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\begin{document}
\title{ERRATUM: Development of Kinetic Energy Density Functional Using Response
Function Defined on the Energy Coordinate}
\author[1]{Hideaki Takahashi}%
\affil[1]{Tohoku University}%
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\date{\today}
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\begin{abstract}
I would like to submit an erratum for the article with the title
`Development of Kinetic Energy Density Functional Using Response
Function Defined on the Energy Coordinate' (Int. J. Quantum Chem.
2022;e26969, https://doi.org/10.1002/qua.26969). The corresponding
author of the article is Hideaki Takahashi. The author found an error in
producing the graph with the legend `OF-DFT' in Fig. 8 in the article.
The error in the graph is attributed to the fact that the atomic
response function was spuriously multiplied by 2. We refer the editor to
the main text of the erratum in more details. The graph was revised
using the amended source code. Fortunately, it was found that the
corrected graph was changed favorably as compared to the original one,
showing better agreement with the reference calculation.%
\end{abstract}%
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\normalsize
The author published an article with the title \textquoteleft Development of Kinetic Energy Density Functional Using Response Function Defined on the Energy Coordinate\textquoteright\; (Int. J. Quantum Chem. 2022;e26969, https://doi.org/10.1002/qua.26969). The author found an error in producing the graph with the legend \textquoteleft OF-DFT\textquoteright\; in Fig. 8 in the article. In the construction of the graph, the author employed the composite response function $\chi_0^e(\epsilon, \epsilon^\prime)$(Eq. (43)), made by a simple sum of two atomic response functions $\chi_0^\text{A}$ and $\chi_0^\text{B}$ corresponding to the hydrogen atoms H$_\text{A}$ and H$_\text{B}$, respectively. Each atomic response function $\chi_0^\text{A,B}(\epsilon, \epsilon^\prime)$ is to be obtained by projecting $\chi_0^\text{A,B}(\bm{r}, \bm{r}^\prime)$ onto the energy coordinate $\epsilon$ through Eq. (19), where the response function $\chi_0^\text{A,B}(\bm{r}, \bm{r}^\prime)$ is explicitly given by Eq. (37), that is,
\begin{equation}
\chi_{0}^\text{A,B}\left(\bm{r}, \bm{r}^{\prime}\right)=\sum_{i}^{\text {occ }} \sum_{a}^{\mathrm{vir}} \frac{1}{\varepsilon_{a}^{0}-\varepsilon_{i}^{0}} \varphi_{i}^{0 *}(\bm{r}) \varphi_{a}^{0}(\bm{r}) \times \varphi_{a}^{0 *}\left(\bm{r}^{\prime}\right) \varphi_{i}^{0}\left(\bm{r}^{\prime}\right) \notag
\end{equation}
The wave functions $\{\varphi_{i,a}^0(\bm{r})\}$ and the eigenvalues $\{\epsilon_{i,a}^0\}$ are those for each reference atomic system H$_\text{A}$ or H$_\text{B}$. In our actual implementation of the approach, however, the right hand side of Eq. (37) for $\alpha$ spin electron was spuriously multiplied by 2 assuming closed shell system although a hydrogen atom has only a single electron. It means that the inverse of the response matrix was spuriously multiplied by 0.5, which gave rise to the decrease in the nonlocal term in the kinetic energy (Eq. (20)). In this Erratum we start over the calculation for \textquoteleft OF-DFT\textquoteright\; in Fig. 8 with the amended source code. The corrected graph is presented in Fig. E.8. As a result that the nonlocal term in Eq. (20) has been calculated soundly, it is seen in the figure that the potential energy curve of \textquoteleft OF-DFT\textquoteright\; has been shifted upward in the entire region of the distance $R$(H$-$H). It should also be noted that the degree of the shift is larger in the small $R$(H$-$H) region. These results are quite reasonable since the nonlocal term in Eq. (20) is always positive and quadratic with respect to the difference $\delta n^e(\epsilon)$. Anyway, it should be stressed that the curve for \textquoteleft OF-DFT\textquoteright\; has been changed favorably by the present correction. Actually, the difference between the minimums of potential energy curves for \textquoteleft OF-DFT\textquoteright\; and \textquoteleft KS-DFT\textquoteright\; has been decreased by $\sim 6.7$ kcal/mol by virtue of the correction although the difference of $17.7$ kcal/mol still remains.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/IJQC-Fig8/IJQC-Fig8}
\caption{{\label{Fig8} Corrected plots of the potential energies of H$_2$ molecule computed by the present OF-DFT approach in comparisons with the one obtained by Kohn-Sham DFT with the BLYP functional. Only the curve \textquoteleft OF-DFT\textquoteright\; has been revised using the corrected source code. The right axis is for the values of OF-DFT and shifted so that the bottoms of the curves almost match that of KS-DFT. The energy region of the right axis is shifted accordingly. The other notations are the same as Figure 8 of the original version.%
}}
\end{center}
\end{figure}
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