As methods for *ab initio* calculations of excited states are becoming
increasingly more routine, questions arise concerning how best to extract
chemical meaning from such calculations. There are several approaches for
analyzing molecular excited states; they are based on reduced one-particle
density matrices (OPDMs). The two objects exploited in this analysis are: (i)
the difference between the ground- and excited-state OPDMs and (ii) the
transition OPDM connecting the ground and excited state. In the case of CIS and
TDDFT/TDA wave functions, both quantities are identical and can be directly
mapped into the CIS amplitudes; however, for correlated wave functions the two
objects are not the same. The most basic analysis includes calculation of
attachment/detachment density^{350} and natural
transition orbitals.^{615} These quantities allow one to arrive to
a most compact description of an excited state. More detailed analysis allows
one to derive additional insight about the nature of the excited state.
Detailed description and illustrative examples can be found in
Refs. 729, 726.

This section describes the theoretical background behind attachment/detachment analysis and natural transition orbitals, while details of the input for creating data suitable for plotting these quantities is described separately in Chapter 11, which also describes additional excited-state analysis tools. For historical reasons, there are duplicate implementations of some features. For example, CIS and TDDFT wave functions can be analyzed using an original built-in code and by using a more recent module, libwfa.