The Li Group

We report the state-averaged spin–orbit driven similarity renormalization group second-order perturbation theory (SA-SO-DSRG-PT2) for efficient computations of spin–orbit coupling (SOC) effects in multireference systems. This approach adopts a set of multiconfigurational reference states obtained with a spin-free relativistic Hamiltonian and introduces SOC during the perturbative treatment of dynamical correlation. Specifically, the zeroth-order Hamiltonian is defined from the spin-free Hamiltonian, while the remaining spin-free contributions and the spin-dependent Hamiltonian enter as first-order perturbations in the DSRG transformation. This formulation generates an SOC-containing SA-DSRG-PT2 effective Hamiltonian while largely preserving the computational cost of the conventional spin-free theory. Two spin-dependent Hamiltonians are examined: the first-order Douglas–Kroll–Hess spin–orbit Hamiltonian with mean-field approximated two-electron contributions and the spin-dependent part of the one-electron X2C Hamiltonian with the screened-nuclear spin–orbit approximation. Benchmarks on main-group atoms and diatomic molecules, transition-metal elements, trivalent lanthanide cations, and actinide dioxide cations show consistent accuracy across the periodic table for both spin-dependent Hamiltonians. Applications to Co(II) single-ion magnets, including systems described with up to 1790 basis functions, further highlight the promise of SA-SO-DSRG-PT2 for efficient SOC calculations in large open-shell molecules.

The accurate prediction of magnetic exchange coupling constants (J) in transition-metal complexes remains challenging for electronic structure theory. In this work, we demonstrate that the state-averaged driven similarity renormalization group second-order perturbation theory (SA-DSRG-PT2), combined with the density matrix renormalization group, provides quantitative J values on a set of 21 3d bimetallic complexes with a focus on dicopper species. Active orbitals are systematically identified using the atomic valence active space approach, ensuring a consistent inclusion of metal 3d and 4d shells, and the ligand orbitals involved in superexchange pathways. Analysis of the SA-DSRG-PT2 effective Hamiltonians indicates that the reduction of on-site Coulomb repulsion is the primary origin of the improved exchange couplings. In addition, we adopt the orbital mutual correlation plot as an intuitive tool to visualize exchange pathways, allowing direct identifications of metal-metal and metal-ligand correlations and their modulation by dynamical correlation. These results establish SA-DSRG- PT2 as a robust and practical multireference approach for studying magnetic interactions in transition-metal complexes.

We present a new implementation of the driven similarity renormalization group (DSRG) based on a density matrix renormalization group (DMRG) reference. The explicit build of high-order reduced density matrices is avoided by forming matrix-product-state compressed intermediates. This algorithm facilitates the application of DSRG second- and third-order perturbation theories to dodecacene with an active space of 50 electrons in 50 orbitals. This active space appears the largest employed to date within the framework of internally contracted multireference formalism. The DMRG-DSRG approach is applied to several challenging systems, including the singlet-triplet gaps (Δ_ST) of oligoacenes ranging from naphthalene to dodecacene, the vertical excitation energies of zeaxanthin, and the ground-state potential energy curve (PEC) of Cr₂ molecule. Our best estimate for the vertical Δ_ST of dodecacene is 0.22 eV, showing an excellent agreement with that of the linearized adiabatic connection method (0.24 eV). For zeaxanthin, all DSRG schemes suggest the order of 2 ¹A_g^– < 1 ¹B_u⁺ < 1 ¹B_u^– for excited states. Both the equilibrium and the shoulder regions of the Cr₂ PEC are reasonably reproduced by the linearized DSRG with one- and two-body operators.

The driven similarity renormalization group (DSRG) provides an alternative way to address the intruder state problem in quantum chemistry. In this review, we discuss recent developments of multireference methods based on the DSRG. We provide a pedagogical introduction to the DSRG and its various extensions and discuss its formal properties in great detail. In addition, we report several illustrative applications of the DSRG to molecular systems.