Local pseudopotentials

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]). 

Having laid the theoretical foimdation for the OF-KEDF s, we now face three technical issues in their numerical implementation how to solve the TF-HK equation efficiently, howto generate suitable local pseudopotentials (LPS s), and most importantly, how to make the entire QF-DFT scheme linear-scaling with respect to the system size. We will address these topics in turn. 

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.)...

The interaction between ions and valence electrons is described in a standard manner by pseudopotentials. Sodium clusters allow to use local pseudopotentials. We employ here a soft shape in terms of error functions... 

The semi-local pseudopotential used for the calculations of Table 3 was based on a parameterization for the neutral atom. Melius, Olafson, and Goddard also included in their paper some calculations based on the single valence electron ion Fe +. As expected, this parameterization leads to far worse results, which differ on average from the all-electron results by 0.05 a.u., thus emphasizing the importance of the contribution of valence-valence interaction to the effective potential [equation (51)]. 

In-situ STM has been approached theoretically. Focus has been on (a) solvent polarization effects on the metallic electronic structure [58, 59] (b) local pseudopotentials [59] (c) electron timnel routes through networks of quantum dots [60] (d) time correlation effects and noise [60] (e) resonance tunneling [62, 63], and... 

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 3D-RISM-MCSCF approach has been applied to carbon monoxide (CO) solute in ambient water [33]. Since it is known that the Hartree-Fock method predicts the electronic structure of CO in wrong character [167], the CASSCF method (2 core, 8 active orbitals, 10 electrons) in the basis sets of double zeta plus polarization (9s5pld/4s2pld) augmented with diffuse functions (s- and p-orbitals) was used. Water was described by the SPC/F model [127] and the site-centered local pseudopotential elaborated by Price and Halley for CP simulation [40]. The 3D-RISM/KH integral equations for the water distributions specified on a grid of 64 points in a cubic supercell of size 20 A were solved at each step of the SCF loop by using the method of modified direct inversion in the iterative subspace (MDIIS) [27, 29] (see Appendix). 

Pi is the projector on the /-space of spherical harmonics. V r) is a function of r describing the potential acting on an electron locaUy of / symmetry. The matrix elements of semi-local pseudopotentials can be expressed in terms of almost analytical expressions, the computational time of which is negligible with respect to that of the two-electron matrix elements. The most general pseudopotentials can be written in the non-local form... 

Chai, J. D. Weeks, J. D. Orbital-free density functional theory kinetic potentials and ab initio local pseudopotentials. Phys. Rev. B 2007, 75, 205122. 

The gap of p-BN increases upon volume compression as in the case of diamond, while in most semiconductors the reverse is true [4]. This is due to the fact that boron and nitrogen atoms have deep and localized pseudopotentials as compared with the atoms of other rows of the periodic table. It can be calculated [5] by the same technique described in [4] that at very high pressures boron nitride should favor the rock salt phase, while the p-BN structure is unstable even at highly compressed volumes [5]. Calculations using a nonlocalized pseudopotential model lead to a band gap of 4.41 eV for p-BN for further details, see [6]. Electronic properties of p-BN have been derived by ab initio, norm-conserving, nonlocal pseudopotential... 

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... 

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