11.2.5 Basic Excited-State Analysis of CIS and TDDFT Wave Functions

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 = $-1$, 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,

 $\mathbf{D}_{ab}^{\mathrm{elec}}=\sum_{i}(\mathbf{X}+\mathbf{Y})^{\dagger}_{ai}% (\mathbf{X}+\mathbf{Y})_{ib}$ (11.10)

and a density matrix for the hole that is left behind in the occupied space:

 $\mathbf{D}_{ij}^{\mathrm{hole}}=\sum_{a}(\mathbf{X}+\mathbf{Y})_{ia}(\mathbf{X% }+\mathbf{Y})^{\dagger}_{aj}$ (11.11)

The quantities $\mathbf{X}$ and $\mathbf{Y}$ are the transition density matrices, i.e., the components of the TDDFT eigenvector.238 The indices $i$ and $j$ denote MOs that occupied in the ground state, whereas $a$ and $b$ index virtual MOs. Note also that $\mathbf{D}^{elec}+\mathbf{D}^{hole}=\Delta\mathbf{P}$, the difference between the ground- and excited-state density matrices.

Upon transforming $\mathbf{D}^{\mathrm{elec}}$ and $\mathbf{D}^{\mathrm{hole}}$ into the AO basis, one can write

 $\Delta q=\sum_{\mu}(\mathbf{D}^{elec}\,\mathbf{S})_{\mu\mu}=-\sum_{\mu}(% \mathbf{D}^{hole}\,\mathbf{S})_{\mu\mu}$ (11.12)

where $\Delta q$ 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 $\Delta q=-1$. For full TDDFT calculations, $\Delta q$ may be slightly different than $-1$.

Comparison of Eq. (11.12) to Eq. (11.3) suggests that the quantities $(\mathbf{D}^{\mathrm{elec}}\,\mathbf{S})$ and $(\mathbf{D}^{\mathrm{hole}}\,\mathbf{S})$ are amenable to population analyses of precisely the same sort used to analyze the ground-state density matrix. In particular, $(\mathbf{D}^{\mathrm{elec}}\,\mathbf{S})_{\mu\mu}$ represents the $\mu$th AO’s contribution to the excited electron, while $(\mathbf{D}^{\mathrm{hole}}\,\mathbf{S})_{\mu\mu}$ is a contribution to the hole. The sum of these quantities,

 $\Delta q_{\mu}=(\mathbf{D}^{\mathrm{elec}}\,\mathbf{S})_{\mu\mu}+(\mathbf{D}^{% \mathrm{hole}}\,\mathbf{S})_{\mu\mu}$ (11.13)

represents the contribution to $\Delta q$ arising from the $\mu$th AO. For the particle/hole density matrices, both Mulliken and Löwdin population analyses available, and are requested by setting CIS_MULLIKEN = TRUE.

CIS_MULLIKEN
Controls Mulliken and Löwdin population analyses for excited-state particle and hole density matrices.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
FALSE Do not perform particle/hole population analysis. TRUE Perform both Mulliken and Löwdin analysis of the particle and hole density matrices for each excited state.
RECOMMENDATION:
Set to TRUE if desired. This represents a trivial additional calculation.

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.)

CIS_AMPL_ANAL
Perform additional analysis of CIS and TDDFT excitation amplitudes, including generation of natural transition orbitals, excited-state multipole moments, and Mulliken analysis of the excited state densities and particle/hole density matrices.
TYPE:
LOGICAL
DEFAULT:
FALSE
OPTIONS:
$n$ Print if $|x_{ia}|$ or $|y_{ia}|$ is larger than $0.1\times n$.