produce the required nodeless pseudo-orbital. If this happens, first try to force the usage of four Bessel functions (this is achieved by setting a small nonzero value of the charge density at the origin, variable `rho0`: unfortunately it works only for s states).

Second-row elements N, O, F, 3d transition metals, rare earths, are typically ``hard'' atoms, i.e. described by NC PP's requiring a high PW cutoff. These atoms are characterized by 2p (N, O, F), 3d (transition metals), 4f (rare earths) valence states with no orthogonalization to core states of the same l and no nodes. In addition, as mentioned in Secs.2.1.2 and 2.1.3, there are case in which you may be forced to include semicore states in valence, thus making the PP hard (or even harder). In all such cases, one should consider ultrasoft pseudization, unless there is a good reason to stick to NC-PP's. For the specific case of rare earths, however, remember that the problem of DFT reliability preempts the (tough) problem of generating a PP. With US-PP's one can give up the NC requirement and get much softer PP's, at the price of introducing an augmentation charge that compensates for the missing charge.

Currently, the `atomic` package generates US-PP's on top of a ``hard'' NC-PP. In order to ensure sufficient transferability, at least two states per angular momentum l are required.

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2.2.1 Pseudization energies

If you stick to single-projector PP's (one potential per angular momentum l , i.e. one projector per l in the separable form), the choice of the electronic configuration automatically determines the reference states to pseudize: for each l , the bound valence eigenstate is pseudized at the corresponding eigenvalue. If no bound valence eigenstate exists, one has to select a reference energy. The choice is rather arbitrary: you may try something between than other valence bound state energies and zero.

If you have semicore states in valence, remember that for each l only the state with lowest n can be used to generate a single-projector PP. The `atomic` package requires that you explicitly specify the configuration for unscreening in the ``test'' configuration: see the detailed input documentation.

It is possible to generate PP's by pseudizing atomic waves, i.e. regular solutions of the radial Kohn-Sham equation, at any energy. More than one such atomic waves of different energy can be pseudized for the same l , resulting in a PP with more than one projector per l (directly produced in the separable form). Note however that the implementation of multiple-projector PP's is correct for US pseudization: NC pseudization is not properly done (a generalized norm-conservation requirement is not accounted for). US pseudization is achieved by setting different NC and US pseudization radii (see Sec.2.2.2),

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2.2.2 Pseudization radii

For NC pseudization, one has to choose, for each state to be pseudized, a NC pseudization radius rc , at which the AE orbital and the corresponding NC-PP orbital match, with continuous first derivative at r = rc . For bound states, rc is typically at the outermost peak or somewhat larger. The larger the rc , the softer the potential (less PW needed in the calculations), but also the less transferable. The rc may differ for different l ; as a rule, one should avoid large differences between the rc 's, but this is not always possible. Also, the rc cannot be smaller than the outermost node.

A big problem in NC-PP's is how strike a compromise between softness and transferability, especially for difficult elements. The basic question: ``how much should I push rc outwards in order to have reasonable results with a reasonable PW cutoff''. has no clear-cut answer. The choice of rc at the outermost maximum for ``difficult'' elements (those described in Sec.2.2.1): typically 0.7-0.8 a.u, even less for 4f electrons, yields very hard PP's (more than 100 Ry needed in practical calculations). With a little bit of experience one can say that for second-row (2p ) elements, rc = 1.1 - 1.2 will yield reasonably good results for 50-70 Ry PW kinetic energy cutoff; for 3d transition metals, the same rc will require > 80 Ry cutoff (highest l have slower convergence for the same rc ). The above estimates are for TM pseudization. RRKJ pseudization will yield an estimate of the required cutoff.

For multiple-projectors PP's, the rc of unbound states may be chosen in the same range as for bound states. Use small rc and don't try to push them outwards: the US pseudization will take care of softness. US pseudization radii can be chosen much larger than NC ones (e.g. 1.3÷ 1.5 a.u. for second-row 2p elements, 1.7÷ 2.2 a.u. for 3d transition metals), but do not forget that the sum of the rc of two atoms should not exceed the typical bond length of those atoms.

Note that it is the hardest atom that determines the PW cutoff in a solid or molecule. Do not waste time trying to find optimally soft PP's for element X if element Y is harder then element X.

### 2.2.3 Choosing the local potential

As explained in Sec. 2.1.3, note 1, one needs in principle angular momentum channels in PP's up to lc + 1 . In the semilocal form, the choice of a ''local'', l -independent potential is natural and affects only seldom-important PW components with l > lc . In PW calculations, however, a separable, fully nonlocal form - one in which the PP's is written as a local potential plus pr ojectors - is used. An arbitrary function can be added to the local potential and subtracted to all l components. Generally one exploits this arbitrariness to remove one l component using it as local potential. The separable form can be either obtained by the Kleinman-Bylander projection [10] applied to single-projector PP's, or directly produced using Vanderbilt's procedure [2] (for single-projector PP's the two approaches are equivalent).

Unfortunately the separable form is not guaranteed to have the correct ground state (unlike the semilocal form, which, by construction, has the correct ground states): ``ghost'' states, having the wrong number of nodes, can appear among the occupied states or close to them, making the PP completely useless. This problem may show up in US-PP's as well.

The freedom in choosing the local part can (and usually must) be used in order to avoid the appearance of ghosts. For PW calculations it is convenient to choose as local part the highest l , because this removes more projectors (2l + 1 per atom) than for low l . According to Murphy's law, this is also the choice that more often gives raise to problems, and one is forced to use a different l . Another possibility is to generate a local potential by pseudizing the AE potential.

Note that ghosts may not be visible to atomic codes based on radial integration, since the algorithm discards states with the wrong number of nodes. Difficult convergence or mysterious errors are almost invariably a sign tha there is something wrong with our PP. A simple and safe way to check for the presence of a ghost is to diagonalize the Kohn-Sham hamiltonian in a basis set of spherical Bessel functions. This can be done together with transferability tests (see Sec.2.4)

Next: 2.3 Generating the pseudopotential Up: 2 Step-by-step Pseudopotential generation Previous: 2.1 Choosing the generation Contents