Characterization of internal rotation of monorotor molecules via bifurcation analysis Hsiu-Yao Cheng a,*, Shyang Chang b a Department of Chemistry, Tunghai University, Taichung, Taiwan, ROC b Department of Electrical Engineering, National Tsinghua University, Hsinchu, Taiwan, ROC Received 11 January 2003; in final form 3 April 2003 Abstract The bifurcation analyses of the internal rotation processes of two different series of monorotor molecules along the reduced reaction coordinate are investigated in this Letter. Our topological analyses indicate that the internal rotation processes are characterized by the properties of critical points. Furthermore, when the energy functions are represented as the third-order generic polynomial of the reduced reaction coordinate, the processes are isomorphic to the ele- mentary fold catastrophe. � 2003 Elsevier Science B.V. All rights reserved. 1. Introduction G�aardenas-Jir�oon et al. [1] have presented a sim- ple analytic form of the potential energy expressed in terms of linear combinations of local properties of monorotor molecules. The resultant potential energy function can be generated from a small number of calculated energy points. When com- bined with the Brønsted coefficient [2], it will also provide a quantitative version of Hammond pos- tulate (HP) [3]. Recently Margalef-Roig et al. [4] have further presented a theoretical formalism based on simple bifurcation theory to study the trans$ cis rota- tional isomerization process of glyoxal and its chlorine derivatives. They determined a local bi- furcation condition on the parameter space of the analytic form of the potential energy presented by G�aardenas-Jir�oon et al. during the internal rotation processes. They claimed that topological regions could be defined for chemical processes that can be analyzed within the HP and principle of maximum hardness (PMH) [5–8] framework. In addition, the behavior of internal rotation process is not iso- morphic to any type of elementary catastrophes. The detailed bifurcation study of the internal rotation of three different series of monorotor molecules by Cheng et al. [9] have indicated that the types of catastrophe are related to the generic form of potential energy function, and the critical points. When the double-barrier energy function is represented as the third-order generic polynomial of the reduced reaction coordinate, the behavior of internal rotation process is isomorphic to the elementary fold catastrophe. However, the Chemical Physics Letters 373 (2003) 62–66 www.elsevier.com/locate/cplett * Corresponding author. Fax: +886-423590426. E-mail address:
[email protected] (H.-Y. Cheng). 0009-2614/03/$ - see front matter � 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00581-5 topological description of the internal rotational processes has not yet been analyzed. In the present work, two series of monorotor molecules (a) OXC–CXO, and (b) XS–SX (X¼H, F, or Cl) are studied. The results will be analyzed by invoking Thom�s catastrophe theory [10–12]. The bifurcations during the internal rotation pro- cesses will be studied by using the generic poly- nomials of potential energy. As compared to the previous approaches, the topological regions can be analyzed much more clearly. 