11.2 Wave Function Analysis

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.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) Ω=𝜸IF2, sum of NTO occupation numbers
E(h) Energy of hole NTO, EI(h)=pqαpIFpqαqI
E(p) Energy of particle NTO, EI(p)=pqβpIFpqβqI
PR_NTO NTO participation ratio PRNTO
Entanglement entropy (S_HE) SH|E=-iλilog2λi
Nr of entangled states (Z_HE) ZHE=2SH|E
Renormalized S_HE/Z_HE Replace λiλi/Ω
<r_h> [Ang] Mean position of hole xhexc
<r_e> [Ang] Mean position of electron xeexc
|<r_e - r_h>| [Ang] Linear e/h distance dhe=xe-xhexc
Hole size [Ang] RMS hole size: σh=(xh 2exc-xhexc2)1/2
Electron size [Ang] RMS elec. size: σe=(xe 2exc-xeexc2)1/2
RMS electron-hole separation [Ang] dexc=(|xe-xh|2exc)1/2
Covariance(r_h, r_e) [Ang^2] COV(xh,xe)=xhxeexc-xhexcxeexc
Correlation coefficient Reh=COV(xh,xe)/σhσe
Table 11.1: 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. Ω equals the square of the norm of the 1TDM.

The transition-density (1TDM) based analyses include the construction and export of natural transition orbitals615 (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 electron-hole entanglement.730 NTOs are obtained by singular value decomposition (SVD) of the 1TDM:

γpqIF =ΨI|pq|ΨF (11.14)
𝜸 =𝜶𝝈𝜷, (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:

𝜸2=pqγpq2=KσK2Ω (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.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 nu=imin(ni,2-ni)
n_u,nl Number of unpaired electrons nu,nl=ini2(2-ni)2
PR_NO NO participation ratio PRNO
p_D and p_A Promotion number pD and pA
PR_D and PR_A D/A participation ratio PRD and PRA
<r_h> [Ang] Mean position of detachment density dD
<r_e> [Ang] Mean position of attachment density dA
|<r_e - r_h>| [Ang] Linear D/A distance dDA=dA-dD
Hole size [Ang] RMS size of detachment density σD
Electron size [Ang] RMS size of attachment density σA
Table 11.2: 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” format372 (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 α (_a) or β (_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.

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

Note:  In Hermitian formalisms, γIF is a Hermitian conjugate of γFI, but in non-Hermitian approaches, such as coupled-cluster theory, the two are slightly different. While for quantitative interstate properties both γIF and γFI are computed, the qualitative trends in exciton properties derived from (γIF) and γFI are very similar. Only one 1TDM is analyzed for EOM-CC.

Note:  In spin-restricted calculations, the libwfa module computes NTOs for the αα 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.

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.