  The numerical analysis of strongly interacting nanostructures requires
powerful techniques. Recently developed methods, such as the time-dependent
density matrix renormalization group (tDMRG) approach or the embedded-cluster
approximation (ECA), rely on the numerical solution of clusters of finite size.
For the interpretation of numerical results, it is therefore crucial to
understand finite-size effects in detail. In this work, we present a careful
finite-size analysis for the examples of one quantum dot, as well as three
serially connected quantum dots. Depending on odd-even effects, physically
quite different results may emerge from clusters that do not differ much in
their size. We provide a solution to a recent controversy over results obtained
with ECA for three quantum dots. In particular, using the optimum clusters
discussed in this paper, the parameter range in which ECA can reliably be
applied is increased, as we show for the case of three quantum dots. As a
practical procedure, we propose that a comparison of results for static
quantities against those of quasi-exact methods, such as the ground-state
density matrix renormalization group (DMRG) method or exact diagonalization,
serves to identify the optimum cluster type. In the examples studied here, we
find that to observe signatures of the Kondo effect in finite systems, the best
clusters involving dots and leads must have a total z-component of the spin
equal to zero.
