and $\gamma$ is the pair damping constant. The crystal volume is given by V. The details of the standard scheme using the direct diagonalization of Eq.(\[eq:4\]) have been discussed elsewhere.[@onidarubioreining; @shirley; @cappellini2; @roedl]

Since the rank of the Hamilton matrix in Eq.(\[eq:4\]) is extremely large (it is given by the number of valence bands times the number of conduction bands times the number of $\bf k$ points) a straightforward diagonalization of this matrix is often not possible due to high CPU but also memory requirements.

Therefore, we mainly refer to a numerically efficient scheme[@smith] to solve the Bethe-Salpeter equation, which has been successfully applied by Bechstedt and coworkers on several systems. This method is based on the calculation of the time evolution of the exciton-state by Fourier transform on the time domain. This scheme delivers optical spectra directly but no exciton eigenvalues and eigenfunctions.[@smith] The calculation of the $\omega$-dependent polarizability can be considered as an initial-value problem.[@smith] This means that it can be written as a Fourier representation within an integral of time-dependent elements. The time evolution of these elements is driven by the pair Hamiltonian \[see Eq.(\[eq:3\])\] and it is operatively performed by using a central-difference method, which requires 