Spin decontamination of broken-symmetry density functional theory calculations: deeper insight and new formulations

This work re-examines the problem of the broken-symmetry Density-Functional Theory (DFT) solutions in diradical systems, in particular for the calculation of magnetic couplings. The Ms = 0 solution is not an eigenfunction of the S-2 spin operator and the evaluation of the singlet state energy requires a spin-decontamination. A popular approximation is provided by the so-called Yamaguchi formula, which operates using the expectation values of S-2 relative to both Ms = 1 and Ms = 0 solutions. Referring to a previous decomposition of the magnetic coupling in terms of direct exchange, kinetic exchange and core polarization, it is shown that this expression will lead to unreliable values of the singlet-triplet energy gap when the spin polarization of the core orbitals becomes large. The here-proposed method of spin-decontamination is based on the Effective Hamiltonian Theory and uses the overlap between the two degenerate Ms = 0 solutions. An approximate and convenient formula, which uses the expectation values of S-2 of the Ms = 0 solutions before and after core polarization is proposed, which is free from the Yamaguchi's formula artefact, as illustrated on an organic diradical presenting a very high value of [S-2] for the Ms = 0 solution, the antiferromagnetic coupling being due to the spin polarization mechanism.