- Email: [email protected]
Helical chirality in model mirror-imaged carbyne trefoil knots
Available online at www.sciencedirect.com
Journal of Molecular Structure 875 (2008) 515–519 www.elsevier.com/locate/molstruc
Helical chirality in model mirror-imaged carbyne trefoil knots Wen-Ye Deng, Wen-Yuan Qiu
*
Department of Chemistry, State Key Laboratory of Applied Organic Chemistry, Lanzhou University, Lanzhou 730000, PR China Received 4 March 2007; received in revised form 22 May 2007; accepted 22 May 2007 Available online 2 June 2007
Abstract A new method for understanding the helical chirality of molecular knots has been developed on the basis of the helix theory. Our results show that the model right-handed carbyne trefoil knot is composed of three small right-handed helices with right-handed chirality, and the left-handed one has three small left-handed helices with left-handed chirality. These two trefoils present mirror-imaged symmetry. Moreover, the rotation strengths of the two carbyne trefoils have been suggested by an analogy to helicene. Helical chirality analysis leads us to infer that helical structure is the essence of molecular knot’s chirality, which is also a possible measure for the chirality of a molecular knot. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Trefoil knot; Carbyne; Chirality; Helix theory; Mirror-image
1. Introduction Originally, knot theory began as a result of a hypothesis made by Lord Kelvin that atoms were rings knotted in different ways to produce different elements. Although chemists lost interest in knots after the hypothesis of Kelvin turned out to be wrong, mathematicians became interested in the study of knots. There are two main applications of knot theory. One is in the study of statistical mechanics and quantum mechanics, the other is the study of the determination of chirality [1,2]. The first topological catenane was synthesized in 1960 [3]. From the time being, the domain of knot theory is fully appreciated by researchers in chemistry and biochemistry. So far, many exciting advances have been made in the discoveries and syntheses of diverse molecular knots [4–12]. Thus, novel challenge and opportunity are raised in the field of chemistry, particularly, the chirality problem of molecular knots. Helix theory [13–17] was developed in the use of molecular chirality. This ideology that a microstructure (possibly helical in nature) is the essence of optical activity was first *
Corresponding author. Tel.: +86 931 8911356; fax: +86 931 8912582. E-mail address: [email protected] (W.-Y. Qiu).
0022-2860/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.molstruc.2007.05.035
suggested by Fresnel [14] in 1827, which was then proved by Tinoco and Freeman [15] and Brewster [16] by experiment. A recent achievement that diverse molecular chiralities generalized on the basis of their inherent helicities has been made by Wang [17]. However, the molecules in the helix theory only include the rigid molecules having the classical stereogenic units (points, axes, helices, and planes), but not the nonplanar molecules, especially the molecular knots. Although some prominent results have been obtained [18–25], the investigation of molecular knot’s chirality still presents enormous difficulties because of the convoluted structures of molecular knots. Fortunately, the circular dichroism (CD) spectrum and X-ray results [4–6] show that helical structures can be found everywhere in the molecular knots, which are concerned to the molecular knots’ chirality and synthesis. Despite this knowledge, the basic factors between the helical structures and molecular knot’s chirality are little understood. As Dobrowolski and Mazurek put, the carbyne knot is intertwined by a sp-hybridized (C„CA)n chain which has been used to the study of the properties of DNA knots [26–28]. Among them, the carbyne trefoil is regarded as one of the simplest and most ideal models in the molecular knot field. A novel research field on the structure and
W.-Y. Deng, W.-Y. Qiu / Journal of Molecular Structure 875 (2008) 515–519
stability of the C30 trefoil has been explored, on which the inherently helical structure and nonplanar conjugated character are shown [29]. The main goal of this paper is to present a method for investigating the relations between the helical structures and chiralities of carbyne knots. The research firstly shows that helical chirality exists in the model left- and righthanded carbyne trefoils by the helix-cutting-reassembling method on the basis of their helical characters and vibrational circular dichroism (VCD) spectra [28–31]; the left–right classification and rotation strengths are also discussed. The significance of this method makes an important contribution to exploring and measuring the chiralities of molecular knots. 2. C30 carbyne trefoil knots In a knot projection, an arc is a segment between two adjacent (without break) under-crossings, and a knot can be viewed as an interrelated collection of arcs [1,18]. Trefoil is the simplest nontrivial knot with three arcs (a, b, and c) as shown in Fig. 1, bottom diagram. In 1914, Dehn [32] firstly proved that the trefoil cannot superimpose upon its mirror image, which is topologically chiral. The left- and right-handed C30 trefoils of our study both belong to D3 point-group of symmetry. Three arcs a, b, and c (Fig. 1, top diagram) in each of them are the same. The structures of the two carbyne trefoils possess three primary features: C„C nonplanar conjugated structure, inherently helical structure, and weak crossing bonds [29]. Although the electron structures of the two carbyne trefoils possess neither asymmetric carbon atom nor axis of chirality, the analysis of the C„C stretching vibrations region of the calculated VCD spectra (Fig. 2) show that these two trefoils are chiral. The most intense VCD m(C„C) bands of the two knots both appear at the 1916 cm1 wavenumber. Obviously, the two spectra exhibit opposite signs but the same values at every wavenumber, while one appears a positive rotatory strength the other has a negative one, vice versa. Globally, the spectral fea-
1500
VCD 1000 Right-handed trefoil 500 Strength
516
0
1800
1900
2000
2100
-500 Left-handed trefoil -1000
-1500
Wavenumbers, cm-1
Fig. 2. Comparison of the B3LYP /6-31G(d) [33] calculated VCD spectra of the left- and right-handed C30 trefoils in the 1700–2100 cm1 regions. The calculated spectra were simulated with Lorentzian band shapes and 5 cm1 half-width in regions. The frequencies were un-scaled.
