DOI:
10.1039/D4CP90138J
(Correction)
Phys. Chem. Chem. Phys., 2024,
26, 2350323504
Correction: Visualizing and characterizing excited states from timedependent density functional theory
Received
30th July 2024
, Accepted 30th July 2024
First published on 23rd August 2024
Abstract
Correction for ‘Visualizing and characterizing excited states from timedependent density functional theory’ by John M. Herbert et al., Phys. Chem. Chem. Phys., 2024, 26, 3755–3794, https://doi.org/10.1039/D3CP04226J.
My recent review^{1} on visualization and characterization methods in timedependent density functional theory (TDDFT) contains a sign error in the definition of the natural transition orbitals. This affects eqn (3.1) and (3.2) and the accompanying text in the same paragraph. The corrected equations are 
 (3.1) 
and 
 (3.2) 
These eigenvalues differ in sign from what was (erroneously) published in ref. 1. The text accompanying these equations should suggest that ΔP^{elec} is positive semidefinite (eigenvalues λ^{2}_{i} ≥ 0) whereas ΔP^{hole} is negative semidefinite (eigenvalues −λ^{2}_{i}). This is consistent with the signs in eqn (2.12) that defines ΔP^{elec} and ΔP^{hole}, and with remarks made concerning the attachment and detachment densities in eqn (2.17).
In addition, eqn (2.13) for the normalization of the electron and hole densities holds only when the Tamm–Dancoff approximation (TDA) is invoked. A more general statement is that
tr(ΔP^{elec}) = −tr(ΔP^{hole}) 
always, consistent with the eigenvalues given above, whereas
holds only within the TDA. Note that
according to eqn (2.12), whereas the normalization condition for linear response TDDFT is
[eqn (2.4)]. Therefore, tr(Δ
P^{elec}) can exceed unity for full linear response TDDFT. Typically 
y ∼ 10
^{−3} for small molecules, so the deviation from the TDA result is not large.
The funding information in the Acknowledgments section of the original article was incomplete, the full funding information is shown here.
The Royal Society of Chemistry apologises for these errors and any consequent inconvenience to authors and readers.
Acknowledgements
This work was supported by National Science Foundation grants CHE1665322, CHE1955282 and CHE2402361.
Notes and references
 J. M. Herbert, Visualizing and characterizing excited states from timedependent density functional theory, Phys. Chem. Chem. Phys., 2024, 26, 3755–3794 RSC .

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