10.2.10 Excited-State Analysis for CIS and TDDFT‣ 10.2 Wave Function Analysis ‣ Chapter 10 Molecular Properties and Analysis ‣ Q-Chem 6.2 User’s Manual

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. ¹⁰⁹³ Richard R. M., Herbert J. M.
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})^{\dagger}_{ai}% (\mathbf{X}+\mathbf{Y})_{ib}

(10.27)

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}

(10.28)

The quantities $\mathbf{X}$ and $\mathbf{Y}$ are the transition density matrices, i.e., the components of the TDDFT eigenvector. ³²⁹ Dreuw A., Head-Gordon M.
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 $\mathbf{D}^{elec}+\mathbf{D}^{hole}=\Delta\mathbf{P}$ , 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}^{\mathrm{elec}}\,\mathbf{S})_{\mu\mu}=-\sum_{% \mu}(\mathbf{D}^{\mathrm{hole}}\,\mathbf{S})_{\mu\mu}

(10.29)

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, ⁵⁴³ Hirata S., Nooijen M., Bartlett R. J.
Chem. Phys. Lett.
(2000), 326, pp. 255. Link $\Delta q=-1$ . For full TDDFT calculations, $\Delta q$ may be slightly different than $-1$ .

Comparison of Eq. (10.29) 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,

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

(10.30)

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

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), ⁵³² Herbert J. M.
Phys. Chem. Chem. Phys.
(2024), 26, pp. 3755. Link which were introduced in Section 7.14.3. (NTOs can also be requested without excited-state population analysis. Some practical aspects of calculating and visualizing NTOs are discussed below, in Section 10.5.3.)

CIS_AMPL_ANAL

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

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 $0.01\times n$ .
RECOMMENDATION:
       Use the default unless you want to see more amplitudes.

EXCIT_ENERGY_COMPONENTS

EXCIT_ENERGY_COMPONENTS
       Compute individual compnents of the CIS/TDDFT excitation energy. ⁹⁸⁷ Pei Z. et al.
J. Phys. Chem. Lett.
(2021), 12, pp. 2712. Link The output is divided into the one-electron components (H); Fock-matrix type components representing the Coulomb (J1), non-local exchange (K1), and xc potentials (XC1); and true two-electron components (J2, K2, XC2). Note that H+J1+K1+XC1 is equivalent to a weighted sum of MO energy differences whereas J2, K2, and XC2 represent the post-MO terms. For more information see Ref. 650 Kimber P., Plasser F.
J. Chem. Theory Comput.
(2023), 19, pp. 2340. Link .
TYPE:
       LOGICAL
DEFAULT:
       FALSE
OPTIONS:
       TRUE Compute excitation energy components. FALSE Do not compute excitation energy components.
RECOMMENDATION:
       Use if more detailed insight into excitation energies is required.

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10.2.10 Excited-State Analysis for CIS and TDDFT