TY - JOUR
T1 - Erratum to
T2 - Development of kinetic energy density functional using response function defined on the energy coordinate (International Journal of Quantum Chemistry, (2022), 122, 20, 10.1002/qua.26969)
AU - Takahashi, Hideaki
N1 - Publisher Copyright:
© 2022 Wiley Periodicals LLC.
PY - 2022/10/15
Y1 - 2022/10/15
N2 - The author published an 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 author found an error in producing the graph with the legend ‘OF-DFT’ in Figure 8 in the article. In the construction of the graph, the author employed the composite response function (Formula presented.) (Equation [43]), made by a simple sum of two atomic response functions (Formula presented.) and (Formula presented.) corresponding to the hydrogen atoms HA and HB, respectively. Each atomic response function (Formula presented.) is to be obtained by projecting (Formula presented.) onto the energy coordinate ϵ through Equation (19), where the response function (Formula presented.) is explicitly given by Equation (37), that is, (Formula presented.) 8E: FIGURE (Figure presented.) Corrected plots of the potential energies of H2 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 ‘OF-DFT’ 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. The wave functions (Formula presented.) and the eigenvalues (Formula presented.) are those for each reference atomic system HA or HB. In our actual implementation of the approach, however, the right hand side of Equation (37) for α 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 (Equation [20]). In this Erratum we start over the calculation for ‘OF-DFT’ in Figure 8 with the amended source code. The corrected graph is presented in Figure E.8. As a result that the nonlocal term in Equation (20) has been calculated soundly, it is seen in the figure that the potential energy curve of ‘OF-DFT’ 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 Equation (20) is always positive and quadratic with respect to the difference δne(ϵ). Anyway, it should be stressed that the curve for ‘OF-DFT’ has been changed favorably by the present correction. Actually, the difference between the minimums of potential energy curves for ‘OF-DFT’ and ‘KS-DFT’ has been decreased by ∼ 6.7 kcal/mol by virtue of the correction although the difference of 17.7 kcal/mol still remains.
AB - The author published an 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 author found an error in producing the graph with the legend ‘OF-DFT’ in Figure 8 in the article. In the construction of the graph, the author employed the composite response function (Formula presented.) (Equation [43]), made by a simple sum of two atomic response functions (Formula presented.) and (Formula presented.) corresponding to the hydrogen atoms HA and HB, respectively. Each atomic response function (Formula presented.) is to be obtained by projecting (Formula presented.) onto the energy coordinate ϵ through Equation (19), where the response function (Formula presented.) is explicitly given by Equation (37), that is, (Formula presented.) 8E: FIGURE (Figure presented.) Corrected plots of the potential energies of H2 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 ‘OF-DFT’ 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. The wave functions (Formula presented.) and the eigenvalues (Formula presented.) are those for each reference atomic system HA or HB. In our actual implementation of the approach, however, the right hand side of Equation (37) for α 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 (Equation [20]). In this Erratum we start over the calculation for ‘OF-DFT’ in Figure 8 with the amended source code. The corrected graph is presented in Figure E.8. As a result that the nonlocal term in Equation (20) has been calculated soundly, it is seen in the figure that the potential energy curve of ‘OF-DFT’ 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 Equation (20) is always positive and quadratic with respect to the difference δne(ϵ). Anyway, it should be stressed that the curve for ‘OF-DFT’ has been changed favorably by the present correction. Actually, the difference between the minimums of potential energy curves for ‘OF-DFT’ and ‘KS-DFT’ has been decreased by ∼ 6.7 kcal/mol by virtue of the correction although the difference of 17.7 kcal/mol still remains.
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U2 - 10.1002/qua.26992
DO - 10.1002/qua.26992
M3 - Comment/debate
AN - SCOPUS:85135443222
SN - 0020-7608
VL - 122
JO - International Journal of Quantum Chemistry
JF - International Journal of Quantum Chemistry
IS - 20
M1 - e26992
ER -