Pseudopotentials weaknesses

There are complicating issues in defmmg pseudopotentials, e.g. the pseudopotential in equation Al.3.78 is state dependent, orbitally dependent and the energy and spatial separations between valence and core electrons are sometimes not transparent. These are not insunnoimtable issues. The state dependence is usually weak and can be ignored. The orbital dependence requires different potentials for different angular momentum components. This can be incorporated via non-local operators. The distinction between valence and core states can be addressed by incorporating the core level in question as part of the valence shell. For... 

For simple monovalent metals, the pseudopotential interaction between ion cores and electrons is weak, leading to a uniform density for the conduction electrons in the interior, as would obtain if there were no point ions, but rather a uniform positive background. The arrangement of ions is determined by the ion-electron and interionic forces, but the former have no effect if the electrons are uniformly distributed. As the interionic forces are mainly coulombic, it is not surprising that the alkali metals crystallize in a body-centered cubic lattice, which is the lattice with the smallest Madelung energy for a given density.46 Diffraction measurements... 

We can understand the behaviour of the binding energy curves of monovalent sodium and other polyvalent metals by considering the metallic bond as arising from the immersion of an ionic lattice of empty core pseudopotentials into a free-electron gas as illustrated schematically in Fig. 5.15. We have seen that the pseudopotentials will only perturb the free-electron gas weakly so that, as a first approximation, we may assume that the free-electron gas remains uniformly distributed throughout the metal. Thus, the total binding energy per atom may be written as... 

The energy-wave-number characteristic, ( )> depends only on the density of the free-electron gas and the nature of the pseudopotential core but not on the structural arrangement of the atoms. Its behaviour as a function of the wave vector, q, is illustrated in Fig. 6.6, where we see that it vanishes at q0 as expected. It also has a weak logarithmic singularity in its slope at q = 2kF. 

The assumption that an electron can be described as a quasi-free particle implies that the interaction between the electron and any atom in the liquid is weak. It is then necessary that the attractive potential of the nucleus experienced by the electron penetrating the atomic core and the long range core polarization potential will be balanced by the electron s increased kinetic energy in the nuclear region. This restriction implies that the pseudopotential of the atom should be small. 

At 4.2 °K. we find /x = 36 cm.2/volt sec. The experimental value is very different, being only 0.03 cm.2/volt sec. This discrepancy and its direction suggest that the electron is much more localized than a delocalized plane wave would allow. The analysis just described is applicable only when the electron atom pseudopotential is weak or attractive, and can therefore be used for studying low field mobility of excess electrons in liquid Ne, Ar, Kr, and Xe. 

The ratio wJEf) of pseudopotential form factor to frec-clcctron Fermi energy for silicon, showing that the direct Joncs-Zone diffraction [220J should be weak. 

In agreement with experiment, pseudopotential density functional theory (DPT) calculations [41] show that Pd-Ni alloying is favored, although there is a weak dependence of the energy on the actual number of Pd-Ni bonds, that is, for a given Pd coverage, there is no substantial difference between different ordering... 

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