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(June 30, 2021)

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.
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J. Chem. Theory Comput.

(2011),
7,
pp. 1296.
Link
To do this, we define a density
matrix for the excited electron,

$${\mathbf{D}}_{ab}^{\mathrm{elec}}=\sum _{i}{(\mathbf{X}+\mathbf{Y})}_{ai}^{\u2020}{(\mathbf{X}+\mathbf{Y})}_{ib}$$ | (10.9) |

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

$${\mathbf{D}}_{ij}^{\mathrm{hole}}=\sum _{a}{(\mathbf{X}+\mathbf{Y})}_{ia}{(\mathbf{X}+\mathbf{Y})}_{aj}^{\u2020}$$ | (10.10) |

The quantities $\mathbf{X}$ and $\mathbf{Y}$ are the transition density
matrices, *i.e.*, the components of the TDDFT eigenvector.
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Chem. Rev.

(2005),
105,
pp. 4009.
Link
The indices $i$ and $j$ denote MOs that occupied in the ground state, whereas
$a$ and $b$ index virtual MOs. Note also that ${\mathbf{D}}^{elec}+{\mathbf{D}}^{hole}=\mathrm{\Delta}\mathbf{P}$, 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

$$\mathrm{\Delta}q=\sum _{\mu}{({\mathbf{D}}^{\mathrm{elec}}\mathbf{S})}_{\mu \mu}=-\sum _{\mu}{({\mathbf{D}}^{\mathrm{hole}}\mathbf{S})}_{\mu \mu}$$ | (10.11) |

where $\mathrm{\Delta}q$ 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,
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Chem. Phys. Lett.

(2000),
326,
pp. 255.
Link
$\mathrm{\Delta}q=-1$. For full TDDFT
calculations, $\mathrm{\Delta}q$ may be slightly different than $-1$.

Comparison of Eq. (10.11) to Eq. (10.3) 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,

$$\mathrm{\Delta}{q}_{\mu}={({\mathbf{D}}^{\mathrm{elec}}\mathbf{S})}_{\mu \mu}+{({\mathbf{D}}^{\mathrm{hole}}\mathbf{S})}_{\mu \mu}$$ | (10.12) |

represents the contribution to $\mathrm{\Delta}q$ 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 7.14.2. (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.)

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:

TRUE
Perform additional amplitude analysis.
FALSE
Do not perform additional analysis.

RECOMMENDATION:

None

CIS_AMPL_PRINT

Sets the threshold for printing CIS and TDDFT excitation amplitudes.

TYPE:

INTEGER

DEFAULT:

15

OPTIONS:

$n$
Print if $|{x}_{ia}|$ or $|{y}_{ia}|$ is larger than $0.1\times n$.

RECOMMENDATION:

Use the default unless you want to see more amplitudes.