We introduce the particle-hole map (PHM), a visualization tool to analyze electronic excitations in molecules in the time or frequency domain, to be used in conjunction with time-dependent density-functional theory (TDDFT) or other ab initio methods. The purpose of the PHM is to give detailed insight into electronic excitation processes which is not obtainable from local visualization methods such as transition densities, density differences, or natural transition orbitals. The PHM is defined as a nonlocal function of two spatial variables and provides information about the origins, destinations, and connections of charge fluctuations during an excitation process; it is particularly valuable to analyze charge-transfer excitonic processes. In contrast with the transition density matrix, the PHM has a statistical interpretation involving joint probabilities of individual states and their transitions, it satisfies several sum rules and exact conditions, and it is easier to read and interpret. We discuss and illustrate the properties of the PHM and give several examples and applications to excitations in one-dimensional model systems, in a hydrogen chain, and in a benzothiadiazole based molecule.
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