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.
||Largest NTO occupation numbers|
||, sum of NTO occupation numbers|
||Energy of hole NTO,|
||Energy of particle NTO,|
||NTO participation ratio|
||Replace in the above two formulas|
||Expec. value of the particle-hole permutation operator,|
||Mean position of hole|
||Mean position of electron|
||Linear e/h distance|
||RMS hole size:|
||RMS elec. size:|
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:
where is diagonal matrix containing singular values and unitary matrices and contain the respective particle and hole NTOs. Note that:
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.
||Number of unpaired electrons|
||Number of unpaired electrons|
||NO participation ratio|
||Promotion number and|
||D/A participation ratio and|
||Mean position of detachment density|
||Mean position of attachment density|
||Linear D/A distance|
||RMS size of detachment density|
||RMS size of attachment density|
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
_nto.mo depending on the type of orbitals they
contain. In case of cube files the state identifier is followed by
_hole for state, difference, transition, attachment,
detachment, electron, or hole densities, respectively. All cube files have the
.cube. In unrestricted calculations an additional part is added
to the file name before
.cube which indicates (
_b) spin. The only exception is the state density for which
_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
Computation of ESPs for state, transition, and electron/hole densities (see Ref. 567) can be
triggered by setting ESP_GRID = .
These are indicated by
_esp as part of the file name.
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
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 and transition densities. Make sure you look at the correct one. The way to figure it out is to remember that in open-shell references , e.g., in doublet references, the unpaired electron is and the hole is . Thus, for transitions of the unpaired electron into the unoccupied orbitals you need block, whereas for the transitions from doubly occupied orbitals into the singly un-occupied orbital (the hole) you need the block.
In Hermitian formalisms, is a Hermitian conjugate
of , but in non-Hermitian approaches, such as
coupled-cluster theory, the two are slightly different. While for quantitative
interstate properties both and are
computed, the qualitative trends in exciton properties derived from
and are very similar. Only
one 1TDM is analyzed for EOM-CC.
In spin-restricted calculations, the libwfa module computes NTOs for
the 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.
$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
$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
Other examples of libwfa uses:
Example 7.3.7 illustrates wave-function analysis of the SF-DFT states in para-benzyne;
Example 126.96.36.199 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.