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Erratum: “Harnessing the meta-generalized gradient approximation for time-dependent density functional theory” [J. Chem. Phys. 137, 164105 (2012)]

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We recently determined that Eq. (55), one of the equations reporting the implementation of current-density corrected metageneralized gradient approximation (cMGGA) density functionals in Ref. 1, is missing a multiplicative factor of 1/2. The correct equation reads(Equations Presented).While the other equations contained in the publication are unaffected by this error, the cMGGA excitation energies and optical rotations (ORs) should be replaced by the values reported below and provided in the updated supplementary material of Ref. 1. While some of the changes are significant, the conclusions of the paper still hold after the correction of these values. The original implementation in TURBOMOLE2 was corrected accordingly and has been released in version V7.6. One of us (J.L.) independently confirmed the implementation of Eq. (55). This implementation is part of Q-Chem version 5.4.3 a. AEX benchmark. Comparing the previous and present current-dependent implementations for cTPSS and cTPSSh over the entire AEX benchmark, the correction resulted in a shift to larger excitation energies by 0.03 eV on average, although the largest change was ~0.2 eV for the 3? state of NH. The correction tends to reduce the effect of the current-dependence and hence leads to slightly larger excitation energies than originally reported. For pyridone–lactam and OMpCA, the values reported in 2012 were too large due to a convergence issue, and the corrected values of the excitation energies display an even smaller effect of including the current-density response than the originally reported ones, in line with the other singlet excitations included in the AEX test set. The statistical error analysis for TPSS, TPSSh, cTPSS, and cTPSSh compared to experiment shows a slight change due to the corrections; see Table II (corrected) and Table III (corrected). Individual results for the benchmark set using cTPSS and cTPSSh are available in the corrected supplementary material of Ref. 1. b. Optical rotations. The corrected ORs are displayed in Table IV (corrected). The current-free results were re-confirmed except for two cases whose ground-state solution was not fully converged. As in the original publication, all results were computed in units of deg [dm(g/cc)]-1, utilizing the length gauge, and obtained at the sodium D-line of 589.3 nm. The effect of the corrections is small for most systems considered here, with the exception of bisnoradamantan-2-one, whose (c)TPSS ORs are significantly more negative than reported previously, while the hybrid results for (c)TPSSh are slightly more positive. The overall conclusion that cMGGAs provide(Table Presented).ORs with an accuracy comparable to that of GGAs remains unaffected. c. Testing strategy. We briefly outline a method to help ascertain the correctness of MGGA and cMGGA implementations using the one- and two-electron limit and an existing GGA implementation. By construction, the generalized Kohn–Sham kinetic energy density ˆt, Eq. (14), reduces to the von Weizsäcker kinetic energy density in the limit of one-electron systems and two-electron singlets. Thus, the substitution(Equations Presented).to the corresponding cMGGA one in this limit. While the resulting GGA generally yields KS potentials, virtual orbitals, and orbital rotation Hessians different from the MGGA and cMGGA ones, the total energy, density, and all properties derived from them are identical. In particular, the above substitution enables the calculation of cMGGA excitation energies for one-electron systems and two-electron singlets using a GGA code. Additional tests performed included the calculation of the second-order correction of the MGGA excitation energy treating the current density response as a perturbation and the calculation of the velocity form of the transition dipole moment µ0n,v from the integration of the transition current density on the molecular integration grid and the (converged) excitation energy O0n according to (Table Presnted).

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