The total pseudopotential

Adding the ionic and the electronic contributions calculated in the previous sections, one obtains the total pseudopotential, which is thus also 

The orthogonalisation hole is then included in the ionic pseudopotential, and both contributions are from now on considered as a single object (the so-called pseudo atom ). The remaining electronic contribution from the 

With the approximation of Abarenkov and Heine, the orthogonalisation hole enters in the expression for in a fairly simple way. A detailed calculation of the orthogonalisation hole from the core wave functions involves matrix elements k + q P k [Ref. 1]. But as mentioned before, the difference between this full treatment and the one proposed by Abarenkov and Heine is quite small in practice. And even in principle, the difference 

This means that a different type of Austin-Heine-Sham 

Note however that for a perturbative treatment, the matrix elements of the pseudopotential have to be sufficiently small. It is thus important that in practice the approximation for the orthogonalisation hole is of the same order of magnitude as in the full treatment. 

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Flere we distinguish between nuclear coordinates R and electronic coordinates r is the single-particle kinetic energy operator, and Vp is the total pseudopotential operator for the interaction between the valence electrons and the combined nucleus + frozen core electrons. The electron-electron and micleus-micleus Coulomb interactions are easily recognized, and the remaining tenu electronic exchange and correlation... 

Let us systematically discuss the effect of the pseudopotential in terms of the empty-core form, Eq. (15-13). We first write the total pseudopotential in the metal as a superposition of the individual pseiidopotcntials w (r — r,) centered at the ion positions r,... 

The concept of quasi-free conduction electrons implies that their scattering by the ion core potential in the solid is rather weak. From here the modem theory of the pseudopotential has been developed. This theory shows that it is possible to reproduce the scattering of electron waves by replacing the deep potential at each site of the ionic core by a very much weaker effective potential, the pseudopotential. Thus the total pseudopotential in the metal or the semiconductor, which the conduction electrons feel, is fairly uniform, and the replacement of the real potential by the pseudopotential is a perfectly rigorous procedure. Furthermore, the fact that the total pseudopotential is fairly flat means that one can apply perturbation theory in order to calculate electron energies, cohesion, optical properties, etc. [20]. 

Subsequently calculate the Hartree and exchange-correlation potentials. Together with the ionic potential the total pseudopotential can then be constructed. ... 

It is found that the consistency between the electron-ion potential and the total pseudopotential is essential in order to satisfy the acoustical sum rule. If this sum rule is not fulfilled the crystal will be unstable against shear forces. In the present work... 

For each angular momentum I, a separate pseudopotential (r) is constructed. The total pseudopotential operator is written as... 

One can quantify the pseudopotential by writing the total crystalline potential for an elemental solid as... 

Figure Al.3.23. Phase diagram of silicon in various polymorphs from an ab initio pseudopotential calculation [34], The volume is nonnalized to the experimental volume. The binding energy is the total electronic energy of the valence electrons. The slope of the dashed curve gives the pressure to transfomi silicon in the diamond structure to the p-Sn structure. Otlier polymorphs listed include face-centred cubic (fee), body-centred cubic (bee), simple hexagonal (sh), simple cubic (sc) and hexagonal close-packed (licp) structures.

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 pseudopotential density-functional technique is used to calculate total energies, forces on atoms and stress tensors as described in Ref. 13 and implemented in the computer code CASTEP. CASTEP uses a plane-wave basis set to expand wave-functions and a preconditioned conjugate gradient scheme to solve the density-functional theory (DFT) equations iteratively. Brillouin zone integration is carried out via the special points scheme by Monkhorst and Pack. The nonlocal pseudopotentials in Kleynman-Bylander form were optimized in order to achieve the best convergence with respect to the basis set size. 5... 

In a second approach of the reactivity, one fragment A is represented by its electronic density and the other, B, by some reactivity probe of A. In the usual approach, which permits to define chemical hardness, softness, Fukui functions, etc., the probe is simply a change in the total number of electrons of A. [5,6,8] More realistic probes are an electrostatic potential cf>, a pseudopotential (as in Equation 24.102), or an electric field E. For instance, let us consider a homogeneous electric field E applied to a fragment A. How does this field modify the intermolecular forces in A Again, the Hellman-Feynman theorem [22,23] tells us that for an instantaneous nuclear configuration, the force on each atom changes by... 

One interesting scheme based on density functional theory (DFT) is particularly appealing, because with the current power of the available computational facilities it enables the study of reasonably extended systems. DFT has been applied with a variety of basis sets (atomic orbitals or plane-waves) and potential formulations (all-electron or pseudopotentials) to complex nu-cleobase assemblies, including model systems [90-92] and realistic structures [58, 93-95]. DFT [96-98] is in principle an ab initio approach, as well as MP2//HF. However, its implementation in manageable software requires some approximations. The most drastic of all the approximations concerns the exchange-correlation (xc) contribution to the total DFT functional. 

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 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 total energy (in Rydbergs) per atom of a NFE metal with valence Z may be written to first order in the pseudopotential as U = ZUeg + where... 

The generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

Clearly any attempt to base FeK on such molecularly defined cores defeats the aims of pseudopotential theory. However, the approximate invariance of atomic cores to molecule formation implies that, of the total of Na electrons which could be associated with the centre A in an atomic calculation, nx are core electrons and n K will contribute to the molecular valence set. Thus we can define a one-centred Fock operator ... 

Open-shell Pseudohamiltonians.—The majority of atoms do not have valence structures which can be represented by the fully closed-shell wavefunction of equation (14), and consequently ab initio pseudopotentials cannot be derived directly from the theory outlined above. Acceptable wavefunctions for such atoms require either more than one determinant or the use of the symmetry-equivalenced or generalized Hartree-Fock method, and usually include partially filled shells. The total all-electron wavefunction may be symbolically expressed in terms of four subspaces,... 

We can see that the non-uniqueness of the pseudopotential and of the open-shell hamiltonian have similar origins. Following Roothaan36 the total open-shell hamiltonian may be written in terms of the basic operator Pa by using projection operators to define the particular form of the operator for each sub-space ... 

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