What limits the accuracy of ab initio methods in predicting the transition temperature of conventional superconductors?

Here I assume that conventional superconductors refer to materials that undergo a superconducting transition at low temperature as described by the Bardeen-Cooper-Schrieffer theory (BCS). For those materials the effective attractive electron-electron interaction is mediated by phonons. How accurately can the superconducting transition temperature be computed from ab initio methods and what are the factors that limit the accuracy of those methods?

• One thing that limits the accuracy of these attempts at predicting $T_c$, is that whether dealing with conventional or high-temperature superconductors, the shear number of atoms/electrons involved is quite large, and often we are dealing with fairly heavy metal elements which require big basis sets, treatment of multi-reference character, relativistic and spin-orbit effects, among other things. Superconductivity is one of the hardest things for ab initio programs to treat accurately! May 30 '20 at 0:08