We presented in the last chapters an electron force field that could describe covalent, ionic, and multicenter bonds between first-row atoms, distinguishing between s-like and p-like electrons. In this chapter, we show that eFF can be modified to describe delocalized or metallic electrons, and we develop a term to account for electron correlation.
Electrons are fermions, and same-spin electrons strive to avoid each other. However, when many electrons are forced into a region with a flat potential, the result can be a lowering of kinetic energy as indistinguishable electrons mix and delocalize over a wider region of space. This effect is responsible for the high conductivity of metals, the stability of benzene and other conjugated pi systems, and the ability of chloroplasts in plants to harvest light energy.
Is it possible for us to model the energetics of delocalized electrons using localized spherical Gaussians? Recall that previously we modeled p electrons using spherical Gaussian functions by modifying the effective interactions between electrons; we use a similar procedure here.
We would also like to develop an eFF expression for electron correlation. eFF uses as its wavefunction a product of three-dimensional orbitals, implicitly assuming that electrons move independently of each other. We know this is not true, and we have already added Pauli terms to the electron force field to account for same spin electrons excluding each other. Additionally all electrons, regardless of spin, repel each other via Coulomb repulsion, which causes electrons to instantaneously avoid each other in space. This instantaneous correlation of electron motions tends to lower the overall energy of the system; the energy difference is termed ``electron correlation''.
To model the delocalization of electrons in a uniform potential and electron correlation effects, it is useful to study the uniform electron gas, which consists of electrons moving in a uniform background charge that exactly neutralizes the electron charge. The system is characterized by a single density parameter , defined such that the density . In the limit of high density (small ), the kinetic energy dominates, and we can take the wavefunction to be the Slater determinant of particle-in-a-box orbitals (Hartree-Fock approximation). The energy per particle in Hartrees is then 
Note that the uniform electron gas at high densities behaves like an ideal gas, which we usually consider to be a valid approximation for atomic gases at low densities. In an atomic gas, the potential energy dominates at high densities, while in an electron gas, the potential energy dominates at low densities. This results in a ``reversal'' of phase changes  -- as the density of a uniform electron gas is decreased, it transitions from a gas to a Fermi liquid, where electrons move freely past each other, but have some affinity for each other. As the density is decreased further, the Fermi liquid becomes a Wigner crystal, where electrons localize and arrange themselves in a crystalline array that minimizes electrostatic potential energy.
The crossover point between electron gas and Fermi liquid occurs roughly when the kinetic and potential energies are the same, = 0.74 bohr. The Wigner crystal was proposed by Wigner  in 1934, but only recently with high-accuracy quantum Monte Carlo computations has it been possible to determine the crossover point from a Fermi liquid; it was found by Ceperley and Adler  to occur at the very low density = 100 bohr.
Metals have an that ranges from 1.87 bohr (Be) to 5.62 bohr (Cs), well within the Fermi liquid range . It has been possible to obtain ``exact'' energies for the uniform electron gas within this regime using diffusion Monte Carlo, and Ceperley and Alder  found that electron correlation effects are significant, with the exact energy greater than the Hartree-Fock energy by as much as 60%. We attempt to use eFF to reproduce both Hartree-Fock and exact uniform electron gas energies as a function of density.
For studying delocalized electrons in metals in molecules, the uniform electron gas serves as model for one extreme -- completely delocalized electrons in a uniform potential -- that contrasts with the systems containing nuclei we have studied thus far. It will serve as the most severe test of eFF's ability to describe delocalized electrons with localized orbitals, and act as an anchor point for interpolation in developing eFF functions applicable to a wide range of potentials and electron localizations.
In regards to developing an eFF correlation function, there has been a long history in the density functional community of developing functionals with the uniform electron gas (such as the local density approximation ) that can be transferred with some modifications to inhomogeneous systems containing nuclei (generalized gradient approximation  and hybrid functionals ). We hope to replicate the success of this approach in the context of the electron force field.
There have been previous efforts to simulate the uniform electron gas using classical particles. Early approaches used screened Coulomb potentials to model electron-electron interactions, with added interactions to account for Pauli repulsion . More recent efforts have focused on reproducing the proper momentum distribution of electrons in dynamics simulations using momentum-dependent potentials [9,10]. Our model is more ambitious in two regards. First, it allows electrons to have different sizes depending on the local electrostatic environment, a degree of freedom useful for describing molecules and different electron packings. Second, we estimate electron correlation, which in the Fermi liquid regime constitutes a large portion of the total energy.
This chapter is organized as follows. We begin by discussing the partition of energy into kinetic energy, electrostatic potential energy, and exchange and correlation components, as in density functional theory. We propose two sets of exchange/correlation functions, one suitable for describing the uniform electron gas, and another suitable for describing systems containing nuclei; we leave the work of interpolating between these two cases for a later date. With these energy functions, we compute the energetics, pair distribution functions, heat capacity, and oscillations of a uniform electron gas; and the energies and geometries of a variety of atoms and molecules with s-like electrons, with correlation included.