irst attempt to control the value of that plateau the initiator approximation has been implemented for walkers that belong to replica 1. We allow the initiator threshold to be different in replica 0 and replica 1. As an illustration we studied the carbon dimer molecule with the cc-pVQZ basis set[@jr_gaussian_1989], with ${\cal H}_{0}$ corresponding to a CAS (8,8) which is the valence space of the molecule. We use a initiator threshold of 3 and a targeted number of walker of 50k for replica 0. In Fig.\[fig1:swalknum\] we present the evolution of the number of walker on replica 0 and on replica 1, the different calculations have been run with a initiator criterion of 3, 1 and no initiator approximation on replica 1.

![Number of walkers in replica 0 (black), and in replica 1 with an initiator approximation of 3 (green), 1 (red) and no initiator approximation (blue) for the $C_{2}$ molecule in the cc-pVQZ basis set. \[fig1:swalknum\]](fig1){width="70.00000%"}

Looking at Fig.\[fig1:swalknum\] it can be seen that with no initiator approximation on $\Ket{\Psi_{1}}$, the number of walkers grows to more than 60 times the number of walkers in the reference. When using the most moderate initiator criteria of 1, this number is already reduced by more than a factor of 2, it can be further decreased by increasing the initiator threshold. However going from a threshold of 1 to 3 only reduced the total number of walkers by roughly 30 %. This is still not satisfying since there is no way to know *a priori* what the number of walkers on $\Ket{\Psi_{1}}$ is going to be. The cost of the calculation cannot be known before running it; for this reason we describe hereafter a way to control the population on replica 1 independently from the initiator threshold.

In Eq.\[eq:ps1equ\] it can be seen that $\Ket{\Psi_{1}}$ scales linearly with the perturbation $\hat{V}$. and thus scale down the perturbation by a real prefactor $\alpha$, that is typically small. This is done in practice by multiplying the matrix elements $H_{ij}$ by this fa