Toward Reliable Prediction of Hyperfine Coupling Constants Using Ab Initio Density Matrix Renormalization Group Method: Diatomic 2 Σ and Vinyl Radicals as Test Cases Tran Nguyen Lan,* ,† Yuki Kurashige, †,‡ and Takeshi Yanai †,‡ † The Graduate University for Advanced Studies, Myodaiji, Okazaki, Aichi 444-8585, Japan ‡ Department of Theoretical and Computational Molecular Science, Institute for Molecular Science, Okazaki, Aichi 444-8585, Japan ABSTRACT: The density matrix renormalization group (DMRG) method is used in conjunction with the complete active space (CAS) procedure, the CAS configuration interaction (CASCI), and the CAS self-consistent field (CASSCF) to evaluate hyperfine coupling constants (HFCCs) for a series of diatomic 2 Σ radicals (BO, CO + , CN, and AlO) and vinyl (C 2 H 3 ) radical. The electron correlation effects on the computed HFCC values were systematically investigated using various levels of active space, which were increasingly extended from single valence space to large-size model space entailing double valence and at least single polarization shells. In addition, the core correlation was treated by including the core orbitals in active space. Reasonably accurate results were obtained by the DMRG-CASSCF method involving orbital optimization, while DMRG- CASCI calculations with Hartree−Fock orbitals provided poor agreement of the HFCCs with the experimental values. To achieve further insights into the accuracy of HFCC calculations, the orbital contributions to the total spin density were analyzed at a given nucleus, which is directly related to the FC term and is numerically sensitive to the level of correlation treatment and basis sets. The convergence of calculated HFCCs with an increasing number of renormalized states was also assessed. This work serves as the first study on the performance of the ab initio DMRG method for HFCC prediction. 1. INTRODUCTION The hyperfine coupling constant (HFCC) is an observation of the interaction between electron spin s and nuclear spin I that plays an important role in understanding molecular electron paramagnetic resonance (EPR) spectra. 1 There are both isotropic and anisotropic contributions to HFCCs. For light element molecules, the spin−orbit coupling (SOC) is small and can be neglected, so that the HFCCs are dominated by the Fermi contact (FC) term 2 and the spin-dipole (SD) interaction term. 3 While the SD term is known to be numerically less sensitive and thus can be relatively easily calculated, the high- accuracy prediction of the FC term is still a challenging task for computational quantum chemistry. Difficulties arise because the FC term is evaluated from a direct numerical measure of the spin density at the position of the nucleus, which is quite sensitive to the quality of the electronic structure calculation. Therefore, high-level treatment of the wave function involving electron correlation and the basis sets is often required to achieve sufficient accuracy in HFCC calculations. The effects of the basis set on the FC term are caused in close proximity to the nuclear centers, so that the tight Gaussian-type functions are also used. The solvent and vibrational effects are also important in extreme cases. 4 However, it is quite difficult and expensive to account for these effects in ab initio quantum chemistry calculations. The nonvibrating gas-phase condition has been widely adopted for HFCC calculations and is also assumed in this study. In addition, we only focus on the nonrelativistic limit in the present work. The restricted open-shell Hartree−Fock (ROHF) method cannot describe the core level spin-polarization; 5 therefore, it is incapable of describing the FC term. In contrast, the spin- polarization is explicitly included in the unrestricted Hartree− Fock (UHF) method. However, the UHF method typically overestimates the FC term to a large extent due to the mean- field treatment of many-electron interactions. Spin contami- nation in the UHF wave function is considered to be another source of errors. To remedy the spin-contamination effect, Nakatsuji et al. 6 developed a modified UHF, the projected UHF (PUHF). Although this method certainly provides an improve- ment of the UHF results, it does not give uniform agreement with experimental results better than approximately 60%, as shown by Chipman. 7 These earlier studies showed that electron correlation effects are largely responsible for accuracy of HFCC calculations. The coupled cluster (CC) approach has been widely applied to HFCC calculations. Bartlett and co-workers developed several approaches based on the CC framework to evaluate the HFCCs, including finite-field CC 8 or analytical derivative CC. 9 Later, they used the two-parameter complete basis set (CBS2) extrapolation scheme in the CC calculations of HFCCs to rectify the error from basis set truncation. 10 Their benchmarks revealed that the calculated FC values are strongly dependent on the choice of basis set and treatment with and without the Received: November 11, 2013 Published: January 28, 2014 Article pubs.acs.org/JCTC © 2014 American Chemical Society 1953 dx.doi.org/10.1021/ct400978j | J. Chem. Theory Comput. 2014, 10, 1953−1967