Spin-free formulation of the multireference driven similarity renormalization group: A benchmark study of first-row diatomic molecules and spin-crossover energetics

We report a spin-free formulation of the multireference (MR) driven similarity renormalization group (DSRG) by employing the ensemble normal ordering of Mukherjee and Kutzelnigg [W. Kutzelnigg and D. Mukherjee, J. Chem. Phys. 107, 432 (1997)]. This ensemble averages over all microstates for a given total spin quantum number and, therefore, it is invariant with respect to SU(2) transformations. As such, all equations may be reformulated in terms of spin-free quantities and they closely resemble those of spin-adapted closed-shell coupled cluster (CC) theory. The current implementation is used to assess the accuracy of various truncated MR-DSRG methods (perturbation theory up to third order and iterative methods with single and double excitations) in computing the constants of thirty-three first-row diatomic molecules. The accuracy trends for these first-row diatomics are consistent with our previous benchmark on a small subset of closed-shell diatomic molecules. We then present the first MR-DSRG application on transition-metal complexes by computing the spin splittings of the [Fe(H$_2$O)$_6$]$^{2+}$ and [Fe(NH$_3$)$_6$]$^{2+}$ molecules. Focal point analysis (FPA) shows that third-order perturbative corrections are essential to achieve reasonably converged energetics. A FPA based on the linearized MR-DSRG theory with one- and two-body operators and up to a quintuple-$\zeta$ basis set predicts the spin splittings of [Fe(H$_2$O)$_6$]$^{2+}$ and [Fe(NH$_3$)$_6$]$^{2+}$ to be $-35.7$ and $-17.1$ kcal mol$^{-1}$, respectively, showing good agreement with results of local CC theory with singles, doubles, and perturbative triples.