Local N-electron valence state perturbation theory using pair-natural orbitals based on localized virtual molecular orbitals

Second-order N-electron valence state perturbation theory (NEVPT2) is an exactly size- consistent and intruder-state-free multi-reference theory. To accelerate the NEVPT2 computation, Guo and Neese combined it with the local pair-natural orbital (PNO) method using the projected atomic orbitals (PAOs) as the underlying local basis [Guo et al., J. Chem. Phys. 144, 094111 (2016)]. In the present paper, we report the further development of the PNO-NEVPT2 method using the orthonormal and non-redundant localized virtual molecular orbitals (LVMOs) instead of PAOs. The LVMOs were previously considered to perform somewhat poorly compared to PAOs because the resulting orbital domains were unacceptably large in size. Our prior work, however, showed that this drawback can be remedied by re-forming the domain construction scheme using differential overlap integrals (DOIs) [Saitow et al., J. Chem. Phys. 157, 084101 (2022)]. In this work, we develop further refinements to enhance the feasibility of using LVMOs. We first developed a two-level semi-local approach for screening out so-called weak-pairs in order to select or truncate the pairs for PNO constructions more flexibly. As a refinement specific to the Pipek–Mezey localization for LVMOs, we introduced an iterative scheme to truncate the Givens rotations using varying thresholds. We assessed the LVMO-based PNO-NEVPT2 method through benchmark calculations for linear phenylalkanes, which demonstrate that it performs comparably well relative to the PAO-based approach. In addition, we evaluated the Co-C bond dissociation energies for the cobalamin derivatives composed of 200 or more atoms, which confirms that the LVMO-based method can recover more than 99.85% of the canonical NEVPT2 correlation energy

Quasi-degenerate extension of local N-electron valence state perturbation theory with pair-natural orbital method based on localized virtual molecular orbitals

Chemical phenomena involving near-degenerate electronic states, such as conical intersections or avoided crossing, can be properly described using quasi-degenerate perturbation theory (QDPT). This study proposed a highly scalable quasi-degenerate second-order N-electron valence state perturbation theory (QD-NEVPT2) using the local pair-natural orbital (PNO) method. Our recent study showed an efficient implementation of the PNO-based state-specific (SS) NEVPT2 method using orthonormal localized virtual molecular orbitals (LVMOs) as an intermediate local basis. This study derived the state-coupling (or off-diagonal) terms to implement QD-NEVPT2 in an alternative manner to enhance efficiency based on the internally contracted basis (ICB) and PNO overlap matrices between different references. To facilitate further acceleration, a local resolution-of-the-identity (RI) three-index integral generation algorithm was developed using LMOs and LVMOs. Although the NEVPT2 theory is considered to be less susceptible to the intruder-state problem (ISP), this study revealed that it can easily suffer from ISP when calculating high-lying excited states. We ameliorated this instability using the imaginary level shift (LS) technique. The PNO-QD-NEVPT2 calculations were performed on small organic molecules for the 30 lowest-lying states, as well as photoisomerization involving the conical intersection of 1,1-dimethyldibenzo[b,f] silepin with a cis-stilbene skeleton. These calculations revealed that the PNO-QD-NEVPT2 method yielded negligible errors compared to the canonical QD-NEVPT2 results. Furthermore, we tested its applicability to a large photoisomerization system using the green fluorescent protein model and the 10-state calculation of the large transition metal complex, showcasing that off-diagonal elements can be evaluated at a relatively low cost

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

Influence of the choice of projection manifolds in the CASPT2 implementation

The Complete Active Space Second-Order Perturbation Theory (CASPT2) is well-established as a high-accuracy electronic structure method. It was originally implemented in the early 1990s to an efficient computer code in the molcas program suite, and this implementation has been extensively used as a standard tool. Here, we report a comparison of it against two independent computer-aided implementations of the CASPT2 method, revealing that the CASPT2 energies provided by the original code of molcas (version 8 or earlier) are inconsistent with the predictions of the lately developed computer-aided implementations. It is shown that this error is associated with the projections of the first-order equation onto the fully internally contracted multireference bases which are partially inconsistent between the left- and right-hand sides. The degree of the errors is assessed by performing illustrative CASPT2 calculations. The errors in total CASPT2 energies are demonstrated to be negligible relative to chemical accuracy in many cases, while there is a difficult case where they may substantially alter chemical description. The incorporation of the consistent projections into molcas has been carried out, which is available in the version 8 sp1

Multistate Complete-Active-Space Second-Order Perturbation Theory Based on Density Matrix Renormalization Group Reference States

We present the development of the multistate multireference second-order perturbation theory (CASPT2) with multiroot references, which are described using the density matrix renormalization group (DMRG) method to handle a large active space. The multistate first-order wave functions are expanded into the internally contracted (IC) basis of the single-state single-reference (SS-SR) scheme, which is shown to be the most feasible variant to use DMRG references. The feasibility of the SS-SR scheme comes from two factors: first, it formally does not require the fourth-order transition reduced density matrix (TRDM) and second, the computational complexity scales linearly with the number of the reference states. The extended multistate (XMS) treatment is further incorporated, giving suited treatment of the zeroth-order Hamiltonian despite the fact that the SS-SR based IC basis is not invariant with respect to the XMS rotation. In addition, the state-specific fourth-order reduced density matrix (RDM) is eliminated in an approximate fashion using the cumulant reconstruction formula, as also done in the previous state-specific DMRG-cu(4)-CASPT2 approach. The resultant method, referred to as DMRG-cu(4)-XMS-CASPT2, uses the RDMs and TRDMs of up to third-order provided by the DMRG calculation. The multistate potential energy curves of the photoisomerization of diarylethene derivatives with CAS­(26e,24o) are presented to illustrate the applicability of our theoretical approach