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.Richard:2011 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.Dreuw:2005 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,Hirata:2000 . For full TDDFT calculations, may be slightly different than .
Comparison of Eq. (10.12) to Eq. (10.3) 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 7.14.2. (NTOs can also be requested without excited-state population analysis. Some practical aspects of calculating and visualizing NTOs are discussed below, in Section 10.5.2.)