Range optimized theory of the density-density response function of electron liquids with application to the homogeneous gas

Tom Banks, Bingnan Zhang

Gabriele F. Giuliani, Giovanni Vignale

Introduction The calculation of the linear response functions of an electron liquid is a very important, but obviously extremely difficult task. Even after many years of study, and in spite of much progress, a complete solution of the problem is still lacking. In preparation to the study of this difficult problem, we consider in this chapter the main response functions of the non-interacting electron gas, namely, the density–density, spin–spin, and current–current response functions, all of which can be calculated analytically. It turns out that understanding the response of the non-interacting electron gas is an essential prerequisite for understanding the richer and more complex response of the interacting liquid. Indeed, one of the most fruitful ideas in many-body physics is that the response of an interacting system can be pictured as the response of a non-interacting system to an effective self-consistent field, which depends on global properties such as the particle density, the density matrix, the current–density etc. ‥ This is true, in particular, for every dynamical mean field theory , which can be derived from the corresponding static mean field theory (such as the HF theory) with the help of the techniques introduced in Section 4.7. In the first part of this chapter we present a detailed study of the response functions of the homogeneous non-interacting electron gas in three, two, and one dimension. Electron–impurity scattering is included only at the most elementary level, and only to demonstrate its main effect, namely, the emergence of diffusion in the dynamics of density fluctuations. In the next chapter we shall present approximations for the effective self-consistent field.

Gabriele F. Giuliani, Giovanni Vignale

1. Introduction to the electron liquid 2. The Hartree-Fock approximation 3. Linear response theory 4. Linear response of independent electrons 5. Linear response of an interacting electron liquid 6. The perturbative calculation of linear response functions 7. Density functional theory 8. The normal Fermi liquid 9. The one-dimensional electron gas and the Luttinger liquid 10. The two-dimensional electron gas at high magnetic field.

Mojmír Tomášek, V. Mikoláš

S. Benner, H. Haug

Katarzyna Pernal

We propose a method that employs functionals of the one-electron reduced density matrix (density matrix) to capture long-range effects of electron correlation. The complementary short-range regime is treated with density functionals. In an effort to find approximations for the long-range density-matrix functional, a modified power functional is applied to the homogeneous electron gas with Coulomb interactions replaced by their corresponding long-range counterparts. For the power $\ensuremath{\beta}=1/2$ and the range-separation parameter $\ensuremath{\omega}=1/{r}_{s}$, the functional reproduces the correlation and the kinetic correlation energies with a remarkable accuracy for intermediate and large values of ${r}_{s}$. Analysis of the Euler equation corresponding to this functional reveals correct ${r}_{s}$ expansion of the correlation energy in the limit of large ${r}_{s}$. The first expansion coefficient is in very good agreement with that obtained from the modified Wigner-Seitz model.