The particle-hole map: formal derivation and numerical implementation

The particle-hole map (PHM) is a tool to visualize electronic excitations, based on representations in a canonical orbital transition space. Introduced as an alternative to the transition density matrix, the PHM has a simple probabilistic interpretation, indicating the origins and destinations of electrons and holes and, hence, the roles of different functional units of molecules during an excitation. We present a formal derivation of the PHM, starting from the particle-hole transition density matrix and projecting onto a set of single-particle orbitals. We implement the PHM using atom-centered localized basis sets and discuss the example of the molecular charge-transfer complex $rm C_2H_4 - C_2F_4$.