The recent advent of the density matrix renormalization group (DMRG) theory has delivered a new capability to compute multireference (MR) wave function with large configuration space, which far exceeds the limitation of conventional approaches. Here, we provide an overview of our recent work on the developments of ab initio DMRG methods in the context of the active space approaches and their applications to MR chemical systems.

We report an extension of our previous development that incorporated quantum-chemical density matrix renormalization group (DMRG) into the complete active space second-order perturbation theory (CASPT2) [Y. Kurashige and T. Yanai, J. Chem. Phys. 135, 094104 (2011)]. In the previous study, the combined theory, referred to as DMRG-CASPT2, was built upon the use of pseudo-canonical molecular orbitals (PCMOs) for one-electron basis. Within the PCMO basis, the construction of the four-particle reduced density matrix (4-RDM) using DMRG can be greatly facilitated because of simplicity in the multiplication of 4-RDM and diagonal Fock matrix in the CASPT2 equation. In this work, we develop an approach to use more suited orbital basis in DMRG-CASPT2 calculations, e.g., localized molecular orbitals, in order to extend the domain of applicability. Because the multiplication of 4-RDM and generalized Fock matrix is no longer simple in general orbitals, an approximation is made to it using the cumulant reconstruction neglecting higher-particle cumulants. Also, we present the details of the algorithm to compute 3-RDM of the DMRG wavefunction as an extension of the 2-RDM algorithm of Zgid et al. [J. Chem. Phys. 128, 144115 (2008)] and Chan et al. [J. Chem. Phys. 128, 144117 (2008)]. The performance of the extended DMRG-CASPT2 approach was examined for large-scale multireference systems, such as low-lying excited states of long-chain polyenes and isomerization potential of {[Cu(NH3)3]2O2}(2+).

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 Δ(9) 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 conditio sine qua non for obtaining well-converged reaction energetics. The derived reaction mechanism involves protonation of the previously characterized 1,2-μ peroxy Fe(III)Fe(III) (P) 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.

Reduction potentials of several M(2+/3+) (M = Ru, Os) octahedral complexes, namely, [M(H2O)6](2+/3+), [MCl6](4-/3-), [M(NH3)6](2+/3+), [M(en)3](2+/3+) [M(bipy)3](2+/3+), and [M(CN)6](4-/3-), were calculated using the CASSCF/CASPT2/CASSI and MRCI methods including spin-orbit coupling (SOC) by means of first-order quasi-degenerate perturbation theory. It was shown that the effect of SOC accounts for a systematic shift of approximately -70 mV in the reduction potentials of the studied ruthenium (II/III) complexes and an approximately -300 mV shift for the osmium(II/III) complexes. SOC splits the sixfold-degenerate (2)T(2g) ground electronic state (in ideal octahedral symmetry) of the M(3+) ions into the E((5/2)g) Kramers doublet and G((3/2)g) quartet, which were calculated to split by 1354-1573 cm(-1) in the Ru(3+) complexes and 4155-5061 cm(-1) in the Os(3+) complexes. It was demonstrated that this splitting represents the main contribution to the stabilization of the M(3+) ground state with respect to the closed-shell (1)A(1g) ground state in M(2+) systems. Moreover, it was shown that the accuracy of the calculated reduction potentials depends on the calculated solvation energies of both the oxidized and reduced forms. For smaller ligands, it involves explicit inclusion of the second solvation sphere into the calculations, whereas implicit solvation models yield results of sufficient accuracy for complexes with larger ligands. In such cases (e.g., [M(bipy)3](2+/3+) and its derivatives), very good agreement between the calculated (SOC-corrected) values of the reduction potentials and the available experimental values was obtained. These results led us to the conclusion that especially for Os(2+/3+) complexes, inclusion of SOC is necessary to avoid systematic errors of approximately 300 mV in the calculated reduction potentials.