2. Conventional approaches Conventionally, to characterize the transition state (TS) of chemical reactions, the quadratic function of the potential energy profile V ½x� ¼ Kvf ðxÞ þ xDV 0 is adopted [1,4,13–19]. Here, Kv stands for the intrinsic property of the reaction, x is the reduced reaction coordinate indicating the reaction progress from reactants ðx ¼ 0Þ via TS ðx ¼ xtsÞ to products ðx ¼ 1Þ, f ðxÞ ¼ xð1� xÞ, and DV 0 is the reaction energy. The activation energy DV z is equal to the single maximum cor- responding to V ½xts�. Hence, by taking the deriv- ative of V ½x� with respect to x, the x of TS is obtained as xts ¼ ð1=2Þ þ ðDV 0=2KvÞ. The explicit formula of activation energy can then be obtained as the same form of Marcus equation as follows: DV z ¼ Kv 4 þ DV 0 2 þ ðDV 0Þ2 4Kv : ð1Þ Subsequently, Margalef-Roig et al. use the Brønsted coefficient, b ¼ oDV z=oDV 0 that is de- fined by Leffler [2], to measure the relative resem- blance of the TS to the product or reactant [13]. According to this definition, the obtained coeffi- cient b ¼ ð1=2Þ þ ðDV 0=2KvÞ is equal to xts exactly and can be adopted to quantify the HP as in [1,4,22]. That is, an early TS ðb ¼ xts < 1=2Þ leads to an exothermic reaction ðDV 0 < 0Þ, and a later TS ðb ¼ xts > 1=2Þ for an endothermic reaction ðDV 0 > 0Þ. Finally, for chemical processes that are difficult to classify within this scheme, they use a third-order polynomial to analyze the reactions. However, we will show in this Letter that the to- pological regions cannot be characterized by the so-called HP, anti-HP, or not in the HP frame- work as mentioned in [4]. 3. Methodology and theory In order to obtain the generic form of the po- tential energy for monorotor molecules, we have performed the Hartree–Fock (HF) calculations with the standard 6-31+G basis set. The energy profiles along the torsional angle h defined with respect to the central C–C or S–S bond of the molecule are obtained at the optimized geometries along the torsional angle. All calculations are performed using the GAUSSIANAUSSIAN 98 package [20]. The generic form of potential function repre- senting trans$ cis rotational isomerization pro- cesses for monorotor molecules (a) OXC–CXO, and (b) XS–SX along the reduced reaction coor- dinate x is represented as follows: V ½x� ¼ a1x þ a2x2 þ a3x3; ð2Þ where x � ð1� cos hÞ=2 is the reduced reaction coordinate measuring the reaction process on going from the trans ðx ¼ 0Þ to the cis ðx ¼ 1Þ confor- mations. Eq. (2) comes from a cosine Fourier ex- pansion up to the third-term, the coefficients a1, a2, and a3 can be accurately determined by least-square fitting of the third-order polynomial V ½x�. Notice that the coefficients a1, a2, and a3 will be equivalent to ðAþ BÞ, �½Aþ 3ðB� CÞ�, and 2ðB� CÞ, in [4]. However, the approximation method used in [1,4] to determine parameters A, B, and C is not as ac- curate when compared with our approach. The bifurcation (BF) conditions can be ob- tained by setting both the first and second deriv- atives of V ½x� to zero with respect to x. Thus, V 0½x� ¼ V 00½x� ¼ 0 occur at a22 ¼ 3a1a3, i.e., y2 ¼ �9ðx� 1Þðx� 1=3Þ > 0 where x ¼ C=B, y ¼ A=B, and x 2 ½1=3; 1�. The critical points xc can be found by solving the critical condition V 0½x� ¼ a1 þ 2a2x þ 3a3x2 ¼ 0. Then, xc can be obtained as ½�a2 � ða22 � 3a1a3Þ1=2�=ð3a3Þ. Since a1, a2, and a3 are real numbers, there- fore, for the region (I) a22 � 3a1a3 > 0, xIc ¼ ½�a2� ða22 � 3a1a3Þ1=2�=ð3a3Þ ¼ 1=2þ ðB� CÞ�1fA� ½A2þ 3ðB� 3CÞ ðB� CÞ�1=2g=6. The bifurcation (BF) condition a22 � 3a1a3 ¼ 0 implies that the root H.