What we can learn from the norms of one-particle density matrices, and what we can't: Some results for interstate properties in model singlet fission systems
The utility of the norms of one-particle density matrices, ||gamma||, for understanding the trends in electronic properties is discussed. Using several model systems that are relevant in the context of singlet fission (butadiene, octatetraene, ethylene dimer), the dependence of interstate properties (such as transition dipole moments and nonadiabatic couplings, NACs) on molecular geometries is investigated. ||gamma|| contains all the information about the changes in electronic states involved, such as varying degree of one-electron character of the transition; thus, it captures leading trends in one-electron interstate properties (i.e., when ||gamma|| is small, the respective interstate matrix elements are also small). However, finer variations in properties that arise due to the dependence of the matrix elements of the respective operators may not be reproduced. Analysis of NACs in ethylene dimer reveals that intermolecular components of NACs follow the trends in ||gamma|| well, as they are determined primarily by the characters of the two wave functions, however, intramolecular components depend on the relative orientation of the two moieties via the dependence in the derivative of the electron-nuclear Coulomb operator. Therefore, intramolecular NACs may exhibit large variations even when the changes in ||gamma|| are small. We observe large NACs at perfectly stacked geometry, however, larger values (by a factor of 1.6) are observed at slip-stacked (along the long axis) geometries. Larger values of NACs at slip-stacked configurations are due to the breaking of symmetry of the local environment of the heavy atoms and not due to the wave function composition. We found that the variations in ||gamma|| for ethylene dimer are due to a varying admixture of the charge-resonance configurations in the S1 state, whereas the 1ME state retains its pure multi-exciton character.