Low-Density Electron Gas Wigner Lattice

Some interpolation-schemes have been proposed for calculating the correlation energy as a smooth function between limits of high and low density of electrons. The high-density system implies free-electron states and the low-density system implies localized electrons. 

Some formulas have been proposed for the correlation energy f corr- The Wigner formula 

At low densities, Wigner supposed that the electrons would become localized and form a regular lattice. The electron lattice is a close-packed structure such as fee or hep, in which electrons vibrate around their equilibrium positions. This lattice is immersed in a positive charged background. 

In metals tg value varies from 1.88 to 5.63. This means that the correlation energy ranges approximately from —0.10 to —0.06 Ry. One may compare this with the exchange energy ranging from —0.57 to —0.23 Ry in the same interval of tg. 

The total ground state energy per electron is given by 

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