-Y. Cheng, S. Chang / Chemical Physics Letters 373 (2003) 62–66 63 xBFc ¼ �a2=3a3 ¼ 1=2þ ðB� CÞ�1A=6 represents a twofold degeneracy. As for the region (II) a22 � 3a1a3 < 0, the roots xIIc are imaginary, hence no real critical point exists. Notice that all the xc�s have to the within range ½0; 1�. Unfortunately, in [4], the values xIIc in region (II) are erroneously claimed to have two non-degenerate critical points. To analyze the bifurcation condition via the Thom�s catastrophe theory, the potential function V ½x� is shifted to a new coordinate xh via the transformation xh ¼ x � h, and the following form is obtained: Vn½xh; u� ¼ x3h þ uxh: ð3Þ Here, h ¼ �a2ð3a3Þ�1, u ¼ a1=a3 � ða2=a3Þ2=3, and �h6xh6 ð1� hÞ. The set of critical points of all first derivatives of Vn, called the catastrophe surface, fðxh; uÞ : V 0n ½xh; u� ¼ 3x2h þ u ¼ 0g is a parabola in the left half plane as shown in Fig. 1. The BF conditions V 0n ½xh; u� ¼ V 00n ½x; u� ¼ 0 occur at the origin ðxh; uÞ ¼ ð0; 0Þ. Notice that this origin if represented by the formulation of Mar- galef-Roig et al. [4] is a parabola in terms of their parameters A, B, and C. As to the critical points, in region (I), u < 0, i.e., a22 � 3a1a3 > 0, there are two critical points, name- ly, a maximum at xIh;c ¼ �ju=3j1=2 and a minimum at xIh;c ¼ ju=3j1=2. The BF condition: u ¼ 0 or a22 ¼ 3a1a3 gives one critical pointxBFh;c at the origin. In region (II), u > 0, i.e., a22 � 3a1a3 < 0, there is no real critical point. Notice that all the critical points xh;c�s have to be within range ½�h; 1� h�. 4. Results and discussions All the numerical analyses of V ½x� and Vn are performed by the Matlab software [21]. The calculated parameters u; h in Eq. (3), the critical points xc, the Brønsted coefficient b, the reaction energy DV 0, and topological region for molecules OXC–CXO, and XS–SX are listed in Table 1. Notice that except for oxalyl-Cl, i.e., OClC–CClO, all the other molecules have the va- lue u < 0 and one critical point. This fact indicates that only oxalyl-Cl lies in the right half plane (which is equivalent to region II in [4]) and the others are on the left half plane (which is equiva- lent to region I in [4]) of Fig. 1. Fig. 1. The catastrophe surface, fðxh; uÞ : V 0n ½xh; u� ¼ 3x2h þ u ¼ 0g. Table 1 The calculated parameters u; h in Eq. (3), the critical points xc, the Brønsted coefficient b, the reaction energy DV 0, and topological region for molecules (a) OXC–CXO, and (b) XS–SX (X ¼ H, F, or Cl) u h xc b DV 0 Top. region OHC–CHO )1.65 1.44 0.70a 0.72 6.06 I OHC–CFO )2.20 1.38 0.53a 0.57 1.84 I OFC–CFO )0.74 0.97 0.47a 0.55 0.99 I OHC–CClO )0.54 0.95 0.53a 0.62 2.41 I OClC–CClO 0.09 0.42 – – 3.31 II HS–SH )2.22 )0.36 0.50b – – I FS–SF )1.06 )0.05 0.54b – – I ClS–SCl )0.56 0.12 0.56b – – I aMaximum. bMinimum. 64 H.-Y. Cheng, S. Chang / Chemical Physics Letters 373 (2003) 62–66 According to Table 1, the xc corresponding to a maximum is slightly different from b. The reason is due to that the xc is derived from our third-order generic polynomial. It actually depends on pa- rameters A, B, and C, not like b which depends only on A and C. Since the position of TS actually corresponds to the value of xc not b, it is better to use the maximal critical point, i.e., xc;max instead of b to measure the relative resemblance of the TS ðx ¼ xc;maxÞ to the product or reactant. In other words, HP should be interpreted as follows: the TS will be closer to the products ðxc;max > 0:50Þ for endothermic reactions ðDV 0 > 0Þ, and it will be closer to the reactants ðxc;max < 0:50Þ for exo- thermic reactions ðDV 0 < 0Þ. Table 1 indicates that the internal rotation processes of glyoxal (OHC–CHO), F-glyoxal (OFC–CHO), Cl-glyoxal (OClC–CHO) are all en- dothermic reactions ðDV 0 > 0Þ and all have a maximum critical point, xIc > 0:50 which corre- sponds to the product-like TS. Therefore, they be- long to the HP behavior. However, the internal rotation process of oxalyl fluoride (OFC–CFO) is also endothermic reactions ðDV 0 > 0Þ but it has a maximum critical point, xIc < 0:50 which corre- sponds to the reactant-like TS. The internal rota- tion process of oxalyl fluoride is classified as anti- HP behavior. Moreover, the internal rotation pro- cesses