Fully Internally Contracted Multireference Configuration Interaction Theory Using Density Matrix Renormalization Group: A Reduced-Scaling Implementation Derived by Computer-Aided Tensor Factorization

We present an extended implementation of the multireference configuration interaction (MRCI) method combined with the quantum-chemical density matrix renormalization group (DMRG). In the previous study, we introduced the combined theory, referred to as DMRG-MRCI, as a method to calculate high-level dynamic electron correlation on top of the DMRG wave function that accounts for active-space (or strong) correlation using a large number of active orbitals. The DMRG-MRCI method is built on the full internal-contraction scheme for the compact reference treatment and on the cumulant approximation for the treatment of the four-particle rank reduced density matrix (4-RDM). The previous implementation achieved the MRCI calculations with the active space (24<i>e</i>,24<i>o</i>), which are deemed the record largest, whereas the inherent <i>N</i>_act⁸ × <i>N</i> complexity of computation was found a hindrance to using further large active space. In this study, an extended optimization of the tensor contractions is developed by explicitly incorporating the rank reduction of the decomposed form of the cumulant-approximated 4-RDM into the factorization. It reduces the computational scaling (to <i>N</i>_act⁷ × <i>N</i>) as well as the cache-miss penalty associated with direct evaluation of complex cumulant reconstruction. The present scheme, however, faces the increased complexity of factorization patterns for optimally implementing the tensor contraction terms involving the decomposed 4-RDM objects. We address this complexity using the enhanced symbolic manipulation computer program for deriving and coding programmable equations. The new DMRG-MRCI implementation is applied to the determination of the stability of the iron­(IV)-oxo porphyrin relative to the iron­(V) electronic isomer (electromer) using the active space (29<i>e</i>,29<i>o</i>) (including four second d-shell orbitals of iron) with triple-ζ-quality atomic orbital basis sets. The DMRG-cu(4)-MRCI+Q model is shown to favor the triradicaloid iron­(IV)-oxo state as the lowest energy state and characterize the iron­(V) electromer as thermally inaccessible, supporting the earlier experimental and density functional studies. This conflicts with the previous MR calculations using the restricted active-space second-order perturbation theory (RASPT2) with the similar-size active space (29<i>e</i>,28<i>o</i>) reported by Pierloot et al. (Radoń, M.; Broclawik, E.; Pierloot, K. J. Chem. Theory Comput. 2011, 7, 898), showing that the hypothetical iron­(V) state indicated by recent laser flash photolysis (LFP) studies is likely thermally accessible because of its underestimated relative energy