Local Noncollinear Spin Analysis

In this work, we generalize the local spin analysis of Clark and Davidson [J. Chem. Phys. 115(16), 7382 (2001)] for the partitioning of the expectation value of the molecular spin square operator, $\langle S^2 \rangle$, into atomic contributions, $\langle S_A \cdot S_B \rangle$, to the noncollinear spin case in the framework of density functional theory (DFT). We derive the working equations and we show applications to the analysis of the noncollinear spin solutions of typical spin-frustrated systems and to the calculation of magnetic exchange couplings. In the former case, we employ the triangular H$_3$He$_3$ test molecule and a Mn$_3$ complex to show that the local spin analysis provides additional information that complements the standard one-particle spin population analysis. For the calculation of magnetic exchange couplings, $J_{AB}$, we employ the local spin partitioning to extract $\langle S_A \cdot S_B \rangle$ as a function of the interatomic spin orientation given by the angle $\theta$. This, combined with the dependence of the electronic energy with $\theta$, provides a methodology to extract $J_{AB}$ from DFT calculations that, in contrast to conventional energy differences based methods, does not require the use of $ad-hoc$ $S_A$ and $S_B$ values.