Within the GAS approach, a given (A, B ← I, J) excitation can lead to a GAS allowed or forbidden determinant D j depending on the starting determinant D i . Hence for GAS, it is not possible to generate probability distributions that only depend on the orbital indices, p(AB; IJ) as for Equation . The concept of supergroups and compositions (Section 2.1), are introduced in our GAS-PCHB excitation generator to circumvent the dependency of the probability distributions on the individual Slater determinants.