10.2 Wave Function Analysis 10.2.8 Basic Excited-State Analysis of CIS and TDDFT Wave Functions 10.3 Interface to the NBO Package

10.2.9 General Excited-State Analysis

(February 4, 2022)

Q-Chem features a new module for extended excited-state analysis, which is interfaced to the ADC, CC/EOM-CC, CIS, and TDDFT/SF-TDDFT methods.^{899, 901, 898, 59, 900, 778} 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 7.14.

Descriptor	Explanation
`Leading SVs`	Largest NTO occupation numbers
`Sum of SVs (Omega)`	$\Omega=\\|\bm{\gamma}^{\text{IF}}\\|^{2}$ , 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 $\mbox{PR}_{\text{NTO}}$
`Entanglement entropy (S_HE)`	$S_{H\|E}=-\sum_{i}\lambda_{i}\log_{2}\lambda_{i}$
`Nr of entangled states (Z_HE)`	$Z_{HE}=2^{S_{H\|E}}$
`Renormalized S_HE/Z_HE`	Replace $\lambda_{i}\rightarrow\lambda_{i}/\Omega$ in the above two formulas
`<Phe>`	Expec. value of the particle-hole permutation operator,
	measuring de-excitations⁵⁶⁷
`<r_h> [Ang]`	Mean position of hole $\langle\vec{x}_{h}\rangle_{\text{exc}}$
`<r_e> [Ang]`	Mean position of electron $\langle\vec{x}_{e}\rangle_{\text{exc}}$
`\|<r_e - r_h>\| [Ang]`	Linear e/h distance $\vec{d}_{h\rightarrow e}=\langle\vec{x}_{e}-\vec{x}_{h}\rangle_{\text{exc}}$
`Hole size [Ang]`	RMS hole size: $\sigma_{h}=(\langle\vec{x}_{h}^{\,2}\rangle_{\text{exc}}-\langle\vec{x}_{h}% \rangle_{\text{exc}}^{2})^{1/2}$
`Electron size [Ang]`	RMS elec. size: $\sigma_{e}=(\langle\vec{x}_{e}^{\,2}\rangle_{\text{exc}}-\langle\vec{x}_{e}% \rangle_{\text{exc}}^{2})^{1/2}$
`RMS electron-hole separation [Ang]`	$d_{\text{exc}}=(\langle\left\|\vec{x}_{e}-\vec{x}_{h}\right\|^{2}\rangle_{\text{% exc}})^{1/2}$
`Covariance(r_h, r_e) [Ang^2]`	$\mbox{COV}\left(\vec{x}_{h},\vec{x}_{e}\right)=\langle\vec{x}_{h}\bm{\cdot}% \vec{x}_{e}\rangle_{\text{exc}}-\langle\vec{x}_{h}\rangle_{\text{exc}}\bm{% \cdot}\langle\vec{x}_{e}\rangle_{\text{exc}}$
`Correlation coefficient`	$R_{eh}=\mbox{COV}\left(\vec{x}_{h},\vec{x}_{e}\right)/\sigma_{h}\bm{\cdot}% \sigma_{e}$
`Center-of-mass size`	$0.5*(\langle\left\|\vec{x}_{e}+\vec{x}_{h}\right\|^{2}\rangle_{\text{exc}}-% \langle\vec{x}_{e}+\vec{x}_{h}\rangle_{\text{exc}}^{2})^{1/2}$

Table 10.1: Descriptors output by Q-Chem for transition density matrix analysis. Note that squares of the SVs, which correspond to the weights of the respective NTO pairs, are printed.

\Omega

equals the square of the norm of the one-electron transition density matrix (1TDM).

One-electron transition-density matrix (1TDM) based analyses include the construction and export of natural transition orbitals⁷⁵⁵ (NTOs) and electron and hole densities,⁹⁰¹ the evaluation of charge transfer numbers,⁸⁹⁹ an analysis of exciton multipole moments,^{59, 900, 778} and quantification of electron-hole entanglement.⁹⁰² NTOs are obtained by singular value decomposition (SVD) of the 1TDM:

	$\displaystyle\gamma^{\text{IF}}_{pq}$	$\displaystyle=\langle\Psi_{I}\|p^{\dagger}q\|\Psi_{F}\rangle$		(10.18)
	$\displaystyle\bm{\gamma}$	$\displaystyle=\bm{\alpha\sigma\beta}^{\dagger}\;,$		(10.19)

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:

\|\bm{\gamma}\|^{2}=\sum_{pq}\gamma_{pq}^{2}=\sum_{K}\sigma_{K}^{2}\equiv\Omega

(10.20)

Furthermore, the formation and export of state-averaged NTOs, and the decomposition of the excited states into transitions of state-averaged NTOs are implemented.⁹⁰¹ 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.⁴³⁴ Furthermore, two measures of unpaired electrons are computed.⁴⁴² 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 10.1 and 10.2. For a detailed description with illustrative examples, see Refs. 901 and 898.

Descriptor	Explanation
`n_u`	Number of unpaired electrons $n_{u}=\sum_{i}\mbox{min}(n_{i},2-n_{i})$
`n_u,nl`	Number of unpaired electrons $n_{u,nl}=\sum_{i}n_{i}^{2}(2-n_{i})^{2}$
`PR_NO`	NO participation ratio $\mbox{PR}_{\text{NO}}$
`p_D` and `p_A`	Promotion number $p_{D}$ and $p_{A}$
`PR_D` and `PR_A`	D/A participation ratio $\mbox{PR}_{D}$ and $\mbox{PR}_{A}$
`<r_h> [Ang]`	Mean position of detachment density $\vec{d}_{D}$
`<r_e> [Ang]`	Mean position of attachment density $\vec{d}_{A}$
`\|<r_e - r_h>\| [Ang]`	Linear D/A distance $\vec{d}_{D\rightarrow A}=\vec{d}_{A}-\vec{d}_{D}$
`Hole size [Ang]`	RMS size of detachment density $\sigma_{D}$
`Electron size [Ang]`	RMS size of attachment density $\sigma_{A}$

Table 10.2: Descriptors output by Q-Chem for difference/state density matrix analysis.

