Exchange & Correlation

Jellium Model

\[ \hat{H} = \sum_{\mathbf{k},\sigma} \frac{\hbar^2 k^2}{2m} c_{\mathbf{k},\sigma}^\dagger c_{\mathbf{k},\sigma} + \frac{1}{2} \sum_{\mathbf{q} \neq 0} v(\mathbf{q}) \sum_{\mathbf{k},\mathbf{k'},\sigma,\sigma'} c_{\mathbf{k+q},\sigma}^\dagger c_{\mathbf{k'-q},\sigma'}^\dagger c_{\mathbf{k'},\sigma'} c_{\mathbf{k},\sigma} \]

where \(v(q) = 4 \pi e^2/q^2\).

Two electronic properties - Self energy of an electron of momentum p - Total Energy of the system: Electrons averaged to obtain the total energy of the system. This average is done at zero temperature. Find: ground state energy per particle \(E_g = E_g(n_o)\).

== Mahan, 5.4

In the homogeneous electron gas, the average kinetic energy of the electrons is going to be proportional to EF ~ (K.E.) ~ k~, which, by dimensional analysis, is inversely proportional to the square of the characteristic length of the system, which is rs. Therefore (K.E.) ()( 1/“;. Similarly, dimensional analysis suggests that the average Coulomb energy per particle will be e’2 divided by the characteristic length, or (P.E.) ()( l/rs’ When the electron gas has suffi- ciently high density, which is small rs , the kinetic energy term will be larger than the potential energy term. In this case, the electrons will behave as free particles, since the potential energy is a perturbation on the dominant kinetic energy. In the high-density limit, the free-particle picture is expected to be valid

Two limits

• High density limit: KE >> PE (free particle)
• Low density limit KE < PE

Energy Contributions

Kinetic

[Mahan 5.7]

Hartree

All the remaining terms in the energy come from the Coulomb interaction between the particles. This contribution is not evaluated exactly. Instead, approximate expressions are used.

Using perturbation theory: the first term which occurs is the Coulomb interaction between the electrons and the uniform positive background, which is called the Hartree interaction given by:

\[ N_e E_o = (e^2/2) \int \frac{d^3r_1 d^3r_2}{|r_1 - r_2|} [\rho_e(r_1) - \rho_i(r_1)][\rho_e(r_2) - \rho_i(r_2)] \]

In the HEG: - the time-averaged electron density is uniform throughout the system, as is the positive background. - these equal and opposite charge densities exactly cancel, so that the net system is charge neutral. - Thus the Hartree energy is zero.

Exchange

The Hartree term corresponds to the following pairing of the operators in (5.1)

This term is called the exchange energy, or the Fock energy. Retaining both terms is called Hartree-Fock.

Exchange energy: gives a contribution to the energy of an individual electron, as well as a contribution to the ground state energy of the collection of electrons:

Hartree-Fock theory: Kinetic energy + the Hartree energy which is zero + the exchange energy

Correlation energy: energy tenus beyond Hartree-Fock. The name is applied both to the additional energy tenus in the self-energy of an electron of wave vector k, and to the ground state energy obtained by averaging over all of the electrons

[Mahan, 5.31] Ee = -0.094 + 0.06221n(rs) + O(rs)

The tenu correlation energy is often applied to other quantities besides the total ground state energy. For example, the correlation energy of a particle of wave vector k are those tenus beyond Hartree-Fock: