Electron-phonon coupling and low-temperature structure of NaV$_2$O$_5$

NaV$_2$O$_5$ is an extraordinary example of a structure where charge, spin and lattice degrees of freedom strongly interact. This low-dimensional compound is characterized by V atoms arranged in the form of ladders. At ambient conditions it is found to be quarter-filled with one electron distributed over one rung of the ladder. Going below T$_c$ = 34K, NaV$_2$O$_5$ undergoes a phase transition involving a reordering of the V charges, a lattice deformation and a spin pairing. In order to investigate the mechanisms driving the phase transition, parameters of electron-phonon and spin-phonon for the $\Gamma$ point phonons are determined from ab initio calculations within density functional theory. They are compared to the corresponding parameters of the isostructural CaV$_2$O$_5$, where no phase transition occurs. Moreover, ab initio results of several candidates for the low-temperature supercell of NaV$_2$O$_5$ are presented and analyzed in terms of total energies and electric field gradients.