Analysis of the singlet-triplet splitting computed by the density functional theory-broken-symmetry method: is it an exchange coupling constant?

We derive an analytical expression of the density functional theory (DFT)-broken-symmetry (BS) estimation J(BS) of the singlet-triplet gap at the "3 sites-4 electrons" level, that is, two S = (1)/(2) metallic sites + one diamagnetic bridge orbital. As originally designed by Noodleman and Davidson (Chem. Phys.1986, 109, 131), J(BS) contains the residual ferromagnetic contribution, single ligand-to-metal and metal-to-metal charge-transfer terms, but no double ligand-to-metal charge-transfer terms or intra/interligand spin-polarization terms. As revealed by the present analysis, the triplet and BS states computed by DFT differ, not only perturbatively (as expected) because of the various physical mechanisms involved (i.e., differential charge-transfer terms) but mainly because of a spurious and unphysical symmetry breaking of the bridge orbitals in the BS state. We examine the consequences of such a difference by deriving two analytical expressions of the exchange coupling constant, one from the BS orbitals designed to match J(BS) and another one from triplet orbitals only. Following and extending on the first paper in the series (J. Phys. Chem. A 2010, 114, 6149), we propose a simple procedure to extract appropriate parameters filling in our analytical expressions. Moreover, we derive the equivalent "3 sites-4 electrons" exchange coupling constant in the configuration-interaction approach, J(CI), for the purpose of comparison. These analytical expressions have been applied to various copper dimers and compared to experimental values.