Spin-contanination in coupled-cluster wavefunctions
The propensity of approximate solutions of the electronic Schrödinger equation to break spin-symmetry is directly related to the quality of the approximate wavefunction, and thus can be used as a diagnostic tool. The quasi-variational nature of the (valence) optimized orbitals coupled-cluster doubles methods, (V)OO-CCD, enables a discussion of the stability of coupled-cluster wavefunctions in terms of both spin-contamination and a corresponding energy lowering relative to the pure spin solutions. The spin-contamination of (V)OO-CCD models has been studied for bond-breaking processes and diradicals. The main findings are: (i) The OO-CCD method is stable for a relatively large range of nuclear distortions and is capable of eliminating even very large spin-contamination of the unrestricted Hartree-Fock solution given that the molecular electronic configuration remains essentially single-reference. When a spin-contaminated solution arises, the energy splitting rapidly becomes large and <Ŝ2> approaches the Hartree-Fock value; (ii) The VOO-CCD method, which is designed to approximate a multi-reference model, remains stable over broader ranges; however, for pure diradicals it becomes unstable. In these cases, spin-contamination is also very large, but the energy lowering for the spin-unrestricted solutions is negligible; (iii) Higher order corrections described by perturbation theory lead to smaller energy splittings between restricted and unrestricted (V)OO-CCD energies. However, in case of spin-contaminated (V)OO-CCD solutions, these corrections may lead to unphysical shapes of the potential energy surfaces. Thus, in order to quantitatively characterize the quality of the wavefunctions, both spin-contamination and energy lowering due to the breaking of spin-symmetry must be considered.