Ab initio calculation of low-energy collective charge-density   excitations in MgB2
  We present {\it ab-initio} time-dependent density-functional theory calculation results for low-energy collective electron excitations in $\textrm{MgB}_2$. The existence of a long-lived collective excitation corresponding to coherent charge density fluctuations between the boron $\sigma$- and $\pi$- bands ($\sigma\pi$ mode) is demonstrated. This mode has a sine-like oscillating dispersion for energies below 0.5 eV. At even lower energy we find another collective mode ($\sigma\sigma$ mode). We show the strong impact of local-field effects on dielectric functions in MgB$_2$. These effects account for the long q-range behavior of the modes. We discuss the physics that these collective excitations bring to the energy region typical for lattice vibrations. 
V. M. Silkin, A. Balassis, P. M. Echenique, E. V. Chulkov