Non-local pseudopotential

One current limitation of orbital-free DFT is that since only the total density is calculated, there is no way to identify contributions from electronic states of a certain angular momentum character /. This identification is exploited in non-local pseudopotentials so that electrons of different / character see different potentials, considerably improving the quality of these pseudopotentials. The orbital-free metliods thus are limited to local pseudopotentials, connecting the quality of their results to the quality of tlie available local potentials. Good local pseudopotentials are available for the alkali metals, the alkaline earth metals and aluminium [100. 101] and methods exist for obtaining them for other atoms (see section VI.2 of [97]). 

The resultant pair potentials for sodium, magnesium, and aluminium are illustrated in Fig. 6.9 using Ashcroft empty-core pseudopotentials. We see that all three metals are characterized by a repulsive hard-core contribution, Q>i(R) (short-dashed curve), an attractive nearest-neighbour contribution, 2( ) (long-dashed curve), and an oscillatory long-range contribution, 3(R) (dotted curve). The appropriate values of the inter-atomic potential parameters A , oc , k , and k are listed in Table 6.4. We observe that the total pair potentials reflect the characteristic behaviour of the more accurate ab initio pair potentials in Fig. 6.7 that were evaluated using non-local pseudopotentials. We should note, however, that the values taken for the Ashcroft empty-core radii for Na, Mg, and Al, namely Rc = 1.66, 1.39, and... 

The structural trends within the group II elements can only be understood by including the influence of the valence-d electrons explicitly through the use of non-local pseudopotentials. This is not unexpected considering our earlier discussion in 5.5 of their densities of states. Figure 6.13 shows the... 

Table 3 A comparison of semi-local and non-local pseudopotential calculations of ionization energies o/Fe (a.u.)...

Van Camp et al [13] also calculated the first and second pressure derivatives by the ab-initio norm-conserving non-local pseudopotential method of the energy differences between the T X-, and L-states of the valence and lowest conduction band and the top of the valence band in ri5 for 3C-SiC. The first-order pressure coefficients for the direct (ri5v-ric) and indirect (T15v - Xu) bandgaps are 61.7 meV GPa 1 and -1.1 meV GPa 1, respectively. The signs of these values are the same for all semiconductors by Paul s rule. However, the magnitudes are quite different from those of other semiconductors. 

The alert reader will have realized that almost all examples given in this chapter are from Na. Of course this was on purpose two of the most important conditions to be fulfilled for the excellent validity of the jellium model, namely (a) that the pseudopotential is local and (b) that the geometrical parts of the ionic arrangement are weak, are best met in NaA. In trying other elements we found only one other material that works comparably well — potassium. But the important point to note is that this simple model serves as a guideline for more complex cases. After electronic shells and plasmons have been found in Na, they have been found in almost all metal clusters. In order to get the same quantitative agreement as in the case of Na one has either to do all-electron calculations or to use non-local pseudopotentials, as has been done in the case of Li [39, 40]. But in these... 

Another approximation is the neglect of intergroup antisymmetry requirement for the total wave function. This implies that important repulsion effects are missing from the above scheme. Appropriately selected non-local pseudopotentials can be useful for surmounting this difficulty, following for example the formalism developed for nonorthogonal group functions. 

We have constructed the non-local pseudopotential of iron from the atomic configuration [Ar]3d 4s and of silver from [Kr]4d °5s In both cases we have used the GGA approximation and non linear core corrections. Two different sets of basis have been used single-Z-simple polarized (SZSP) and double-Z-simple-polarized (DZSP), with a single polarizing p-orbital in both cases. The cutoff radii for the pseudoorbitals have been fitted to assure the minimization in energy for the bcc Fe bulk and for the fee Ag bulk. 

We will describe the calculation of the total energy and its relation to the band structure in the framework of Density Functional Theory (DFT in the following) see chapter 2. The reason is that this formulation has proven the most successful compromise between accuracy and efficiency for total-energy calculations in a very wide range of solids. In the following we will also adopt the pseudopotential method for describing the ionic cores, which, as we discussed in chapter 2, allows for an efficient treatment of the valence electrons only. Furthermore, we will assume that the ionic pseudopotentials are the same for all electronic states in the atom, an approximation known as the local pseudopotential this will help keep the discussion simple. In realistic calculations the pseudopotential typically depends on the angular momentum of the atomic state it represents, which is known as a non-local pseudopotential . 

Pseudopotentials in DMC computations lead to special consideration owing to their non-locality. There is no difficulty with VMC calculations using a non-local potential. As long as one has a method to evaluate the operation of the pseudopotential operator on the trial function then it contributes to the local energy as any other term. In the DMC method, however, one does not have a method of evaluating the effect of the pseudopotential operator on the fixed-node wave function. This problem has led to the introduction of a localization approximation in which the non-local pseudopotential operator is replaced by a local potential. 

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