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Model space quantum Monte Carlo (MSQMC) is an extension of full configurationinteraction QMC (FCIQMC) that allows us to calculate quasi-degenerate andexcited electronic states by sampling the effective Hamiltonian in the modelspace. We introduce a novel algorithm based on the state-selective partitioningfor the effective Hamiltonian using left eigenvectors to calculate severalelectronic states simultaneously at much less computational cost than theoriginal MSQMC with the energy dependent partitioning. The sampling of walkersin MSQMC is analyzed in the single reference limit using a stochastic algorithmfor higher-order perturbation energies by the analogy of the deterministic caseutilizing a full configuration interaction program. We further developsize-consistency corrections of the initiator adaptation (i-MSQMC) in threedifferent ways, i.e. the coupled electron pair approximation, a posteriori, andsecond-order pertrubative corrections. It is clearly demonstrated that most ofthe initiator error is originating from the deficiency of proper scaling ofcorrelation energy due to its truncated CI nature of the initiatorapproximation, and that the greater part of the error can be recovered by thesize-consistency corrections developed in this work.

Multi-state effective Hamiltonian and size-consistency corrections in stochastic configuration interactions