## 1 Introduction
In contrast, the energy of 4_H_-pyrans (**3**) is very close to 2_H_-pyrans (**11**) [24] which are the precursors of 1,3-butadienes (**4**) shown in Figure 2 to attribute the preparation of both **3** and **11** to the thermodynamically controlled reaction of **7a** with aromatic compounds. Relatively high temperature (the temperature of refluxing acetonitrile) required for the reaction of **1** giving **3** and **4** is consistent with the thermodynamically controlled reaction of pyrylium (**7a**) with aromatic compounds. Next, we estimated the activation energy for the ring-opening process from 2_H_-pyrans (**11**) resulted by the reaction of pyrylium (**7a**) with aromatic compounds at 6-position (Figure 3). The optimized transition state structure (**TS11**; Ar = Ph) [25] and the most stable structure of **4** (Ar = Ph) are illustrated together with their energies. The energy difference between **11** (Ar = Ph) [16] and **TS11** (Ar = Ph) is estimated to be _ca._ 15 kcal/mol, which corresponds to the activation energy of this process. The (_E_,_Z_)-dienes (**4'**) given by ring-opening of **11** readily isomerize to thermodynamically more stable (_E_,_E_)-dienes (**4**) via protonation and deprotonation processes (Figure 3). The dienes **4'  
Figure 2: Optimized structures and energy values of intermediates (**8a**,**b**, **2**, and **11**). (Ar = Ph) and **4** (Ar = Ph) are calculated to be _ca._ 8 kcal/mol and _ca._ 9 kcal/mol more stable than **11** (Ar = Ph), respectively. The above results suggest that the irreversible ring-opening of the intermediates (**11**) will easily occur at acetonitrile reflux temperature to afford **4**. In addition, we examined the process from 4-cation (**8a**) to pyrylium (**7a**) to support the mechanism presented in our previous report [16] for the reaction of dihydropyran (**1**) with aromatic compounds in trifluoroacetic acid at ambient temperature giving dihydro-4_H_-pyrans (**2**) solely (Scheme 1). As is mentioned before, this elimination is an endothermic reaction requiring the external energy of 11.3 kcal/mol [22]. Therefore, this result predicts that the reaction of the first intermediate, 4-cation (**8a**), with aromatic compounds directly affording dihydro-4_H_-pyrans (**2**) have precedence over the course via 4_H_-pyrans (**3**) comprised of the reaction of pyrylium (**7a**) with aromatic compounds given that the dihydropyran (**1**) would undergo the reaction at ambient temperature. Even though 4_H_-pyrans (**3**) was given by the _p_-toluenesulfonic acid catalyzed reaction, the reaction of **1** carried out in trifluoroacetic acid with aromatic compounds [6] resulted in the failure of the formation of **3**. These experimental evidences provide us with compatible conclusion as to the above calculated prediction. Figure 3: Ring-opening process from 2_H_-pyrans (**11**) to (_E_,_E_)-dienes (**4**). Finally, we examined the reaction of dihydropyran (**5**) with aromatic compounds giving butadiene derivatives (**6**) shown in Scheme 1. As for this reaction, an alternative pathway including the reaction of 6-cation (**9b**) with aromatic compounds directly affording 2\(H\)-pyrans (**10**) is possible in addition to the pathway via pyrylium (**7b**) illustrated in Scheme 2. To elucidate such alternative pathway, we considered as to figuring out dihydropyran (**5**), 4-cation (**8b**), and 6-cation (**9b**). The results are summarized in Figure 4.  
Based on the energy values for **5**, **8b**, and **9b**, the ionization process from **5** to 6-cation (**9b**) is estimated to require _ca._ 20 kcal/mol more energy [26] compared with such ionization to 4-cation (**8b**). The exclusive formation of **8b** from **5** in trifluoroacetic acid is attributed to this value. Hence, the reaction of **5** with aromatic compounds giving **6** does not proceed along the pathway via **9b**. In other words, butadiene derivatives (**6**) are assumed to be derived from pyrylium (**7b**) which formed via 4-cation (**8b**) along the pathway shown in Scheme 2 as we reported previously [16, 27].