Correction: Visualizing and characterizing excited states from time-dependent density functional theory

John M. Herbert
Department of Chemistry & Biochemistry, The Ohio State University, Columbus, Ohio 43210, USA. E-mail: herbert@chemistry.ohio-state.edu

First published on 23rd August 2024

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My recent review¹ on visualization and characterization methods in time-dependent density functional theory (TD-DFT) 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 λ²_i ≥ 0) whereas ΔP^hole is negative semidefinite (eigenvalues −λ²_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(ΔPelec) = −tr(ΔPhole)

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 TD-DFT is [eqn (2.4)]. Therefore, tr(ΔP^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 CHE-1665322, CHE-1955282 and CHE-2402361.

Notes and references

1. J. M. Herbert, Visualizing and characterizing excited states from time-dependent density functional theory, Phys. Chem. Chem. Phys., 2024, 26, 3755–3794 RSC .

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