Projector Augmented Wave Method Incorporated into Gauss-Type Atomic Orbital Based Density Functional Theory

The Projector Augmented Wave (PAW) method developed by Blöchl is well recognized as an efficient, accurate pseudopotential approach in solid-state density functional theory (DFT) calculations with the plane-wave basis. Here we present an approach to incorporate the PAW method into the Gauss-type function (GTF) based DFT implementation, which is widely used for molecular quantum chemistry calculations. The nodal and high-exponent GTF components of valence molecular orbitals (MOs) are removed or pseudized by the ultrasoft PAW treatment, while there is elaborate transparency to construct an accurate and well-controlled pseudopotential from all-electron atomic description and to reconstruct an all-electron form of valence MOs from the pseudo MOs. The smoothness of the pseudo MOs should benefit the efficiency of GTF-based DFT calculations in terms of elimination of high-exponent primitive GTFs and reduction of grid points in the numerical quadrature. The processes of the PAW method are divided into basis-independent and -dependent parts. The former is carried out using the previously developed PAW libraries libpaw and atompaw. The present scheme is implemented by incorporating libpaw into the conventional GTF-based DFT solver. The details of the formulations and implementations of GTF-related PAW procedures are presented. The test calculations are shown for illustrating the performance. With the near-complete GTF basis at the cc-pVQZ level, the total energies obtained using our PAW method with suited frozen core treatments converge to those with the conventional all-electron GTF-based method with a rather small absolute error

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

Computational Evidence of Inversion of ¹L_a and ¹L_b‑Derived Excited States in Naphthalene Excimer Formation from <i>ab Initio</i> Multireference Theory with Large Active Space: DMRG-CASPT2 Study

The naphthalene molecule has two important lowest-lying singlet excited states, denoted ¹L_a and ¹L_b. Association of the excited and ground state monomers yields a metastable excited dimer (<i>excimer</i>), which emits characteristic fluorescence. Here, we report a first computational result based on <i>ab initio</i> theory to corroborate that the naphthalene excimer fluorescence is ¹L_a parentage, resulting from inversion of ¹L_a and ¹L_b-derived dimer states. This inversion was hypothesized by earlier experimental studies; however, it has not been confirmed rigorously. In this study, the advanced multireference (MR) theory based on the density matrix renormalization group that enables using unprecedented large-size active space for describing significant electron correlation effects is used to provide accurate potential energy curves (PECs) of the excited states. The results evidenced the inversion of the PECs and accurately predicted transition energies for excimer fluorescence and monomer absorption. Traditional MR calculations with smaller active spaces and single-reference theory calculations exhibit serious inconsistencies with experimental observations

More π Electrons Make a Difference: Emergence of Many Radicals on Graphene Nanoribbons Studied by <i>Ab Initio</i> DMRG Theory

Graphene nanoribbons (GNRs), also seen as rectangular polycyclic aromatic hydrocarbons, have been intensively studied to explore their potential applicability as superior organic semiconductors with high mobility. The difficulty arises in the synthesis or isolation of GNRs with increased conjugate length, GNRs being known to have radical electrons on their zigzag edges. Here, we use a most advanced <i>ab initio</i> theory based on density matrix renormalization group (DMRG) theory to show the emerging process of how GNRs develop electronic states from nonradical to radical characters with increasing ribbon length. We show the mesoscopic size effect that comes into play in quantum many-body interactions of π electrons, which is responsible for the polyradical nature. An analytic form is presented to model the size dependence of the number of radicals for arbitrary-length GNRs. These results and associated insights deepen the understanding of carbon-based chemistry and offer useful information for the synthesis and design of stable and functional GNRs

Multireference Ab Initio Density Matrix Renormalization Group (DMRG)-CASSCF and DMRG-CASPT2 Study on the Photochromic Ring Opening of Spiropyran

The photochromic ring-opening reaction of spiropyran has been revisited at the multireference CASSCF and CASPT2 level with a CAS­(22e,20o) active space, in combination with density matrix renormalization group (DMRG) methods. The accuracy of the DMRG-CASSCF and DMRG-CASPT2 calculations, with respect to the number of renormalized states, the number of roots in state-averaged wave functions, and the number basis functions, was examined. For the current system, chemically accurate results can be obtained with a relatively small number of renormalized states. The nature and vertical excitation energies of the excited (S₁ and S₂) states are consistent with conventional CAS­(or RAS)­PT2 with medium active spaces. The capability of the DMRG-CASSCF method in the optimization of molecular geometry is demonstrated for the first time. The computation costs (several hours per optimization cycle) are comparable with that of the conventional CASSCF geometry optimization with small active space. Finally, the DMRG-PT2 computed S₁-MEP for the C–O and C–N bond-cleavage processes show good agreement with our previous calculations with a CAS­(12e,10o) active space [Liu, F.; Morokuma, K. <i>J. Am. Chem. Soc.</i> <b>2013</b>, <i>135</i>, 10693–10702]. Especially, the role of the HOOP valleys in the S₁ → S₀ nonadiabatic decay has been confirmed

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

Reactivity of the Binuclear Non-Heme Iron Active Site of Δ⁹ Desaturase Studied by Large-Scale Multireference <i>Ab Initio</i> Calculations

The results of density matrix renormalization group complete active space self-consistent field (DMRG-CASSCF) and second-order perturbation theory (DMRG-CASPT2) calculations are presented on various structural alternatives for the O–O and first C–H activating step of the catalytic cycle of the binuclear nonheme iron enzyme Δ⁹ desaturase. This enzyme is capable of inserting a double bond into an alkyl chain by double hydrogen (H) atom abstraction using molecular O₂. The reaction step studied here is presumably associated with the highest activation barrier along the full pathway; therefore, its quantitative assessment is of key importance to the understanding of the catalysis. The DMRG approach allows unprecedentedly large active spaces for the explicit correlation of electrons in the large part of the chemically important valence space, which is apparently <i>conditio sine qua non</i> for obtaining well-converged reaction energetics. The derived reaction mechanism involves protonation of the previously characterized 1,2-μ peroxy Fe^IIIFe^III (<b>P</b>) intermediate to a 1,1-μ hydroperoxy species, which abstracts an H atom from the C₁₀ site of the substrate. An Fe^IV-oxo unit is generated concomitantly, supposedly capable of the second H atom abstraction from C₉. In addition, several popular DFT functionals were compared to the computed DMRG-CASPT2 data. Notably, many of these show a preference for heterolytic C–H cleavage, erroneously predicting substrate hydroxylation. This study shows that, despite its limitations, DMRG-CASPT2 is a significant methodological advancement toward the accurate computational treatment of complex bioinorganic systems, such as those with the highly open-shell diiron active sites