For a given spin-projection σ we write the primed σ for the opposite spin-projection. We conclude that the particle selection p PCHB 2 (j σ |i σ ) with the modified weight becomes proportional to the spin penalty, i.e. opposite spin pairs are more likely to be picked. The increased probability of exchange interactions depends continuously on J and still incorporates the value of the integrals, e.g. g ijji . The same applies to the subsequent hole-selections which are more likely to sample exchange excitations. Hence, the modified hamiltonian is efficiently sampled, with a continuous dependence on J.