10.2.5 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.⁸¹⁹ To do this, we define a density matrix for the excited electron,

$${𝐃}_{ab}^{\mathrm{elec}}=\sum _i{(𝐗+𝐘)}_{ai}^{†}{(𝐗+𝐘)}_{ib}$$

(10.10)

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

$${𝐃}_{ij}^{\mathrm{hole}}=\sum _a{(𝐗+𝐘)}_{ia}{(𝐗+𝐘)}_{aj}^{†}$$

(10.11)

The quantities $𝐗$ and $𝐘$ are the transition density matrices, i.e., the components of the TDDFT eigenvector.²³³ The indices $i$ and $j$ denote MOs that occupied in the ground state, whereas $a$ and $b$ index virtual MOs. Note also that ${𝐃}^{elec}+{𝐃}^{hole}=\mathrm{\Delta }𝐏$, the difference between the ground- and excited-state density matrices.

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

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

(10.12)

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,³⁸⁶ $\mathrm{\Delta }q=-1$. For full TDDFT calculations, $\mathrm{\Delta }q$ may be slightly different than $-1$.

Comparison of Eq. (10.12) to Eq. (10.3) suggests that the quantities $({𝐃}^{\mathrm{elec}}𝐒)$ and $({𝐃}^{\mathrm{hole}}𝐒)$ are amenable to population analyses of precisely the same sort used to analyze the ground-state density matrix. In particular, ${({𝐃}^{\mathrm{elec}}𝐒)}_{\mu \mu }$ represents the $\mu$th AO’s contribution to the excited electron, while ${({𝐃}^{\mathrm{hole}}𝐒)}_{\mu \mu }$ is a contribution to the hole. The sum of these quantities,

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

(10.13)

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.

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 7.15.2. (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.)