To facilitate an accurate description of the localized d and f electrons in the solids, the self interaction correction is introduced. Within the LSDA, the spurious self-interaction tends to underestimate their localization. In SIC-LDA formalism, the localized electrons see a potential (SIC potential) different from that of the delocalized ones for which the self-interaction vanishes and whose potential therefore reduces to that of the LSDA. It is known that self-interaction is important only for localized states in the solid. The correction for extended states is proportional to 1/V, where V is the volume of the system. For extended states in an infinite system, the correction will vanish identically.
In the SIC-LSD approximation [73], the …show more content…
The SIC approach generates an orbital dependent potential which can be significant for localized states, yielding a much-improved description of the static Coulomb correlation effect compared to that provided by LSDA.
Another advantage of the SIC-LSD method is that the minimization of total energy, with respect to the number of localized electrons, leads to a determination of the nominal valence defined as the integer number of electrons available for band formation. (1.37) where Z is the atomic number, and is the number of core and localized states, respectively. This information is important to the analysis of the various properties of solid …show more content…
The explicit expression of the first gradient can be written as: (1.43)
The lowest terms in the expansion of Fx havebeen calculated analytically [82] and are expressed as: (1.44)
Numerous forms of Fx (n, s) , where s=s1, can be illustrated by three widely used forms of potentials: Becke [78], Perdew and Wang [79], and Perdew, Burke and Ernzerhof [80]. Many GGA methods are tailored for specific classes of problems and have therefore a limited general applicability. For the work presented in this thesis where we deal with transition metal systems, Perdew and Wang ’91[81] is used because of its correctness to find true ground state for the transition metals. Under this approximation, the exchange energy Ex and correlation energy Ec are given as: (1.45) (1.46)
With , , , ,
and
In the SIC-LSD approximation [73], the …show more content…
The SIC approach generates an orbital dependent potential which can be significant for localized states, yielding a much-improved description of the static Coulomb correlation effect compared to that provided by LSDA.
Another advantage of the SIC-LSD method is that the minimization of total energy, with respect to the number of localized electrons, leads to a determination of the nominal valence defined as the integer number of electrons available for band formation. (1.37) where Z is the atomic number, and is the number of core and localized states, respectively. This information is important to the analysis of the various properties of solid …show more content…
The explicit expression of the first gradient can be written as: (1.43)
The lowest terms in the expansion of Fx havebeen calculated analytically [82] and are expressed as: (1.44)
Numerous forms of Fx (n, s) , where s=s1, can be illustrated by three widely used forms of potentials: Becke [78], Perdew and Wang [79], and Perdew, Burke and Ernzerhof [80]. Many GGA methods are tailored for specific classes of problems and have therefore a limited general applicability. For the work presented in this thesis where we deal with transition metal systems, Perdew and Wang ’91[81] is used because of its correctness to find true ground state for the transition metals. Under this approximation, the exchange energy Ex and correlation energy Ec are given as: (1.45) (1.46)
With , , , ,
and