4. Results and Discussions
In this investigation, we will determine whether a reaction is non-competitive or competitive as well as distinguish the torquoselectivity along the reaction coordinate of given reactions in order to understand either TSIC or TSOC preferences. To begin with, we propose to use chemical topological properties of reactions instead of using a universally conventional method, which is based on activation energy, see Table 1 and Figures (1-4(c)) . According to TS < 1.0 kcal/mol shown in this table results thatReaction 1 is competitive while the other three reactions,Reactions (2-4) , are non-competitive reactions. As mentioned earlier, we decide to use some plots such as, metallicity ξ(r b), stress tensor polarizability ℙσ, ellipticity ε, total local energy densityH (r b), and stress tensor λ that contain chemical topological properties of reactions. From the examination of these plots, consider that the TSIC and TSOC reactions of the ring-opening C2-C3/C2–C3 BCP s ofReactions (1-4) exhibit degeneracy or non-degeneracy that could assist in predicting either non-competitive or competitive reactions, see Figures (1-4) . The plots of the variation of the metallicity ξ(r b) with the IRC related to the ring-opening C2-C3/C2–C3 BCP s, found for ξ(r b) > 1, indicate degeneracy forReaction 1 and non-degeneracy for remain reactions,Reactions (2-4) , see Figures (1-4(d)) respectively. This degenerate and non-degenerate behavior in the metallicity ξ(r b) suggests that in fact Reaction 1is a competitive reaction and Reaction 2 , Reaction 3 , and Reaction 4 are non-competitive reactions on the chemical basis of metallicity being a key factor in understanding the electronic reorganization of the ring-opening BCP along the reaction pathways. In addition, based on the transition state theory and previous researches outcomes, we obtain that for all four reactions, the transition state and maximum metallicity ξ(r b) relevant to the ring-opening C2-C3/C2–C3 BCP s do not simultaneously occur 48. Subsequently, the degeneracy results from other QTAIM measures such as ellipticity ε and the total local energy density H (r b) are in confirmation with metallicity ξ(r b) results which indicate that Reaction 1 is a competitive reaction andReaction 2 , Reaction 3 , and Reaction 4 are non-competitive reactions, see Figures (1-4(e)) andFigures (1-4(f)) respectively.
In the following, investigation the plots of variation of stress tensor polarizability ℙσ with the IRC demonstrate an apparent differentiation among all four reactions. In other words, we can conclude from the degenerate and non-degenerate behavior of the stress tensor polarizability ℙσ corresponding to the ring-opening C2-C3/C2–C3 BCP s that Reaction 1 is a competitive reaction and Reaction 2 , Reaction 3 , andReaction 4 are non-competitive reactions, see Figures (1-4(g)) . Additionally, we obtain the same behavior as was happened for the metallicity ξ(r b) that the saddle point and transition state do not coincide precisely with each other. Moreover, the plots of examination of the variation of the stress tensor λ with the IRC for all four reactions are considered which display similar evaluation with QTAIM measures mentioned above; see Figures (1-4(h)) .
From this part of our survey, we can conclude that besides activation energy which is a conventional method, we introduce some different measures such as, ξ(r b), ε,H (r b), ℙσ, λ, which could be suitable indicators to predict that each reaction is non-competitive or competitive.
Inspired by the theoretical predictions, we decided to evaluate the tendency of each non-competitive reaction to determine if it is either TSIC or TSOC reactions. Hence, using the eigenvector-following path with length, which is nominated as helicity length H relevant to the shared-shell ring-opening BCP of each of the Reactions (1-4) pathways was determined, see Figures (1-4(j))respectively.
According to what has been discussed in the theory section, the ellipticity εi values along the bond-paths (r ) associated with shared-shell BCP s could demonstrate if the bond is a single or double bond character not the same for the closed-shell BCP s. This has been done to decide the inclination for either the transition state (TS) outward conrotatory (TSOC) or the transition state inward conrotatory reaction pathway (TSIC). To gain predictions of the TSIC or TSOC product preference, we will analyze the bond paths of the shared-shell ring-opening BCP s of the TSIC and TSOC reaction pathways before they alter to closed-shell BCP s or rupture completely. Subsequently, the plots of the variation of the bond-path length (BPL) with the IRC for both TSIC and TSOC reactions of the ring-opening C2-C3/C2–C3 BCP were evaluated. In addition, the non-degeneracy and degeneracy trend of each reaction shows thatReaction 1 is degenerate while in opposite Reaction 2 ,Reaction 3 , and Reaction 4 show significant differences for the corresponding variations and exhibit non-degenerate values. Therefore, in agreement with the QTAIM and stress tensorBCP properties, we demonstrate Reaction 1 as a competitive reaction and Reaction 2 , Reaction 3 , andReaction 4 as non-competitive reactions, see Figures (1-4(i)) respectively. The favored TSIC or TSOC pathway is determined by the greater length H, which is constructed from thee2 eigenvector tip paths alongside the bond- path (r ) for a pair of TSIC and TSOC pathways that are being compared. Larger deviations of the tips are associated with greater deviations of the tips from the original orientation of thee2 at the BCP of TSIC or TSOC pathways. Notice that e1 ande2 eigenvectors are associated with the least and most preferred direction of ρ (r ) accumulation. As a result, a larger value of H is relating to the most easily distorted molecular graph and accordingly determines the preference for either the TSIC or TSOC pathways. Consequently, we introduce that for the shared-shell ring-opening BCP s, greater H lengths predict the preferred TSIC or TSOC reaction pathway. Besides, the variation of the H with the IRC for the TSIC and TSOC for all four reactions were display similar behavior like QTAIM and stress tensor measures that are discussed above. According to the results that obtained from the variation of the H plots, we suggest that Reaction 1 is a competitive reaction and Reaction 2 , Reaction 3 , andReaction 4 are non-competitive reactions, see Figures (1-4(j)) respectively. However, the helicity length H values demonstrate subtle differences amidst TSIC and TSOC reaction pathways. Consequently, the torquoselectivity preferences of Reaction 2 ,Reaction 3 , and Reaction 4 are TSOC, TSOC, and TSIC respectively based on the Longer H values, which is in confirmation with the experimental results shown in Table 1 , Figures (1-4) .