next up previous contents
Next: Dynamic properties of the Up: Development of an electron Previous: Exchange and correlation functions   Contents

Static properties of the uniform electron gas

We model the uniform electron gas as a periodic lattice of same-size electrons, as in a Wigner lattice, but at a density well within the Fermi liquid regime. We consider three structures initially (Figure 6.1): a close packed face-centered cubic structure (fcc), where electrons are spin-paired on top of each other; and two open-shell structures (NaCl and sphalerite), which fill interstices in the fcc lattice with electrons of opposite spin, and have octahedral and tetrahedral coordinations, respectively.

Figure 6.1: Uniform electron gas represented as different packings of localized electrons.

For comparison purposes, we calculate the Hartree-Fock energy per electron using equation 6.1, and the exact energy per electron using an analytic expression due to Perdew and Wang [15], fit to the quantum Monte Carlo calculations of Ceperley and Alder for an unpolarized uniform electron gas [4]. Over the range $ r_{s}$ = 1 to 10 bohr, we find that all of the eFF (exchange only) energies agree with the Hartree-Fock energies to within 0.01 hartrees per electron, and all of the eFF (exchange + correlation) energies agree with the exact energies to within 0.005 hartrees per electron (Figure 6.2).

Figure 6.2: Uniform electron gas energy versus density. eFF with exchange matches Hartree-Fock, while eFF with exchange and correlation matches exact quantum Monte Carlo energies.

The different lattices are very close in energy to each other, with the eFF (exch only) energies within 0.01 hartrees per electron of each other, and the eFF (exchange + correlation) energies within 0.005 hartrees per electron of each other. Hence the uniform electron gas is fluxional, varying easily from one lattice type to another.

The energies are similar because the electrons vary in size to accommodate different packing arrangements (Figure 6.3). In general, lower densities create larger electrons. Open-shell lattices pack the electrons together more tightly, which reduces their size and increases their kinetic energy. Counteracting this increase in kinetic energy is the fact that electrons of opposite spin are no longer placed on top of each other, which reduces the electron-electron repulsion. For eFF (exch only), the electron-electron repulsion lowering dominates, and NaCl is the most stable, followed by sphalerite, and then fcc. The differences in energy are the greatest at low densities (high $ r_{s}$), where potential energy dominates and differences in electron-electron repulsion are made most apparent.

Figure 6.3: Density versus electron size. Adding correlation causes the electrons to grow larger.

Adding correlation tends to equalize the energies of the different lattices. Correlation acts as a stabilizing factor that favors the overlap of opposite-spin electrons, which causes the electrons in the uniform electron gas to expand slightly. The closed-shell lattices are preferentially stabilized since their opposite spin electrons have more overlap with each other. Without correlation, open-shell lattices were slightly more stable than closed-shell lattices; adding correlation counteracts this preference and makes open-shell and closed-shell lattices have nearly the same energy, even at low densities.

The electrons in sphalerite have the most room to expand, and as a result, the eFF (exchange + correlation) energy of the sphalerite lattice is slightly below the others. Is it possible that less tightly packed lattices would see even more correlation stabilization? We consider other lattices with a variety of packing fractions and coordinations (Figure 6.4), and compare their energies to Hartree-Fock and exact values (Figure 6.5).

For eFF (exchange only), nearly all the lattices have the same energy (within 0.01 hartrees per electron), with the exception of diamond with its very low packing fraction. For eFF (exchange and correlation), all of the close-packed structures have similar energy (within 0.005 hartrees per electron), but the correlation function lowers the energy of closed-shell non-close-packed structures too much, and raises the energy of open-shell non-close-packed structures too much as well. The last effect is probably an artifact of the electrons expanding too much when correlation is added.

Figure 6.4: Survey of electron packings considered, showing a variety of packing fractions and coordinations.

Figure 6.5: Energetics of different electron packings, with non-close-packed configurations marked red, and close-packed configurations marked black. All close-packed arrangements have similar energies which are near the exact values.

It is worthwhile to investigate the origin of exchange and correlation stabilizations in eFF. In configuration interaction methods, the exchange and correlation energies arise from the explicit form of the wavefunction. The electron-electron repulsion is lower in a CI wavefunction than in a HF wavefunction, for example, because the CI wavefunction increases the average distance between electrons. In contrast, in density functional theory the exchange and correlation energies arise mostly implicitly, from a functional that is applied to a wavefunction that does not necessarily segregate electrons from each other to the extent they would be separated in an exact description. The term ``mostly'' is used here because some self-consistent variation of the orbitals that is dependent on the correlation energy is allowed, and so a limited amount of explicit electron segregation may take place.

We suspect that eFF, like DFT, falls into the category of methods that compute exchange and correlation energies implicitly. To check whether this is the case, we compare the electron-electron pair distribution functions in eFF with Hartree-Fock pair distribution functions [16], and exact pair distribution functions [17] fit to quantum Monte Carlo results. Disregarding spin, we would expect electrons to be further apart at lower density (higher $ r_{s}$), where the average electron size is larger. Looking at the spin-averaged pair distribution function for eFF (exchange + correlation), we find some partial segregation of electrons that becomes larger at higher $ r_{s}$ (Figure 6.6) -- the remainder of the difference must be made up implicitly, as discussed above.

Figure 6.6: Spin-averaged electron-electron pair distribution function, showing partial explicit segregation of electrons in eFF.

In computing pair distribution functions of electrons in Gaussian orbitals, we have used the relation

$\displaystyle \rho(r_{12}) = \frac{1}{2} \left( \frac{1}{\pi s_{avg}^{2}} \righ...
...s_{avg}^{2}) \cdot \frac{\sinh 2 r r_{c} / s_{avg}^{2}}{r r_{c} / s_{avg}^{2}}

where $ r_{12}$ is the distance between electrons, and $ r_{c}$ is the midpoint between two Gaussian orbitals.

Looking at the spin-resolved pair distribution functions (Figure 6.7), we find that eFF (exchange only) causes same-spin electrons to avoid each other, but in a way that exaggerates the effects of different electron sizes. Adding correlation damps out some of the oscillations present in the pair distribution functions by increasing the electron sizes, but the exaggerated effect of different electron sizes on same-spin exclusion remains. In contrast, opposite-spin electrons with correlation do not avoid each other as they should. When the spin-average is taken, it looks as if we have the right dependence of segregation on electron size, but it is in reality the result of an error cancellation between same- and opposite-spin pair distributions.

Figure 6.7: Spin-resolved electron-electron pair distribution functions, showing that eFF keeps same-spin electrons apart, but allows opposite-spin electrons to mingle.

We have shown that using a localized electron model, we obtain good energies for the uniform electron gas over a range of Fermi liquid densities ($ r_{s}$ = 1 to 10 bohr). eFF with exchange agrees well with Hartree-Fock energies, while eFF with exchange and correlation agrees well with exact QMC-derived energies. Different close-packed lattices have very similar eFF (exchange + correlation) energies, supporting the view that the uniform electron gas at Fermi liquid densities has a fluxional structure. An analysis of electron-electron pair distribution functions reveals that there is some explicit segregation of electrons that increases with increasing electron size; however, the exchange and correlation stabilizations still come mostly from the exchange and correlation functions rather than any explicit optimization of electron positions or sizes.

next up previous contents
Next: Dynamic properties of the Up: Development of an electron Previous: Exchange and correlation functions   Contents
Julius 2008-04-29