## 1 Introduction  
In recent years, a number of researches have been reported about the development of new methodologies for syntheses of various kinds of fluorine-containing heterocycles. These compounds have put emphasis on the interest of high biological activities especially in life-science fields due to the unique character that could contribute to the exploration of new active ingredients [1; 2; 3; 4]. In the course of our researches concerning syntheses and reactions of novel fluorine-containing heterocycles, we found that 2,4-diethyloxy-6-trifluoromethyl-3,4-dihydro-2\(H\)-pyran (**1**) prepared in a simple step of hetero-Diels-Alder reaction of 4-ethoxy-1,1,1-trifluorobut-3-en-2-one with ethyl vinyl ether [5] reacted with aromatic compounds in trifluoroacetic acid at ambient temperature to give 4-aryl-2-ethoxy-6-trifluoromethyl-3,4-dihydro-2\(H\)-pyrans (**2**) as a sole product (Scheme 1) [6]. In contrast to this, it was found that the reaction of **1** with 1,3-dimethoxybenzene in refluxing acetonitrile in the presence of catalytic amounts (0.3 equiv.) of \(p\)-toluenesulfonic acid afforded _ca._ 1:1 mixture of the corresponding 4-aryl-2-trifluoromethyl-4\(H\)-pyrans (**3**) and the ring-opening product, 6-aryl-1,1,1-trifluorohexa-3,5-dien-2-ones (**4**) [6]. Similar ring-opening reaction of fluorine-containing dihydro-2\(H\)-pyrans giving 6-aryl-1,1,1,5,5,5-hexafluoro-3-[(\(E\))-3-propylidene]pentane-2,4-diones (**6**) was also found in the reaction of 2-ethoxy-4-isobutoxy-5-trifluoroacetyl-6-trifluoromethyl-3,4-dihydro-2\(H\)-pyran (**5**) [7] with aromatic compounds in refluxing trifluoroacetic acid [8]. **Scheme 1. ** Acid catalyzed reactions of 2,4-dialkoxy-3,4-dihydro-2\(H\)-pyrans with aromatic compounds. **Scheme 2. ** Acid catalyzed reactions of 2,4-dialkoxy-3,4-dihydro-2\(H\)-pyrans with aromatic compounds. **Scheme 3. ** Acid catalyzed reactions of 2,4-dialkoxy-3,4-dihydro-2\(H\)-pyrans with aromatic compounds. **Scheme 4. ** Acid catalyzed reactions of 2,4-dialkoxy-3,4-dihydro-2\(H\)-pyrans with aromatic compounds. **Scheme 5. ** Acid catalyzed reactions of 2,4-dialkoxy-3,4-dihydro-2\(H\)-pyrans with  
Derivatives having the skeletons of dihydropyrans (**2**), 4_H_-pyrans (**3**), and 1,3-butadienes (**4** and **6**) have a high potential use as synthetic intermediates to access a variety of heterocycles [9, 10, 11, 12, 13, 14, 15]. Hence, highly important practices for the constructions of various kinds of novel fluorine-containing heterocyclic systems would be provided by the above reactions of dihydropyrans, **1** and **5**. As we proposed in our previous report [16], the selective formation of dihydropyrans (**2**) from **1** and of butadiene derivatives (**6**) from **5** could be explained by the kinetically controlled reactions of pyrylium (**7a**) at C-4 and the thermodynamically controlled reaction of **7b** at C-6, respectively (Figure 1). Such pyryliums (**7a,b**) are assumed to form easily from **1** and **5** in the strong acid, trifluoroacetic acid. Moreover, the unexpected formation of **7a** in the course of the reaction of dihydropyran (**1**) giving 4_H_-pyrans (**3**) and butadiene derivatives (**4**) is also figured out under weaker acidic conditions such as the presence of catalytic _p_-toluenesulfonic acid in refluxing acetonitrile. In this case, it is probable that **3** and **4** are directly derived from 4-cation (**8a**) and 6-cation (**9a**), respectively, which are the precursors of pyrylium (**7a**). Here we wish to report our DFT calculation study for these acid catalyzed reactions of dihydropyrans, **1** and **5**, with aromatic compounds. The mechanisms giving 4_H_-pyrans (**3**) and 1,3-butadienes (**4**) from **1** in the presence of _p_-toluenesulfonic acid catalyst were elucidated by making use of benzene as a model of aromatic compounds.