$MgB_2$ under pressure; band-filling, phonon hardening and electrical anisotropy

The electrical two-dimensional character has been accepted as an important factor in the high $T_c$ superconductors. In $MgB_2$ it is the almost two dimensional $\sigma$-bands that are responsible for the superconductivity. On the other hand in $MgB_2$ the band-filling and phonon hardening have been found to be the responsible for the high $T_c$ [1]. But previous calculations [2] have shown that with pressure both, electrical anisotropy and $T_c$ are reduced. Thus the question arises: Is the electrical anisotropy, together with band-filling and phonon hardening, also responsible of the high $T_c$ in $MgB_2$? Using the $WIEN2k$ package the $MgB_2$ superconductor is analyzed as function of pressure. At each pressure the cell parameters are optimized and the $\sigma$-$DOS$, the $E_{2g}$-phonons and the electrical anisotropy of the $\sigma$-bands are calculated and, with the use of the Hopfield expression, are analyzed to see what the correlation of these elements with $T_c$ is. [1] J Kortus, \textit{Physica C} \textbf{456} (2007) 54-62 [2] U Esteves and P de la Mora, \textit{Rev. Mex. Fis.} \textbf{53} (2007) 95-98