Theoretical study on aromatic and open-shell characteristics of carbon nanobelts composed of indeno[1,2-b]fluorene units: dependence on the number of units and charge states

In this study, we theoretically investigate the aromatic and open-shell characteristics of carbon nanobelts (CNBs) composed of five- and six-membered rings. We have designed nanobelts composed of indeno[1,2-b]fluorene ([1,2-b]IF) units, which are referred to as [N]IF-CNB (N: the number of five-membered rings). The number of π-electrons, nπ, in neutral [N]IF-CNB is 7N, and thus depending on N and charge states, nπ can be 4n + 2 and 4n. Quantum chemical calculations on neutral [6]IF-CNB and [8]IF-CNB and dicationic [8]IF-CNB2+ have revealed that they are expected to exhibit unique aromatic and open-shell characteristics depending on nπ, there are several analogies of the electronic structures in [N]IF-CNB to those in [N]annulene. Delocalized and intermediate open-shell electronic structures of [N]IF-CNB are also useful to drastically change the third-order nonlinear optical properties. These results suggest that theoretically designed [N]IF-CNB can be attractive and challenging targets of organic synthesis for realizing novel open-shell functional conjugated macrocycles.


Dependence of the optimized geometries on the calculation method.
Exploring appropriate choice of theoretical method for geometry optimizations of intermediate di-and multi-radicaloids with extended π-conjugations is one of the most important topics in the chemistry of open-shell molecules. A given approximate method will sometimes predict totally different geometric features and resulting physico-chemical properties depending on how electron correlation effect is treated. In our previous studies [1][2][3] , we have carefully compared the optimized geometries for linear oligomers of On the other hand, from the viewpoint of cyclic structures, π-conjugation lengths of the present conjugated macrocycles are regarded as infinite, and thus, their electronic structures are closely related to those of one-dimensional polymers, where the description of electron correlation is also considered to be important for the predictions of geometries and properties. Indeed, BLA patterns of fully π-conjugated macrocycles and of onedimensional polymers have also been studied actively.

S3
As is explained in the main text of the manuscript, RB3LYP is found to predict a local minimum structure where CC bonds around the vertex of five-membered ring (bonds 5 and 6) are BLA-less, meaning that two canonical forms in Fig. 3a contribute equivalently in the resonance structure. RPBE0 also predicted a BLA-less local minimum structure, namely, RB3LYP and RPBE0 tend to predict the delocalized electronic structures. On the other hand, such a BLA-less structure is predicted to be saddle-point structures on the potential energy surface (PES) at the RHF, RM06-2X and RCAM-B3LYP levels. At these levels of approximation, we found a finite BLA around the bonds 5 and 6, meaning that one of the canonical forms in Fig. 3a  level, starting from both the alternated and less-alternated initial geometries. On the other hand, description of dynamical correlation is also important to characterize the localized/delocalized nature of electrons. In order to examine the effect of dynamical correlation on the total energies, we have performed additional single point calculations using the CASSCF optimized geometries at the strongly-contracted n-electron valence state perturbation theory (SC-NEVPT2) level. [4] These multi-reference calculations were performed using ORCA 4.2 program package. [5] In order to reduce the computational efforts, resolution of the identity (RI) approximation for the integral evaluations was employed during the CASSCF and SC-NEVPT2 calculations, where the automatic generation procedure for auxiliary basis functions implemented in ORCA (with the keyword "AutoAux") was employed. No symmetry constraint was imposed during the calculations.
We have numerically constructed the Hessian matrix for each local minimum in order to perform CASSCF frequency analysis. Table S1 shows the summary of calculation results at the CASSCF and SC-NEVPT2 levels. We obtained both alternated and less-alternated stationary point structures from the CASSCF geometry optimizations. The CASSCF total energy of alternated structure is found to be lower than that of less-alternated structure, and from the results of frequency analysis, alternated and less-alternated structures are predicted to be a local minimum and saddle-point structures, respectively. Since y 0 /y 1 values at the CASSCF level are found to be smaller than 0.1, such a BLA features of CASSCF geometry at the local minimum is similar to that of single reference RHF method. Actually, CASSCF local minimum geometry is found to be very similar to RHF one (see Fig. S2). However, when we evaluated the single point SC-NEVPT2 energies on these stationary point structures, the order of the total energy becomes inverted, namely, less-alternated structure is energetically more stable than the alternated one. Even though the present single point S5 SC-NEVPT2 calculations may not be sufficient enough to describe the electronic structures of these systems, these results suggest that the description of dynamical correlation is very important in the present case.  [6] . Again, RI approximation was employed during the RMP2 calculations using ORCA program package.
Calculation results are shown in Fig. S4. Unfortunately, RMP2 Hessian calculation for this system is found to demand high computational effort, and thus we could not have obtained the results of frequency analysis at the RMP2 level. However, we should note that we could not have obtained a stationary point with alternated structure, even though we started from both the alternated initial geometries. From these results, RMP2 is considered to predict the less-alternated structure. Furthermore, the BLA features of RMP2 geometry are considered to be reproduced well by the RB3LYP method, although slight deviations of bond-lengths (~ 0.01 Å at the maximum) exist.
Judging from the calculation method dependence of the optimized geometry discussed here, we have employed the RB3LYP method for the geometry optimizations of [N]IF-CNBs studied here. Of course, when N is increasing, the electronic structure of [N]IF-CNB is considered to become closer to its one-dimensional counterpart (polymer).
In such case, whether less-alternated structure still becomes a local minimum or not should S6 be carefully discussed by comparing results at several possible approximation methods. Figure S4. Comparison of bond-lengths of selected CC bond for [6]IF-CNB by ab initio correlation methods.