11.2.6 General Excited-State Analysis

Q-Chem features a new module for extended excited-state analysis, which is interfaced to the ADC, CC/EOM-CC, CIS, and TDDFT methods.^{734, 736, 733, 54, 735, 637} These analyses are based on the state, transition and difference density matrices of the excited states; the theoretical background for such analysis is given in Chapter 7.13.

The transition-density (1TDM) based analyses include the construction and export of natural transition orbitals⁶²⁰ (NTOs) and electron and hole densities,⁷³⁶ the evaluation of charge transfer numbers,⁷³⁴ an analysis of exciton multipole moments,^{54, 735, 637} and quantification of electron-hole entanglement.⁷³⁷ NTOs are obtained by singular value decomposition (SVD) of the 1TDM:

	${\gamma }_{pq}^{\text{IF}}$	$=⟨{\mathrm{\Psi }}_I|p^{†}q|{\mathrm{\Psi }}_F⟩$		(11.14)
	$𝜸$	$=𝜶𝝈{𝜷}^{†},$		(11.15)

where $𝝈$ is diagonal matrix containing singular values and unitary matrices $𝜶$ and $𝜷$ contain the respective particle and hole NTOs. Note that:

$${\parallel 𝜸\parallel }^2=\sum _{pq}{\gamma }_{pq}^2=\sum _K{\sigma }_K^2\equiv \mathrm{\Omega }$$

(11.16)

Furthermore, the formation and export of state-averaged NTOs, and the decomposition of the excited states into transitions of state-averaged NTOs are implemented.⁷³⁶ The difference and/or state densities can be exported themselves, as well as employed to construct and export natural orbitals, natural difference orbitals, and attachment and detachment densities.³⁵³ Furthermore, two measures of unpaired electrons are computed.³⁶¹ In addition, a Mulliken or Löwdin population analysis and an exciton analysis can be performed based on the difference/state densities. The main descriptors of the various analyses that are printed for each excited state are given in Tables 11.1 and 11.2. For a detailed description with illustrative examples, see Refs. 736 and 733.

To activate any excited-state analysis STATE_ANALYSIS has to be set to TRUE. For individual analyses there is currently only a limited amount of fine grained control. The construction and export of any type of orbitals is controlled by MOLDEN_FORMAT to export the orbitals as MolDen files, and NTO_PAIRS which specifies the number of important orbitals to print (note that the same keyword controls the number of natural orbitals, the number of natural difference orbitals, and the number of NTOs to be printed). Setting MAKE_CUBE_FILES to TRUE triggers the construction and export of densities in “cube file” format³⁷⁵ (see Section 11.5.4 for details). Activating transition densities in $plots will generate cube files for the transition density, the electron density, and the hole density of the respective excited states, while activating state densities or attachment/detachment densities will generate cube files for the state density, the difference density, the attachment density and the detachment density. Setting IQMOL_FCHK = TRUE (equivalently, GUI = 2) will export data to the “.fchk” formatted checkpoint file, and switches off the generation of cube files. The population analyses are controlled by POP_MULLIKEN and LOWDIN_POPULATION. Setting the latter to TRUE will enforce Löwdin population analysis to be employed for regular populations as well as CT numbers, while by default the Mulliken population analysis is used.

Any MolDen or cube files generated by the excited state analyses can be found in the directory plots in the job’s scratch directory. Their names always start with a unique identifier of the excited state (the exact form of this human readable identifier varies with the excited state method). The names of MolDen files are then followed by either _no.mo, _ndo.mo, or _nto.mo depending on the type of orbitals they contain. In case of cube files the state identifier is followed by _dens, _diff, _trans, _attach, _detach, _elec, or _hole for state, difference, transition, attachment, detachment, electron, or hole densities, respectively. All cube files have the suffice .cube. In unrestricted calculations an additional part is added to the file name before .cube which indicates $\alpha$ (_a) or $\beta$ (_b) spin. The only exception is the state density for which _tot or _sd are added indicating the total or spin-density parts of the state density. Analysis of relaxed CIS/TDDFT densities can be triggered by CIS_RELAXED_DENSITY=True. The corresponding output files are marked by _rlx. Computation of ESPs for state, transition, and electron/hole densities can be triggered by setting ESP_GRID=-3. These are indicated by _esp as part of the file name.

The _ctnum_atomic.om files created in the main directory serve as input for a charge transfer number analysis, as explained, e.g., in Refs. 734, 636. Use the external TheoDORE program (theodore-qc.sourceforge.net) to create electron/hole correlation plots and to compute fragment based descriptors.

When doing excited-state calculations from an open-shell reference, libwfa will perform the analysis for both $\alpha \alpha$ and $\beta \beta$ transition densities. Make sure you look at the correct one. The way to figure it out is to remember that in open-shell references $N_{\alpha }>N_{\beta }$, e.g., in doublet references, the unpaired electron is $\alpha$ and the hole is $\beta$. Thus, for transitions of the unpaired electron into the unoccupied orbitals you need $\alpha \alpha$ block, whereas for the transitions from doubly occupied orbitals into the singly un-occupied orbital (the hole) you need the $\beta \beta$ block.

Note:  In Hermitian formalisms, ${\gamma }^{\mathrm{\text{IF}}}$ is a Hermitian conjugate of ${\gamma }^{\mathrm{\text{FI}}}$, but in non-Hermitian approaches, such as coupled-cluster theory, the two are slightly different. While for quantitative interstate properties both ${\gamma }^{\mathrm{\text{IF}}}$ and ${\gamma }^{\mathrm{\text{FI}}}$ are computed, the qualitative trends in exciton properties derived from ${\mathrm{(}{\gamma }^{\mathrm{\text{IF}}}\mathrm{)}}^{\mathrm{†}}$ and ${\gamma }^{\mathrm{\text{FI}}}$ are very similar. Only one 1TDM is analyzed for EOM-CC.

Note:  In spin-restricted calculations, the libwfa module computes NTOs for the $\alpha \mathbf{}\alpha$ block of transition density. Thus, when computing NTOs for the transitions between open-shell EOM-IP/EA states make sure to specify correct spin states. For example, use EOM_EA_ALPHA to visualize transitions involving the extra electron.

$rem
basis = def2-sv(p)
n_frozen_core = fc
method = adc(2)
ee_singlets = [0,1,1,0]
state_analysis = true
$end

$molecule
0 1
6 0 0 0.523383
8 -0 0 -0.671856
1 0.931138 0 1.11728
1 -0.931138 0 1.11728
$end

$rem
method = pbe0
basis = def2-sv(p)
cis_n_roots = 4
cis_singlets = true
cis_triplets = true
rpa = false
state_analysis = true
molden_format = true
nto_pairs = 3
make_cube_files = true
esp_grid = -3
$end

$molecule
0 1
6 1.19438 1.10251 -0
6 -0.00836561 1.69243 -0
7 -1.1696 0.978035 -0
6 -1.21206 -0.402293 -0
7 0.0346914 -0.97914 0
6 1.28159 -0.348737 0
8 -2.24342 -1.02375 -0
8 2.29918 -0.995854 0
1 -0.12316 2.76714 -0
1 -2.06144 1.4441 -0
1 0.0448178 -1.98999 0
1 2.10472 1.67984 -0
$end

$plots
Write cube files for all 4 states
70 -3.5 3.5
70 -3.5 3.5
30 -1.5 1.5
0 4 0 0
1 2 3 4
$end