The multicopper oxidases (MCOs) couple the four-electron reduction of dioxygen to water with four one-electron oxidations of various substrates. Extensive spectroscopic studies have identified several intermediates in the MCO catalytic cycle, but they have not been able to settle the structures of three of the intermediates, viz. the native intermediate (NI), the peroxy intermediate (PI), and the peroxy adduct (PA). The suggested structures have been further refined and characterized by quantum mechanical/molecular mechanical (QM/MM) calculations. In this paper, we try to establish a direct link between theory and experiment, by calculating spectroscopic parameters for these intermediates using multireference wave functions from the multistate CASPT2 and MRDDCI2 methods. Thereby, we have been able to reproduce low-spin ground states (S = 0 or S = 1/2) for all the MCO intermediates, as well as a low-lying (approximately 150 cm-1) doublet state and a doublet-quartet energy gap of approximately 780 cm-1 for the NI. Moreover, we reproduce the zero-field splitting (approximately 70 cm-1) of the ground 2E state in a D3 symmetric hydroxy-bridged trinuclear Cu(II) model of the NI and obtain a quantitatively correct quartet-doublet splitting (164 cm-1) for a mu3-oxo-bridged trinuclear Cu(II) cluster. All results support the suggestion that the NI has an O2- atom in the center of the trinuclear cluster, whereas both the PI and PA have an O22- ion in the center of the cluster, in agreement with the QM/MM results and spectroscopic measurements.

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.

The O-O (oxygen-oxygen) bond formation is widely recognized as a key step of the catalytic reaction of dioxygen evolution from water. Recently, the water oxidation catalyzed by potassium ferrate (K2FeO4) was investigated on the basis of experimental kinetic isotope effect analysis assisted by density functional calculations, revealing the intramolecular oxo-coupling mechanism within a di-iron(vi) intermediate, or diferrate [Sarma et al., J. Am. Chem. Soc., 2012, 134, 15371]. Here, we report a detailed examination of this diferrate-mediated O-O bond formation using scalable multireference electronic structure theory. High-dimensional correlated many-electron wave functions beyond the one-electron picture were computed using the ab initio density matrix renormalization group (DMRG) method along the O-O bond formation pathway. The necessity of using large active space arises from the description of complex electronic interactions and varying redox states both associated with two-center antiferromagnetic multivalent iron-oxo coupling. Dynamic correlation effects on top of the active space DMRG wave functions were additively accounted for by complete active space second-order perturbation (CASPT2) and multireference configuration interaction (MRCI) based methods, which were recently introduced by our group. These multireference methods were capable of handling the double shell effects in the extended active space treatment. The calculations with an active space of 36 electrons in 32 orbitals, which is far over conventional limitation, provide a quantitatively reliable prediction of potential energy profiles and confirmed the viability of the direct oxo coupling. The bonding nature of Fe-O and dual bonding character of O-O are discussed using natural orbitals.

EPR spectroscopy has proven to be an indispensable tool in elucidating the structure of metal sites in proteins. In recent years, experimental EPR data have been complemented by theoretical calculations, which have become a standard tool of many quantum chemical packages. However, there have only been a few attempts to calculate EPR g tensors for exchange-coupled systems with more than two spins. In this work, we present a quantum chemical study of structural, electronic, and magnetic properties of intermediates in the reaction cycle of multicopper oxidases and of their inorganic models. All these systems contain three copper(II) ions bridged by hydroxide or O2− anions and their ground states are antiferromagnetically coupled doublets. We demonstrate that only multireference methods, such as CASSCF/CASPT2 or MRCI can yield qualitatively correct results (compared to the experimental values) and consider the accuracy of the calculated EPR g tensors as the current benchmark of quantum chemical methods. By decomposing the calculated g tensors into terms arising from interactions of the ground state with the various excited states, the origin of the zero-field splitting is explained. The results of the study demonstrate that a truly quantitative prediction of the g tensors of exchange-coupled systems is a great challenge to contemporary theory. The predictions strongly depend on small energy differences that are difficult to predict with sufficient accuracy by any quantum chemical method that is applicable to systems of the size of our target systems.