For CIS, TDHF, and TDDFT excited-state calculations, we have already mentioned that Mulliken population analysis of the excited-state electron densities may be requested by setting POP_MULLIKEN = , and multipole moments of the excited-state densities will be generated if CIS_MOMENTS = TRUE. Another useful decomposition for excited states is to separate the excitation into “particle” and “hole” components, which can then be analyzed separately.782 To do this, we define a density matrix for the excited electron,
and a density matrix for the hole that is left behind in the occupied space:
The quantities and are the transition density matrices, i.e., the components of the TDDFT eigenvector.238 The indices and denote MOs that occupied in the ground state, whereas and index virtual MOs. Note also that , the difference between the ground- and excited-state density matrices.
Upon transforming and into the AO basis, one can write
where is the total charge that is transferred from the occupied space to the virtual space. For a CIS calculation, or for TDDFT within the Tamm-Dancoff approximation,379 . For full TDDFT calculations, may be slightly different than .
Comparison of Eq. (11.12) to Eq. (11.3) suggests that the quantities and are amenable to population analyses of precisely the same sort used to analyze the ground-state density matrix. In particular, represents the th AO’s contribution to the excited electron, while is a contribution to the hole. The sum of these quantities,
represents the contribution to arising from the th AO. For the particle/hole density matrices, both Mulliken and Löwdin population analyses available, and are requested by setting CIS_MULLIKEN = TRUE.
Although the excited-state analysis features described in this section require very little computational effort, they are turned off by default, because they can generate a large amount of output, especially if a large number of excited states are requested. They can be turned on individually, or collectively by setting CIS_AMPL_ANAL = TRUE. This collective option also requests the calculation of natural transition orbitals (NTOs), which were introduced in Section 7.13.2. (NTOs can also be requested without excited-state population analysis. Some practical aspects of calculating and visualizing NTOs are discussed below, in Section 11.5.2.)