Abstract
In this work, we demonstrate how to efficiently compute the one- and two-body reduced density matrices within the spin-adapted full configuration interaction quantum Monte Carlo (FCIQMC) method, which is based on the graphical unitary group approach (GUGA). This allows us to use GUGA-FCIQMC as a spin-pure configuration interaction (CI) eigensolver within the complete active space self-consistent field (CASSCF) procedure and hence to stochastically treat active spaces far larger than conventional CI solvers while variationally relaxing orbitals for specific spin-pure states. We apply the method to investigate the spin ladder in iron–sulfur dimer and tetramer model systems. We demonstrate the importance of the orbital relaxation by comparing the Heisenberg model magnetic coupling parameters from the CASSCF procedure to those from a CI-only (CASCI) procedure based on restricted open-shell Hartree–Fock orbitals. We show that the orbital relaxation differentially stabilizes the lower-spin states, thus enlarging the coupling parameters with respect to the values predicted by ignoring orbital relaxation effects. Moreover, we find that, while CASCI results are well fit by a simple bilinear Heisenberg Hamiltonian, the CASSCF eigenvalues exhibit deviations that necessitate the inclusion of biquadratic terms in the model Hamiltonian.
Highlights
The complete active space self-consistent field (CASSCF) method is a well-established approach in quantum chemistry for the treatment of strongly correlated electron systems with substantial multireference character.[1−8] Important static correlation effects are rigorously described within the active space, consisting of the most important orbitals and electrons, while the effect of the environment is accounted for at the mean-field level via a variational orbital optimization
In our earlier works,[62,63] we have demonstrated via theoretical arguments, and shown with calculations, that the choice of orbital representation and reordering greatly affect the sparsity of the configuration interaction (CI) wave function within the graphical extension of the UGA (GUGA) formalism
We show in a radar plot (Figure 10) the reference weight, the sum of all metal-to-metal charge transfer (MMCT), local d → d′ radial excited configurations (Radial), ligand-tometal charge transfer (LMCT), and local Hund’s rule-violating configurations for the CAS configuration interaction (CASCI) and CASSCF singlet state in the (22e,26o) active space
Summary
The complete active space self-consistent field (CASSCF) method is a well-established approach in quantum chemistry for the treatment of strongly correlated electron systems with substantial multireference character.[1−8] Important static correlation effects are rigorously described within the active space, consisting of the most important orbitals and electrons, while the effect of the environment (electrons not included in the active space) is accounted for at the mean-field level via a variational orbital optimization (the SCF procedure). While GAS was designed with the same GUGA framework discussed in the present work to enforce spin-adaptation,[7] ORMAS is Slaterdeterminant-based and recently made use of the spin-flip configuration interaction method[56] to ensure the correct spin multiplicity.[57] These approaches allow the study of much larger active spaces.[6,39,40,55,58−64] The use of FCIQMC as the CASSCF CI eigensolver within the super-CI framework, termed stochasticCASSCF,[6] has been developed in our group and used to study a number of strongly correlated systems, such as model systems of Fe(II)-porphyrins and the correlation mechanisms that differentially stabilize the intermediate spin states over the high-spin. We supply coordinate and orbital files, computational details, and comparisons with available exact results for small active spaces, a table with the data used in Figure 10, a study on improved convergence due to stochastic noise, the protocol on how we compared the orbitals in Figure 11, details on interface and the RDM storage convention in OpenMolcas, and a quick access literature overview of computational results for the Fe2S2 system in the Supporting Information (SI)
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