Hartree Coulomb! term

Thus, once we know the various contributions in equation (5-15) we have a grip on the potential Vs which we need to insert into the one-particle equations, which in turn determine the orbitals and hence the ground state density and the ground state energy by employing the energy expression (5-13). It should be noted that Veff already depends on the density (and thus on the orbitals) through the Coulomb term as shown in equation (5-13). Therefore, just like the Hartree-Fock equations (1-24), the Kohn-Sham one-electron equations (5-14) also have to be solved iteratively. 

A relativistic Hartree-Fock-Wigner-Seitz band calculation has been performed for Bk metal in order to estimate the Coulomb term U (the energy required for a 5f electron to hop from one atomic site to an adjacent one) and the 5f-electron excitation energies (143). The results for berkelium in comparison to those for the lighter actinides show increasing localization of the 5f states, i.e., the magnitude of the Coulomb term U increases through the first half of the actinide series with a concomitant decrease in the width of the 5f level. 

A different approach to treat correlation effects which are not well described within the LSDA consists in incorporating self-interaction corrections (SIC) [111-114] in electron structure methods for solids, Svane et al. [115-120]. In the Hartree-Fock (HF) theory the electron-electron interactions are usually divided into two contributions, the Coulomb term and the exchange term although they both are Coulomb interactions. The separation though, is convenient because simplifications of self-consistent-field calculations can be obtained by including in both terms the interaction of the electron itself. In the HF theory this has no influence on the solutions because these selfinteractions in the Coulomb and exchange terms exactly cancel each other. However, when the exchange term is treated... 

Functional forms based on the above ideas are used in the HFD [127] and Tang-Toermies models [129], where the repulsion term is obtained by fitting to Hartree-Fock calculations, and in the XC model [92] where the repulsion is modelled by an ab initio Coulomb term f and a semi-empirical exchange-repulsion term Current versions of all these models employ an individually damped dispersion series for the attractive... 

Another disadvantage of the LDA is that the Hartree Coulomb potential includes interactions of each electron with itself, and the spurious term is not cancelled exactly by the LDA self-exchange energy, in contrast to the HF method (see A1.3I. where the self-interaction is cancelled exactly. Perdew and Zunger proposed methods to evaluate the self-interaction correction (SIC) for any energy density functional [40]. However, full SIC calculations for solids are extremely complicated (see, for example [41. 42 and 43]). As an alternative to the very expensive GW calculations, Pollmann et al have developed a pseudopotential built with self-interaction and relaxation corrections (SIRC) [44]. 

This method should lead to results which are just as accurate as the results of the methods described in the previous sections, and can be used as a check on the computed potential-energy minimum E(R ) at R = Re if fl is determined from curve-fitting of the Morse potential with the computed R and De and this leads to a wrong we and/or w, then it can be assumed that De and/or Rg are/is wrong. It is to be emphasized (12) that the Morse curve can mostly not be used with essentially ionic compounds like NaF because the attraction given by the Coulomb term extends out in space to greater distances than the Morse exponential part for these compounds many other types of potential have been postulated (e.g. the Hellmann-potential or the Bom-Landd potential (77)). The reader can try to calculate cog, etc. of NaF from the SCF— LCAO—MO calculation of Matcha (72) in the Roothaan-Hartree-Fock approximation, using the Morse curve (E = —261.38 au, R =3.628 au experimental values Rg = 3.639 au, a)g=536 cm i, >g g=3.83 cm-i). 

The coulomb term has a simple interpretation. In an exact theory, the coulomb interaction is represented by the two-electron operator r j. In the Hartree or Hartree-Fock approximation, as Eq. (3.4) shows, electron-one in Xa experiences a one-electron coulomb potential... 

The exchange term in (3.4), arising from the antisymmetric nature of the single determinant, has a somewhat strange form and does not have a simple classical interpretation like the coulomb term. We can, however, write the Hartree-Fock equation (3.4) as an eigenvalue equation... 

These are the Hartree-Fock equations. The first summation term (the coulomb potential) is the repulsive potential experienced by an electron in orbital j at ri due to the presence of all the other electrons in orbitals k at r2. Note however that this summation also contains a term corresponding to an electron s interaction with itself (i.e., when j=k) and this self-interaction must be compensated for. The second summation is called the exchange potential. The exchange potential modifies the interelectronic repulsion between electrons with like spin. Because no two electrons with the same spin can be in the same orbital j, the exchange term removes those interactions from the coulombic potential field. The exchange term arises entirely because of the antisymmetry of the determinental wavefunctions. The exchange term also acts to perform the self-interaction correction since it is equal in magnitude to the coulomb term when j =k. 

Including the Breit term for the electron-electron interaction in a scalar basis requires extensive additions to a Dirac-Hartree-Fock-Coulomb scheme. It is not possible to achieve the same reductions as for the Coulomb term, and the derivation of the Fock matrix contributions requires considerable bookkeeping. We will not do this in detail, but will provide the development for the Gaunt interaction as we did for the 2-spinor case. 

Figure 2.6. The basic steps in constructing a pseudopotential. F is the operator in the singleparticle hamiltonian 7fP that contains all other terms except the ionic (external) potential, that is, F consists of the kinetic energy operator, the Hartree potential term and the exchange-correlation term. and are the Coulomb potential and the pseudopotential of the ion.

Hartree, DR. Wave mechanics of an atom with a non-coulomb central field. HI. Term values and intensities in series in optical spectra. Proc Cambridge Phil Soc 1928 24 (Pt. 3) 426-37. 

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