Link to HTML version of Power Point Slides for Lecture 4

Reading: Text: Chapter 5.

The homogeneous electron gas is the simplest model that captures the essence of the quantum state of electrons in condensed matter. It illustrates many aspects of the Hartree and Hartree-Fock approximations, where the properties can be calculated analytically. Electron correlation can be treated by numerical methods and the resulting correlation functions illustrate salient features of the many-electron problem.

1. The homogeneous electron gas
• Simplest example of the many-electron problem in condensed matter
• Hamiltonian - nuclei replaced by "jellium" background
• Characterized by the density parameter r_s
• Large r_s (low density) - PE dominates
• Small r_s (high density) - KE dominates
• Real materials - 1.3 < r_s < 6
2. Hartree Approximation
• The only possible potential is V(r) = constant - can set V = 0
• Result: ε_k = (1/2) k²; ψ_k = (1/volume)^1/2 e^ikr
• States occupied up to Fermi energy; Fermi surface defined by k = k_Fermi
3. Hartree-Fock approximation
• Include exact electron-electron interaction but assume many-body wave function is a single determinant
• Smeared nuclear potential exactly cancels direct part of electron-electron interaction
• Exchange gives a k-dependent effect upon the eigenvalue
• Result - ψ_k = (1/volume)^1/2 e^ikr as before, but ε_k = (1/2) k² + V_exchange(k)
• V_exchange(k) leads to a broadened occupied part of the band and a singularity at the Fermi energy - shown by Bardeen - fundamental failure of Hartree-Fock that occurs in all metals
• Analytic formulas for density matrix and "exchange hole" around each electron
4. Beyond Hartree-Fock - correlation
• Correlation energy and functions
• Results of quantum Monte Carlo and many-body perturbation theory calculations
• Indicates size of effects in real materials
5. How to predict band widths in a crystal knowing only atomic wave functions
6. Understanding of alkali metals in terms of the homogeneous gas
• Energy bands - close to the simple kinetic energy approximation - experimental proof that the effect upon the eigenvalues due to exchange is fundamentally wrong in Hartree-Fock
• Total energy - close to Hartree-Fock - evidence that the total exchange energy is basically correct in Hartree-Fock
7. Conclusions