10.2 Wave Function Analysis

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

(May 16, 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. 935 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,

𝐃abelec=i(𝐗+𝐘)ai(𝐗+𝐘)ib (10.9)

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

𝐃ijhole=a(𝐗+𝐘)ia(𝐗+𝐘)aj (10.10)

The quantities 𝐗 and 𝐘 are the transition density matrices, i.e., the components of the TDDFT eigenvector. 285 Dreuw A., Head-Gordon M.
Chem. Rev.
(2005), 105, pp. 4009.
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The indices i and j denote MOs that occupied in the ground state, whereas a and b index virtual MOs. Note also that 𝐃elec+𝐃hole=Δ𝐏, the difference between the ground- and excited-state density matrices.

Upon transforming 𝐃elec and 𝐃hole into the AO basis, one can write

Δq=μ(𝐃elec𝐒)μμ=-μ(𝐃hole𝐒)μμ (10.11)

where Δq 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, 456 Hirata S., Nooijen M., Bartlett R. J.
Chem. Phys. Lett.
(2000), 326, pp. 255.
Link
Δq=-1. For full TDDFT calculations, Δq may be slightly different than -1.

Comparison of Eq. (10.11) to Eq. (10.3) suggests that the quantities (𝐃elec𝐒) and (𝐃hole𝐒) are amenable to population analyses of precisely the same sort used to analyze the ground-state density matrix. In particular, (𝐃elec𝐒)μμ represents the μth AO’s contribution to the excited electron, while (𝐃hole𝐒)μμ is a contribution to the hole. The sum of these quantities,

Δqμ=(𝐃elec𝐒)μμ+(𝐃hole𝐒)μμ (10.12)

represents the contribution to Δq arising from the μth 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 |xia| or |yia| is larger than 0.1×n.
RECOMMENDATION:
       Use the default unless you want to see more amplitudes.