In this section, we describe how the PCHB excitation generation and the concept of supergroups in GAS can be combined to derive an efficient algorithm, that we call GAS-PCHB, for performing Stochastic-GASCI and Stochastic-GASSCF calculations within the FCIQMC framework. Stochastic-RAS and Stochastic-MRCI calculations are obvious extensions of the Stochastic-GAS scheme, considering that RAS and MRCI wave functions are special cases of GAS constructions.