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Exchange and correlation functions for the uniform electron gas

In the simplest approximation, we take the eFF exchange energy to be the pairwise sum of kinetic energy changes upon pairwise orbital orthogonalization:

$\displaystyle E_{exch} = \sum_{\sigma_{i} = \sigma_{j}} \frac{S_{ij}^{2}}{1 - S_{ij}^{2}} \left(t_{11} + t_{22} - \frac{2 t_{12}}{S_{ij}}\right).
$

The formula can be interpreted as quantifying the effect of moving electron density from the region between the two electrons to the electron centers. For systems with nuclei, we found it necessary to modify this function with scaling factors and additional terms to prevent electron-electron coalescence, in order to obtain stable atoms and bonds. For the uniform electron gas, we obtain good agreement with Hartree-Fock energies versus density if we scale the function by $ 1/4$ and neglect the $ t_{12}$ term entirely:

$\displaystyle E_{exch} = \sum_{\sigma_{i} = \sigma_{j}} \frac{1}{4} \frac{S_{ij}^{2}}{1 - S_{ij}^{2}} \left(t_{11} + t_{22}\right).
$

In developing the correlation function, we assumed that the function had the form

$\displaystyle E_{corr} = \sum_{\sigma_{i} \neq \sigma_{j}} f(s_{avg}) \cdot g(S_{ij})
$

where $ s_{avg} = \sqrt{(s_{i}^{2} + s_{j}^2) / 2}$. We experimented with different functional forms for $ g(S)$, then evaluated the energy of an fcc lattice of electrons for different electron densities. Since all the electrons were the same size, we could factor out $ f(s_{avg})$ and determine what $ f$ had to be in order to fit the correlation energy. We found that if $ g(S)$ varied too quickly, electrons tended to expand to maximize their overlap with each other in an unphysical way. However, if $ g(S)$ varied too slowly, an unphysically high $ f(s_{avg})$ was needed to obtain the correct correlation energy. Through trial and error, we found optimal forms for $ f(s_{avg})$ and $ g(S)$:
$\displaystyle f(s_{avg})$ $\displaystyle =$ $\displaystyle \frac{-a_{corr}}{1 + b_{corr} s_{avg}}$  
$\displaystyle g(S_{ij})$ $\displaystyle =$ $\displaystyle S_{ij}^{1/2}$  

where the parameters are listed in the previous section.


next up previous contents
Next: Static properties of the Up: Development of an electron Previous: On the exchange and   Contents
Julius 2008-04-29