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” format⁴⁶⁰ (see Section 10.5.5 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 or 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 (see Ref. 567) can be triggered by setting ESP_GRID = $-3$ . These are indicated by _esp as part of the file name.

The ctnum_*.om file created in the main directory serves as input for a charge transfer number analysis, as explained, e.g., in Refs. 899, 777. 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, $\gamma^{\text{IF}}$ is a Hermitian conjugate of $\gamma^{\text{FI}}$ , but in non-Hermitian approaches, such as coupled-cluster theory, the two are slightly different. While for quantitative interstate properties both $\gamma^{\text{IF}}$ and $\gamma^{\text{FI}}$ are computed, the qualitative trends in exciton properties derived from $(\gamma^{\text{IF}})^{\dagger}$ and $\gamma^{\text{FI}}$ are very similar. Only one 1TDM is analyzed for EOM-CC.

Note: In spin-restricted calculations, the libwfa module computes NTOs for the $\alpha\alpha$ 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.

STATE_ANALYSIS

STATE_ANALYSIS
       Triggers the general state analysis via libwfa.
TYPE:
       LOGICAL
DEFAULT:
       FALSE
OPTIONS:
       FALSE Do not run excited state analysis. TRUE Activate excited state analysis.
RECOMMENDATION:
       This analysis produces only minimal computational overhead (as long as no cube files are produced) and can be activated whenever some additional information about the excited state is required.

WFA_LEVEL

WFA_LEVEL
       Master variable for controlling the amount of output produced by libwfa.
TYPE:
       INTEGER
DEFAULT:
       3
OPTIONS:
       1 Only perform some population analyses. 2 Also perform exciton analysis and compute natural (transition/difference) orbitals. 3 Also perform charge transfer number analysis. 4 Maximal output (this is needed to reproduce Ref. 567)
RECOMMENDATION:
       Reduce if you want less print-out.

NTO_PAIRS

NTO_PAIRS
       Controls how many hole/particle NTO pairs and frontier natural orbital pairs and natural difference orbital pairs are printed in the standard output.
TYPE:
       INTEGER
DEFAULT:
       0
OPTIONS:
       $N$ Write $N$ NTO/NO/NDO pairs per excited state.
RECOMMENDATION:
       This controls the print-out to the standard output. Use WFA_ORB_THRESH if you want to modify the number of orbitals exported.

WFA_ORB_THRESH

WFA_ORB_THRESH
       Controls the number of hole/particle NTO pairs and frontier natural orbital pairs and natural difference orbital pairs exported to the Molden files.
TYPE:
       INTEGER
DEFAULT:
       3
OPTIONS:
       $N$ Export all NTO/NO/NDO pairs with a weight above $10^{-N}$ .
RECOMMENDATION:

WFA_REF_STATE

WFA_REF_STATE
       Controls the reference state for the transition and difference density matrices used by libwfa. This keyword works for CIS/TDDFT/SF-DTDDFT computations. Use CC_STATE_TO_OPT for EOM-CC.
TYPE:
       INTEGER
DEFAULT:
       -1
OPTIONS:
       -1 Use default: ground-state for standard CIS/TDDFT computations, first response state for SF-TDDFT. 0 Reference state N $N$ th excited state/response state.
RECOMMENDATION:
       NONE

Example 10.4 Basic excited-state analysis example for formaldehyde at the ADC(2)/def2-SV(P) level.

$molecule
   0 1
   C     0.000000   0.000000    0.523383
   O    -0.000000   0.000000   -0.671856
   H     0.931138   0.000000    1.11728
   H    -0.931138   0.000000    1.11728
$end

$rem
   METHOD           adc(2)
   BASIS            def2-sv(p)
   EE_SINGLETS      [0,1,1,0]
   STATE_ANALYSIS   true
   N_FROZEN_CORE    fc
$end

View output

Example 10.5 Uracil computed at the PBE0/def2-SV(P) level. Activation of the full libwfa functionality: export of NOs, NTOs and NDOs in MolDen format, densities in cube format, and computation of the ESPs of these densities.

$molecule
   0 1
   C    1.194380    1.102510    0.000000
   C   -0.008366    1.692430    0.000000
   N   -1.169600    0.978035    0.000000
   C   -1.212060   -0.402293    0.000000
   N    0.034691   -0.979140    0.000000
   C    1.281590   -0.348737    0.000000
   O   -2.243420   -1.023750    0.000000
   O    2.299180   -0.995854    0.000000
   H   -0.123160    2.767140    0.000000
   H   -2.061440    1.444100    0.000000
   H    0.044818   -1.989990    0.000000
   H    2.104720    1.679840    0.000000
$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

$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

View output

Other examples of libwfa uses:

•

Example 7.3.7 illustrates wave-function analysis of the SF-DFT states in para-benzyne;
•

Example 7.10.8.2 illustrates wave-function analysis of XAS transitions within CVS-EOM-EE;
•

Example 7.10.29 illustrates wave-function analysis for transitions between EOM-IP states;
•

Example 7.10.19 illustrates wave-function analysis of complex-valued densities within CAP-EOM-CCSD.

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10.2.9 General Excited-State Analysis