  Model space quantum Monte Carlo (MSQMC) is an extension of full configuration
interaction QMC (FCIQMC) that allows us to calculate quasi-degenerate and
excited electronic states by sampling the effective Hamiltonian in the model
space. We introduce a novel algorithm based on the state-selective partitioning
for the effective Hamiltonian using left eigenvectors to calculate several
electronic states simultaneously at much less computational cost than the
original MSQMC with the energy dependent partitioning. The sampling of walkers
in MSQMC is analyzed in the single reference limit using a stochastic algorithm
for higher-order perturbation energies by the analogy of the deterministic case
utilizing a full configuration interaction program. We further develop
size-consistency corrections of the initiator adaptation (i-MSQMC) in three
different ways, i.e. the coupled electron pair approximation, a posteriori, and
second-order pertrubative corrections. It is clearly demonstrated that most of
the initiator error is originating from the deficiency of proper scaling of
correlation energy due to its truncated CI nature of the initiator
approximation, and that the greater part of the error can be recovered by the
size-consistency corrections developed in this work.
