  In a recent paper, we proposed the adaptive shift method for correcting the
undersampling bias of the initiator-FCIQMC. The method allows faster
convergence with the number of walkers to the FCI limit than the normal
initiator method, particularly for large systems. In its application to
strongly correlated molecules, however, the method is prone to overshooting the
FCI energy at intermediate walker numbers, with convergence to the FCI limit
from below. In this paper, we present a solution to the overshooting problem in
strongly correlated molecules, as well as further accelerating convergence to
the FCI energy. This is achieved by offsetting the reference energy to a value
typically below the Hartree-Fock energy but above the exact energy. This
offsetting procedure does not change the exactness property of the algorithm,
namely convergence to the exact FCI solution in the large-walker limit, but at
its optimal value greatly accelerates convergence. There is no overhead cost
associated with this offsetting procedure, and is therefore a pure and
substantial computational gain. We illustrate the behavior of this offset
adaptive shift method by applying it to the N$_2$ molecule, the ozone molecule
at three different geometries (equilibrium open minimum, a hypothetical ring
minimum, and a transition state) in three basis sets (cc-pV$X$Z, $X$=D,T,Q),
and the chromium dimer in cc-pVDZ basis set, correlating 28 electrons in 76
orbitals. We show that in most cases the offset adaptive shift method converges
much faster than both the normal initiator method and the original adaptive
shift method.
