  We studied the effects of the incommensurate structural modulations on the ladder subsystem of the $Sr\_{14-x}Ca\_{x}Cu\_{24}O\_{41}$ family of compounds using ab-initio explicitly-correlated calculations. From these calculations we derived $t-J$ model as a function of the fourth crystallographic coordinate $\tau$ describing the incommensurate modulations. It was found that in the highly calcium-doped system, the on-site orbital energies are strongly modulated along the ladder legs. On the contrary the two sites of the ladder rungs are iso-energetic and the holes are thus expected to be delocalized on the rungs. Chain-ladder interactions were also evaluated and found to be very negligible. The ladder superconductivity model for these systems is discussed in the light of the present results. 
Influence of the structural modulations and the Chain-ladder interaction   in the $Sr\_{14-x}Ca\_{x}Cu\_{24}O\_{41}$ compounds