large. Importantly, this assumption is physically self-consistent: neglecting correlations outside the sufficiently large extended molecule implies that they are significant only inside it, ˆ Σ$_{corr}$ ≃ ˆ Σ$_{i,corr}$ , i. e., the parenthesis of Eq. (11) also vanishes. The mathematical equivalence of these two physical conditions is guaranteed by Eqs. (11) and (12), which show that the two parentheses vanish simultaneously, and yield a unique equation in the case of a sufficiently large extended molecule. Ultimately, the above physical assumption of a sufficiently large extended molecule is justified by just the problem that one wants to describe. If the opposite were true, the separation in uncorrelated left and right electrodes, and an extended (possibly strongly) correlated molecule would be impossible, and the problem would be no longer that of the transport through a single molecule. 

Practically, it is important that the extended molecule be not too large, such that accurate ab initio quantum-chemical calculations to correlated molecules of sizes L$_{c}$ be feasible. As any approximation, the present approach also has a limited applicability. Importantly is however that — unlike other approximate schemes justified mathematically rather than physically — its limitation is physically transparent. The present approach can not reliably tackle situations characterized by too large correlation lengths L$_{c}$ . Fortunately, such cases are rare. We can only mention a single case: the Kondo effect, extensively studied from various persepctives (e. g., Refs. 20–24 and citations therein). As it is widely accepted, the Kondo anomaly in the zero-bias conductance is due to electron correlations related to coherent spin fluctuations within the Kondo cloud of linear size L$_{K}$ ∼ ℏ v$_{F}$ / ( k$_{B}$T$_{K}$ ). For typical Kondo temperatures T$_{K}$ ∼ 1 mK − 1 K and v$_{F}$ = 1 . 4 × 10 $^{6}$m/s (Fermi velocity in gold electrodes), L$_{K}$ ∼ 10 3 − 10 $^{6}$nm, a size which is well beyond the possibilities of any nontrivial ab initio study. As discussed recently [24], Kondo-related properties obtained by treating exactly clusters of (much) smaller sizes, L$_{c}$ ≪ L$_{K}$ , should be considered with very special care. 

We do not attempt to give a general estimate of the size L$_{c}$ , a hardly feasible task for real systems. Instead, we shall benchmark the approach based on a sufficiently large extende molecule for a non-trivial correlated model. On this basis, we argue that this approach can provide results, which are relevant for the transport in correlated molecular systems. 

So, if the extended molecule is sufficiently large, one can reliably determine the matrix 

