The effective atomic orbitals have been realized in the framework of Bader's atoms in molecules theory for a general wavefunction. This formalism can be used to retrieve from any type of calculation a proper set of orthonormalized numerical atomic orbitals, with occupation numbers that sum up to the respective Quantum Theory of Atoms in Molecules (QTAIM) atomic populations. Experience shows that only a limited number of effective atomic orbitals exhibit significant occupation numbers. These correspond to atomic hybrids that closely resemble the core and valence shells of the atom. The occupation numbers of the remaining effective orbitals are almost negligible, except for atoms with hypervalent character. In addition, the molecular orbitals of a calculation can be exactly expressed as a linear combination of this orthonormalized set of numerical atomic orbitals, and the Mulliken population analysis carried out on this basis set exactly reproduces the original QTAIM atomic populations of the atoms. Approximate expansion of the molecular orbitals over a much reduced set of orthogonal atomic basis functions can also be accomplished to a very good accuracy with a singular value decomposition procedure

Financial help has been furnished by the Spanish MICINN Project No. CTQ2011-23441/BQU and Accion Complementaria del MCI (PCI2006-A7-0631). Financial support from MICINN and the FEDER fund (European Fund for Regional Development) was also provided by grant UNGI08-4E-003. Financial support from the Generalitat de Catalunya (SGR528 and Xarxa de Referncia en Qumica Terica i Computacional) is also acknowledged. E.R-C. acknowledges support from the Spanish FPU program (Grant No. AP2008-01231)