Sample-based quantum diagonalization approach for open-shell transition-metal complexes in gas and implicit-solvent

Open-shell $3d$ transition-metal complexes challenge electronic-structure methods because competing spin states, charge transfer, and solvation jointly determine their energetics. Here, we combine sample-based quantum diagonalization (SQD) with the integral-equation-formalism polarizable continuum model (IEF-PCM), extending SQD to correlated open-shell transition-metal systems in a dielectric environment. We investigate the octahedrally coordinated $\mathrm{[Co(H_2O)_5CO_2]^{2+/3+}}$ complex across two oxidation states, four spin multiplicities, and a metal-ligand dissociation coordinate. We study the Co(III) singlet and quintet states and the Co(II) doublet and quartet states, incorporating open-shell references into SQD-IEF-PCM through an outer self-consistent reaction-field loop. Using samples collected on an IBM Heron quantum processor and active spaces of up to 50 qubits, SQD reproduces coupled-cluster and heat-bath configuration-interaction benchmarks within the same active space in the gas phase and implicit solvent, with a largest observed deviation below 9 $mE_h$. Along the dissociation coordinate of high-spin quintet $\mathrm{[Co(H_2O)_5CO_2]^{3+}}$, SQD resolves an avoided crossing caused by internal charge transfer; this feature is absent in the singlet and the lower oxidation state of the complex. Relative to the gas phase, implicit solvation stabilizes for the quintet state the neutral CO$_2$ dissociation and suppresses the avoided-crossing feature. To our knowledge, this is the first hardware demonstration of SQD for an open-shell $3d$ transition-metal complex in gas phase and implict solvent. These results establish SQD as a robust quantum-centric approach for transition-metal chemistry where spin state ordering, charge transfer, and environmental effects are strongly intertwined.