QChem features a new module for extended excitedstate analysis, which is interfaced to the ADC, CC/EOMCC, CIS, and TDDFT methods.^{727, 729, 726, 53, 728, 632} 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.
Descriptor  Explanation 

Leading SVs$^2$ 
Largest NTO occupation numbers 
Sum of SVs$^2$ (Omega) 
$\mathrm{\Omega}={\parallel {\bm{\gamma}}^{\text{IF}}\parallel}^{2}$, sum of NTO occupation numbers 
E(h) 
Energy of hole NTO, ${E}_{I}(h)={\sum}_{pq}{\alpha}_{pI}{F}_{pq}{\alpha}_{qI}$ 
E(p) 
Energy of particle NTO, ${E}_{I}(p)={\sum}_{pq}{\beta}_{pI}{F}_{pq}{\beta}_{qI}$ 
PR_NTO 
NTO participation ratio ${\text{PR}}_{\text{NTO}}$ 
Entanglement entropy (S_HE) 
${S}_{HE}={\sum}_{i}{\lambda}_{i}{\mathrm{log}}_{2}{\lambda}_{i}$ 
Nr of entangled states (Z_HE) 
${Z}_{HE}={2}^{{S}_{HE}}$ 
Renormalized S_HE/Z_HE 
Replace ${\lambda}_{i}\to {\lambda}_{i}/\mathrm{\Omega}$ 
<r_h> [Ang] 
Mean position of hole ${\u27e8{\overrightarrow{x}}_{h}\u27e9}_{\text{exc}}$ 
<r_e> [Ang] 
Mean position of electron ${\u27e8{\overrightarrow{x}}_{e}\u27e9}_{\text{exc}}$ 
<r_e  r_h> [Ang] 
Linear e/h distance ${\overrightarrow{d}}_{h\to e}={\u27e8{\overrightarrow{x}}_{e}{\overrightarrow{x}}_{h}\u27e9}_{\text{exc}}$ 
Hole size [Ang] 
RMS hole size: ${\sigma}_{h}={({\u27e8{\overrightarrow{x}}_{h}^{\mathrm{\hspace{0.17em}2}}\u27e9}_{\text{exc}}{\u27e8{\overrightarrow{x}}_{h}\u27e9}_{\text{exc}}^{2})}^{1/2}$ 
Electron size [Ang] 
RMS elec. size: ${\sigma}_{e}={({\u27e8{\overrightarrow{x}}_{e}^{\mathrm{\hspace{0.17em}2}}\u27e9}_{\text{exc}}{\u27e8{\overrightarrow{x}}_{e}\u27e9}_{\text{exc}}^{2})}^{1/2}$ 
RMS electronhole separation [Ang] 
${d}_{\text{exc}}={({\u27e8{\left{\overrightarrow{x}}_{e}{\overrightarrow{x}}_{h}\right}^{2}\u27e9}_{\text{exc}})}^{1/2}$ 
Covariance(r_h, r_e) [Ang^2] 
$\text{COV}({\overrightarrow{x}}_{h},{\overrightarrow{x}}_{e})={\u27e8{\overrightarrow{x}}_{h}\mathbf{\cdot}{\overrightarrow{x}}_{e}\u27e9}_{\text{exc}}{\u27e8{\overrightarrow{x}}_{h}\u27e9}_{\text{exc}}\mathbf{\cdot}{\u27e8{\overrightarrow{x}}_{e}\u27e9}_{\text{exc}}$ 
Correlation coefficient 
${R}_{eh}=\text{COV}({\overrightarrow{x}}_{h},{\overrightarrow{x}}_{e})/{\sigma}_{h}\mathbf{\cdot}{\sigma}_{e}$ 
The transitiondensity (1TDM) based analyses include the construction and export of natural transition orbitals^{615} (NTOs) and electron and hole densities,^{729} the evaluation of charge transfer numbers,^{727} an analysis of exciton multipole moments,^{53, 728, 632} and quantification of electronhole entanglement.^{730} NTOs are obtained by singular value decomposition (SVD) of the 1TDM:
${\gamma}_{pq}^{\text{IF}}$  $=\u27e8{\mathrm{\Psi}}_{I}{p}^{\u2020}q{\mathrm{\Psi}}_{F}\u27e9$  (11.14)  
$\bm{\gamma}$  $=\bm{\alpha}\bm{\sigma}{\bm{\beta}}^{\u2020},$  (11.15) 
where $\bm{\sigma}$ is diagonal matrix containing singular values and unitary matrices $\bm{\alpha}$ and $\bm{\beta}$ contain the respective particle and hole NTOs. Note that:
$${\parallel \bm{\gamma}\parallel}^{2}=\sum _{pq}{\gamma}_{pq}^{2}=\sum _{K}{\sigma}_{K}^{2}\equiv \mathrm{\Omega}$$  (11.16) 
Furthermore, the formation and export of stateaveraged NTOs, and the decomposition of the excited states into transitions of stateaveraged NTOs are implemented.^{729} 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.^{350} Furthermore, two measures of unpaired electrons are computed.^{358} 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. 729 and 726.
Descriptor  Explanation 

