On Singlet-Triplet Transitions Induced by Exchange with Paramagnetic Molecules and the Intermolecular Coupling of Spin Angular Momenta

Abstract

The theoretical basis of the singlet‐triplet transition mechanism by exchange coupling with neighboring paramagnetic molecules is examined. Attention is drawn to the two implicit assumptions of this exchange coupling model: (1) the validity of ``perfect'' permutational exchange of electrons between the neighboring molecules, and (2) the validity of the conservation of total spin angular momentum of the two neighboring molecules. With respect to the first assumption, it is found that the finiteness (vs perfection) of exchange overlap has not been sufficiently allowed for in the conventional (electrostatic) exchange mixing of allowed singlet and charge‐transfer states and in the conventional transitions only among the composite states of the same total spin angular momentum (e.g., $\mathrm{triplet}\leftarrow \mathrm{triplet}$ ). It is derived that there exists a direct transition moment (between the two molecules) weighted by overlap integrals, which may not be much smaller than the allowed transition moment weighted by (the indirect) exchange mixing coefficients, and which may exhibit interference effect with the latter. It is pointed out that if exchange energy is finite and spin interactions are sizable, transitions to other states of different spin multiplicities (e.g., $\mathrm{singlet}\leftarrow \mathrm{triplet}$ and $\mathrm{quintet}\leftarrow \mathrm{triplet}$ ), which are not split sufficiently far apart, may contribute to intensity through spin interactions. With respect to the second assumption, it is shown that the small but finite spin interactions, especially interelectronic spin interactions, which do not commute with the assertedly conserved total angular momentum S², have not been accounted for. A comprehensive formalism is developed to treat simultaneously exchange interaction along with spin interactions and to provide for various ranges of their relative magnitudes. These spin interactions are also shown to exhibit interference effects with the exchange interaction in the total intensity expression. All transitions, including $\text{Δ S ≠ 0}$ and ${\mathrm{\Delta \; M}}_S\text{≠ 0}$ ,, are considered. Matrix elements of the direct transition, of the exchange mixing of allowed singlet and charge‐transfer states, and of the spin‐orbit, spin‐other‐orbit, and spin‐spin interactions are computed for general molecular orbitals and for the case of three and four electrons. The commutation and noncommutation of spin operators with S² are traced to permutation symmetry and are treated by isomorphism to C_3v (for three electrons) and T_d (for four electrons) point groups. Irreducible representations of the linear as well as quadratic forms of spin and orbital operators and their interaction tensors are derived. Their relationship to spin eigenfunctions of given permutational symmetries is discussed.