Assessing the accuracy of Kohn-Sham conductances using the Friedel sum rule

We use the Friedel sum rule (FSR) to discuss the accuracy of exact and approximated Kohn-Sham density-functional theory for the calculation of the electrical conductance. For a two-level molecular junction model we prove that if the Kohn-Sham Hamiltonian reproduces the density of the interacting system calculated in some approximation then it also reproduces the conductance at the same level of approximation. This result is argued to be general for single-channel molecular conductors and is confirmed by means of the exchange-only and GW approximations. The former is found to underestimate the GW conductance. Using the FSR we show how small errors in the description of the density can lead to relatively large errors in the conductance.