of oxalyl-Cl and molecules XS–SX cannot be classified within the HP framework because there are no TS between trans and cis conformations. Notice that all the internal rotation processes of the aforementioned molecules except oxalyl-Cl are located in the same region I ðu < 0Þ irrespective of their HP, anti-HP behavior or not in the HP framework. In addition, the internal rotation processes of oxalyl-Cl and molecules XS–SX that are not in the HP framework are actually located in separate regions. These results are not revealed by the topological analysis presented in [4]. Moreover, our critical-point results of glyoxal and chlorine derivatives are different from that of [4]. We have found no critical point obtained for the internal rotation of oxalyl-Cl in our calculation while Margalef-Roig et al. found two critical points (see Eq. (11) of [4]). This discrepancy is due to their misinterpretation of the two complex roots as two critical points, hence result in the wrong conclusion of not being isomorphic to any ele- mentary catastrophes known. The HF/6-31+G energy profiles along the variable x are given in Figs. 2a and b for molecules (a) OXC–CXO, and (b) XS–SX, respectively. Ac- cording to Fig. 2a, the internal rotation processes of glyoxal, F-glyoxal, Cl-glyoxal, and oxalyl fluo- ride, all have a maximum critical point corre- sponding to the TS. There is no critical point for the internal rotation process of oxalyl-Cl. Fig. 2b indicates that all the internal rotation processes of XS-SX have a minimum critical point corre- sponding to a stable conformation. According to Fig. 2. HF/6-31+G potential energy profiles along the vari- ables x for molecules (a) OXC–CXO, and (b) XS–SX (X ¼ H; F, or Cl), respectively. H.-Y. Cheng, S. Chang / Chemical Physics Letters 373 (2003) 62–66 65 our study, the behavior of internal rotation pro- cess for molecules (a) OXC–CXO and (b) XS–SX are isomorphic to the elementary fold catastrophe that has no real critical point in region (II). 5. Conclusions The bifurcation analyses of the rotational isomerization processes of monorotor molecules along the reduced reaction coordinate have given us the bifurcation conditions on the reaction pathway space. Analyses have also indicated that the internal rotation processes that are conform- able to HP, anti-HP or not in the HP framework may locate in the same topological region. For internal rotation processes that are not in the HP framework may actually be located in separate regions. These results have fortified the fact that the topological regions of internal rotation pro- cesses are characterized by the properties of critical points. In addition, results have also revealed that if the energy functions are represented as the third- order generic polynomial of the reduced reaction coordinate, the processes are isomorphic to the elementary fold catastrophe. Acknowledgements We thank the referees for helpful comments about this Letter and National Science Council of the Republic of China for financial support under the Grant No. NSC 91-2113-M029-004. References [1] G.I. G�aardenas-Jir�oon, A. Toro-Labb�ee, C.W. Bock, J. Maruani, in: Y.G. Smeyers (Ed.), Structure and Dynamics of Non-Rigid Molecular Systems, Kluwer Academic Pub- lishers, Dordrecht, The Netherlands, 1995, p. 97. [2] J.E. Leffler, Science 117 (1953) 340. [3] G.S. Hammond, J. Am. Chem. Soc. 77 (1955) 334. [4] J. Margalef-Roig, S. Miret-Art�ees, A. Toro-Labb�ee, J. Phys. Chem. A 104 (2000) 11589. [5] R.G. Pearson, Chemical Hardness, Wiley-VCH, Oxford, 1997. [6] R.G. Parr, P.K. Chattaraj, J. Am. Chem. Soc. 113 (1991) 1854. [7] P.K. Chattaraj, G.H. Liu, R.G. 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