The eigenvectors, X P N O,ij µ ij a ij , define the transformation matrix from PAOs to PNOs, and the eigenvalues n occ,ij a ij represent the natural orbital occupation numbers. The PNOs are then truncated by occupation number (T CUT PNO , defaulting to 3.33 • 10 -7 , or 10 -7 using tight cutoffs), where all PNOs with corresponding eigenvalue less than the cutoff are eliminated.