  Recent developments of experimental techniques in the field of ultra-cold
gases open a path to study the crossover from 'few' to 'many' on the quantum
level. In this case, accurate description of inter-particle correlations is
very important since it is believed that they can be utilized by quantum
engineers in quantum metrology, quantum thermometry, quantum heat engines, {\it
etc}. Unfortunately, a theoretical description of these correlations is very
challenging since they are far beyond any variational approaches. By contrast,
the exact many-body description rapidly hits numerical limitations due to an
exponential increase of the many-body Hilbert space. In this work, we brush up
a very effective method of constructing a many-body basis which originates in
the physical argumentation. We show that, in contrast to the commonly used
approach of a straightforward cut-off, it enables one to perform exact
calculations with very limited numerical resources. As examples, we study
quantum correlations in systems of spinless bosons and two-component mixtures
of fermions confined in a one-dimensional harmonic trap being far from the
perturbative regime.
