11.2 Wave Function Analysis

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,

𝐃abelec=i(𝐗+𝐘)ai(𝐗+𝐘)ib (11.10)

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

𝐃ijhole=a(𝐗+𝐘)ia(𝐗+𝐘)aj (11.11)

The quantities 𝐗 and 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 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 (11.12)

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,379 Δq=-1. For full TDDFT calculations, Δq may be slightly different than -1.

Comparison of Eq. (11.12) to Eq. (11.3) 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 (11.13)

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 7.13.2. (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:
       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.