n_u 
Number of unpaired electrons ${n}_{u}={\sum}_{i}\text{min}({n}_{i},2{n}_{i})$ 
n_u,nl 
Number of unpaired electrons ${n}_{u,nl}={\sum}_{i}{n}_{i}^{2}{(2{n}_{i})}^{2}$ 
PR_NO 
NO participation ratio ${\text{PR}}_{\text{NO}}$ 
p_D and p_A

Promotion number ${p}_{D}$ and ${p}_{A}$ 
PR_D and PR_A

D/A participation ratio ${\text{PR}}_{D}$ and ${\text{PR}}_{A}$ 
<r_h> [Ang] 
Mean position of detachment density ${\overrightarrow{d}}_{D}$ 
<r_e> [Ang] 
Mean position of attachment density ${\overrightarrow{d}}_{A}$ 
<r_e  r_h> [Ang] 
Linear D/A distance ${\overrightarrow{d}}_{D\to A}={\overrightarrow{d}}_{A}{\overrightarrow{d}}_{D}$ 
Hole size [Ang] 
RMS size of detachment density ${\sigma}_{D}$ 
Electron size [Ang] 
RMS size of attachment density ${\sigma}_{A}$ 
To activate any excitedstate 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”
format^{372} (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, 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 spindensity parts
of the state density.
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. 727, 631. These files are processed by the
external TheoDORE
program () to
create electron/hole correlation plots and to compute fragment based
descriptors.
When doing excitedstate calculations from an openshell 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 openshell 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 unoccupied 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 nonHermitian approaches, such as
coupledcluster 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
${\mathrm{(}{\gamma}^{\mathrm{\text{IF}}}\mathrm{)}}^{\mathrm{\u2020}}$ and ${\gamma}^{\mathrm{\text{FI}}}$ are very similar. Only
one 1TDM is analyzed for EOMCC.
Note:
In spinrestricted calculations, the libwfa module computes NTOs for
the $\alpha \mathbf{}\alpha $ block of transition density. Thus, when computing NTOs for
the transitions between openshell EOMIP/EA states make sure to specify
correct spin states. For example, use EOM_EA_ALPHA to visualize
transitions involving the extra electron.
NTO_PAIRS
Controls how many hole/particle NTO pairs and frontier natural orbital pairs and
natural difference orbital pairs are computed for excited states.
TYPE:
INTEGER
DEFAULT:
0
OPTIONS:
$N$
Write $N$ NTO/NO/NDO pairs per excited state.
RECOMMENDATION:
If activated ($N>0$), a minimum of two NTO pairs will be printed for each state.
Increase the value of $N$ if additional NTOs are desired. By default, one pair of frontier
natural orbitals is computed for $N=0$.