The centerpiece of this chapter was the uniform electron gas, a system characterized by a single density parameter that can exist as a uniform gas, Fermi liquid, or Wigner crystal, depending on density. We saw the uniform electron gas as both a model for electron delocalization, and a starting point for developing new exchange and correlation functions. In developing previous electron force fields, we had focused on reproducing the best known ``exact'' energy with our Pauli function. In this chapter, we sought to reproduce two energies: the Hartree-Fock energy, based on a combination of eFF kinetic energy, electrostatic potential energy, and exchange energy; and the exact energy, based on the above combination with correlation energy added.
We were successful in describing the uniform electron gas using localized electrons. With the appropriate exchange and correlation functions, we found that we could reproduce Hartree-Fock and exact energies of the uniform electron gas as a function of density. As a further surprise, we discovered that the energies of many different lattices were similar, hinting that the uniform electron gas at Fermi liquid densities had a fluxional structure. As a caveat, we noted that our potentials in their current form could only reproduce a classical and not a quantum distribution of momenta.
After extending and modifying the exchange and correlation functions, we were able to describe systems with nuclei and s-like electrons, with good agreement between eFF (with exchange) and Hartree-Fock, and eFF (with exchange and correlation) and exact energies. For example, by fitting parameters to reproduce repulsion, the bond length of lithium hydride, and the long range decay of dynamic correlation energy, we obtained exchange/correlation functions that could reproduce the barrier to the reaction (20.3 kcal/mol eFF with exchange versus 24.3 kcal/mol unrestricted Hartree-Fock; 9.1 kcal/mol eFF with exchange and correlation versus 9.7 kcal/mol exact).
We still have separate versions of exchange/correlation functions, one for the uniform electron gas, and another for systems with nuclei. Future versions of eFF should handle both extremes using a single set of functions, interpolating smoothly between uniform and non-uniform electrostatic potentials, just as the eFF in the previous chapter interpolated between s-like and p-like electron shapes. We hope the ideas provided in this chapter serve as a useful first step in that direction.