Quantitative El-Sayed Rules for Many-Body Wave Functions from Spinless Transition Density Matrices

One-particle transition density matrices and natural transition orbitals enable quantitative description of electronic transitions and interstate properties involving correlated many-body wave functions within the molecular orbital framework. Here we extend the formalism to the analysis of tensor properties, such as spin-orbit couplings (SOCs), which involve states of different spin projection. By using spinless density matrices and Wigner-Eckart's theorem, the approach allows one to treat the transitions between states with arbitrary spin projections in a uniform way. In addition to a pictorial representation of the transition, the analysis also yields quantitative contributions of hole-particle pairs into the overall many-body matrix elements. In particular, it helps to rationalize the magnitude of computed SOCs in terms of El-Sayed's rules. The capabilities of the new tool are illustrated by the analysis of the equation-of-motion coupled-cluster calculations of two transition metal complexes.