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<math>\zeta = 0\,</math> corresponds to the diamagnetic spin-unpolarized situation with equal
<math>\alpha\,</math> and <math>\beta\,</math> spin densities whereas <math>\zeta = \pm 1</math> corresponds to the ferromagnetic situation where one spin density vanishes. The spin correlation energy density for a given values of the total density and relative polarization, ''ε''<sub>c</sub>(''ρ'',''ς''), is constructed so to interpolate the extreme values. Several forms have been developed in conjunction with LDA correlation functionals.
== Exchange-correlation potential ==
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:<math>v_{xc}^{\mathrm{LDA}}(\mathbf{r}) = \frac{\delta E^{\mathrm{LDA}}}{\delta\rho(\mathbf{r})} = \epsilon_{xc}(\rho(\mathbf{r})) + \rho(\mathbf{r})\frac{\partial \epsilon_{xc}(\rho(\mathbf{r}))}{\partial\rho(\mathbf{r})}\ .</math>
In finite systems, the LDA potential decays asymptotically with an exponential form. This is in error; the true exchange-correlation potential decays much slower in a Coulombic manner. The artificially rapid decay manifests itself in the number of Kohn–Sham orbitals the potential can bind (that is, how many orbitals have energy less than zero). The LDA potential can not support a Rydberg series and those states it does bind are too high in energy. This results in the [[HOMO]] energy being too high in energy, so that any predictions for the [[ionization potential]] based on [[Koopmans' theorem]] are poor. Further, the LDA provides a poor description of electron-rich species such as [[anion]]s where it is often unable to bind an additional electron, erroneously predicating species to be unstable.
== References ==
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