Interpretation of Exchange Interaction through Orbital Entanglement

Recently, the analysis of single-orbital entropy and mutual information has been introduced as a tool for the investigation of contributions to the exchange (J) coupling between open-shell metal ions [Stein et at J. Phys. Chem. Lett. 2019, 10, 6762-6770]. Here, we show that this analysis may lead to an incorrect interpretation of the J-coupling mechanism. Instead, we propose an orbital-entanglement analysis that is based on the two-electron density and that provides a coherent picture of the contributing exchange pathways, which seems fully consistent with the available J values. For this purpose, we used a prototypical bis-mu-oxo binuclear manganese complex ([Mn2O2 (NH3)(8)](4+)) and demonstrated that its antiferromagnetism (J < 0), calculated by using the active space composed of all valence p(O) and d(Mn) orbitals, correlates well with the largest elements in the differential low-spin vs high-spin entanglement map. These elements correspond to interactions between the pairs of d(Mn) orbitals mediated by the oxo-bridging out-of-plane p orbitals, representing the pi superexchange pathway. We also show that the reduction of active space to manifold of the singly occupied magnetic orbitals does not lead to discrepancy between the calculated J values and entanglement maps. This contrasts to analysis of mutual information, which suggests the direct d(Mn)-d(Mn) interactions to play a dominant role for the J coupling, irrespective of the size of active space as well as of the antiferromagnetism expected. The failure is attributed to the large contribution of spin entanglement contained in the mutual information of the low-spin state, which may be regarded as the origin of the different complexity of its wave function and electron density.

Automated summary

The analysis of entropy and mutual information may lead to incorrect interpretations of the J-coupling mechanism when investigating exchange interactions between open-shell metal ions. Instead, a new orbital-entanglement analysis based on two-electron density provides a coherent picture of exchange pathways that correlates well with experimental J values. This study highlights the importance of considering spin entanglement in understanding the complexity of wave functions and electron density, particularly in the context of exchange interactions in metal complexes.