Corrigendum: Open‐Shell Characters, Aromaticities and Third‐Order Nonlinear Optical Properties of Carbon Nanobelts Composed of Five‐ and Six‐Membered Rings

Abstract

In the original article,1 we reported that electronic structures and optical and magnetic response properties of an open-shell carbon nanobelt (IF-CNB) are evaluated by the spin-unrestricted density functional theory (U-DFT) method using the long-range corrected LC-BLYP exchange correlation functional, namely, at the LC-UBLYP/6-311+G** level. However, only in the calculations of magnetic response properties, we found that the results are obtained by the spin-restricted DFT (R-DFT) solution with <S2>=0. Therefore, the magnetically-induced current (MIC) density maps given in Figures 6 and S6 (in the Supporting Information) as well as the nucleus-independent chemical shift (NICS) values given in Figure S5 for the open-shell IF-CNB are found to be those evaluated at the LC-RBLYP/6-311+G** level. We sincerely apologize for this mistake. In a previous study,2 on the other hand, we compared the MIC density results from the R- and U-DFT solutions for oligoacenes and dicyclopenta-fused oligoacenes by using the gauge-including MIC (GIMIC) method. We found that the diatropic (aromatic) contributions of aromatic (4n+2)π systems with open-shell singlet electronic structures tend to be exaggerated when the R-DFT solution was employed primarily because of its very small energy gaps for the orbital transitions contributing to the diatropic current. Therefore, we have recalculated the MIC density map for IF-CNB at the LC-UBLYP/6-311+G** level of approximation and compared the result with that at the LC-RBLYP/6-311+G** level. The results are shown in Figure 1. Even though strong contributions of diatropic (aromatic) ring current were predicted everywhere around the belt at the LC-RBLYP level (Figure 1a), such strong aromatic contributions of ring current are found to be suppressed at the LC-UBLYP level (Figure 1b). In particular, paratropic (anti-aromatic) contributions are found to be predominant inside the belt whereas diatropic contributions are found to appear outside the belt at the LC-UBLYP level. Such paratropic/diatropic contributions inside/outside the hoop are similar to those of [6]annulene (benzene). However, the amplitudes of integrated MIC values of IF-CNB (1-3 nA/T) are found to be much smaller than that of the closed-shell [6]annulene (11.9 nA/T) evaluated at the LC-RBLYP/6-311+G** level [2]. Figure 2 shows the side-views of the MIC density map and the integrated values of MIC density for each bond calculated at the LC-RBLYP and LC-UBLYP levels. Even though the amplitudes of integrated MIC values are decreased, the net induced ring currents along the belt are predicted to flow clockwise at the LC-UBLYP level. Magnetically induced current (MIC) density vector plots of IF-CNB evaluated at the LC-RBLYP/6-311+G** (a) and LC-UBLYP/6-311+G** (b) levels on the plane perpendicular to the z-axis and 1 Å above the center of mass. Diatropic currents rotate clockwise, while the paratropic ones anticlockwise. The values in the color bar are given in a.u. The current vectors are scaled with a factor of 30 Å a.u.-1, where 1 a.u.=100.63 nA T−1 Å−2. Integrated MIC values [nA/T] of each bond with along with MIC maps for IF-CNB at the LC-RBLYP/6-311+G** (a) and LC-UBLYP/6-311+G** (b) levels. Positive and negative values represent clockwise (diatropic) and counter-clockwise (paratropic) contributions of ring current, respectively. Spatial integration of MIC densities was performed on the bisector plane of the bond with a width of 2.0 Å. The length of the plane is 8.0 Å from the center of the mass of the belt. Here, the current vectors are scaled with a factor of 20 Å a.u.−1. Figure 3 shows the results of NICS(0) values on the five- and six-membered rings, which characterize aromatic nature on each ring. In both R- and U-DFT cases, positive (negative) NICS values at the center of five- (six-) membered ring are obtained, whereas, the U-DFT results are found to exhibit more antiaromatic nature than the R-DFT results. This feature may be associated with the emergence of the odd electron density in the U-DFT solution. NICS(0) values [ppm] at the centers of five- and six-membered rings calculated at the LC-RBLYP/6-311+G** (a) and LC-UBLYP/6-311+G** levels. NICS(0) values of symmetry-independent positions are only shown. In summary, the MIC density map of IF-CNB in Figure 6 in the original article should be replaced by Figure 1b, the correct integrated MIC values corresponding to Figure S6 in the Supporting Information are shown in Figure 2b, and the NICS(0) values in Figure S5 (Supporting Information) are presented in Figure 3b. Judging from the results at the LC-UBLYP level, IF-CNB is expected to show not so strong, but fairly weak aromatic nature over the belt in the open-shell singlet state.