Effects of Exchange and Correlation in the Electron Bandstructure Problem

Abstract

The typical example of an elementary excitation in a solid is the one-electron quasi-particle, that is the type of excitation that we are concerned with in a bandstructure calculation. We call it quasi-particle to remember that the excitation is something more general and complicated than just an electron moving in some periodic potential. To bring out the essential points in the simplest way, let us discuss the electron gas. The Hamiltonian in second quantization is

$$\text{H=}\sum\limits_{\text{k}}{{{\text{ }\!\!\varepsilon\!\!\text{ }}_{\text{k}}}\text{c}_{\text{k}}^{\text{+}}}{{\text{c}}_{\text{k}}}\text{+}\frac{\text{1}}{\text{2 }\!\!\Omega\!\!\text{ }}\sum\limits_{\text{q}}{\text{ }\!\!\upsilon\!\!\text{ }\left( \text{q} \right){{\text{ }\!\!\rho\!\!\text{ }}_{\text{q}}}{{\text{ }\!\!\rho\!\!\text{ }}_{\text{-q}}}}$$

(1)

where εk = h ² k ²/2m is the kinetic energy, Ω the volume of the electron gas, ν(q) = 4 e ²/q ² the Coulomb interaction and ρ_q the density Fourier component

$${{\rho }_{q}}=\sum\limits_{k}{c_{k+q}^{+}{{c}_{k}}}$$

(2)

We will not write out spin indices explicitly nor use fat letters for vectors, but leave that to the imagination of the reader. We have further written the electron-electron interaction on the form ρ_qρ_-q which differs from the correct expression $c_k^\dag + q^{c_{k'}^\dag } - q^c k'^c k$ by the (infinite) constant $\text{N/2 }\!\!\Omega\!\!\text{ }\sum\limits_{\text{q}}{\text{ }\!\!\upsilon\!\!\text{ }\left( \text{q} \right)}$. Those who objects to infinite constants may flatten the Coulomb potential for r< r _c which gives a convergence factor sin(gr _c)/qr _c. In the final expressions we can then take r _c → 0.