tures present mirror-imaged symmetry, which suggests that the two C30 trefoils are enantiomers. In the absence of any chiral inducing agents, the synthesized trefoil knots always exist as a mixture of stereoisomers [4–6]. It is worth noting that [6]helicene [13,34], as one of the most successful demonstrations by helix theory, displays unique nonplanar conjugated and inherently helical chiral structure. These structural characters similar to the C30 trefoils [29] provide a critical insight with regard to the validity of helical chirality analysis for the carbyne trefoils. 3. Helical chirality According to the helix theory and the structures of C30 trefoils, a helix-cutting-reassembling method is applied to analyze the helical chiralities of the two trefoils. This approach can be mainly characterized by two operations: cutting and reassembling. Cutting involves the disconnection of the carbyne trefoil in the three trefoil arcs with the three crossings. The goal of the reassembling operation is to stick these three helical arcs together to form a net helicity. A detailed description of the approach for the right-handed trefoil can be found as follows. Step 1. Cutting (in Fig. 3B): Cutting the crossing bonds (C(8)AC(9),C(18)AC(19), and C(28)AC(29): marked with blue lines1), three equal 12-carbon (saving two end bonds) chains labeled with a, b, and c arcs (I) can be obtained, respectively. Then, rotating each arc clockwise about C2-axis of the
Fig. 1. The B3LYP/6-31G(d) [33] optimized structure (top) and projections (bottom) of the C30 trefoils and the arcs (a, b, and c): (A) left-handed trefoil and (B) right-handed trefoil.
1 For interpretation of colour, the reader is referred to the web version of this article.
W.-Y. Deng, W.-Y. Qiu / Journal of Molecular Structure 875 (2008) 515–519
517
Fig. 3. The process of helix-cutting of the C30 trefoils: (A) left-handed trefoil and (B) right-handed trefoil.
arc, the top view (II) and side view (III) can be observed. It is obvious that each arc twists into just a one-turn small helix, and there are three small right-handed helices. Note that if this process is reversed: sticking a, b, and c arcs in proper order, a right-handed trefoil can be rebuilt. Step 2. Reassembling (in Fig. 4B): Placing these arc chains in three cylinders, respectively, the helices could be more clearly viewed from IV. Finally, these helices can be reassembled into a big right-handed helix (V). Now, the right-handed trefoil has been transformed into a net right-handed helicity. This might account for its right-handed chirality. Similarly, the left-handed trefoil forms a net left-handed helicity, as illustrated in part A of Figs. 3 and 4, which has left-handed chirality. Analysis of the C„C stretching vibrations regions of the theoretical VCD spectra (Fig. 5) show that the chirality of the a arcs of the two C30 trefoils can be detected. At the same time, their VCD spectra exhibit mirror-imaged symmetry. Furthermore, the most intense VCD band of the arcs is nearly twice larger than that of the trefoils (mentioned above). It was reported that the VCD spectrum of trefoil molecule decreases with the increase of the molecular size [28]. It means that there are some quantitative relations between the chiralities of the carbyne trefoils and the chiralities of their arcs.
4. Rotation strength In this section, we will explore the totalizing characters of the chiralities of the carbyne trefoils, and predict their rotation strengths. As discussed previously [13–17,34], the optical intensities of the diverse helical structures are related merely to their helical parameters, such as radius (r), pitch (S), length (L), and polarizability. Here we will focus on the a helix of right-handed trefoil. In the view of structural property, the polarizability of the (C„CA)n is similar to the polycyclic aromatic system. Moreover, since the carbyne chain has an approximately curvilinear shape, and the helix length (L) is equal to the sum of the lengths of the carbon–carbon bonds in the arc, as shown in IV. With the following relationship: 2
L2 ¼ ð2prÞ þ S 2 ; The values of the parameters used are shown as below: ˚ , L = 14.69 A ˚ , and r = 2.31 A ˚. S = 2.27 A Then, the ratio S/L = 0.1546 < 0.5774, the optical intensity can be increased with the value of S/L [13]. Compared with the right-handed [6]helicene (Fig. 6) [13,33,34], a helix ˚ ), but they have the similar has a small pitch (2.27 < 3.81 A ˚ lengths (14.69 14.60 A) and approximate radius (2.31 ˚ ). It can be seen that a helix has a relatively smaller 2.24 A rotation than that of [6]helicene whose rotation is 12,000°. According to helix theory, totalizing a, b, and c helices is
518
W.-Y. Deng, W.-Y. Qiu / Journal of Molecular Structure 875 (2008) 515–519
Fig. 4. The process of helix-reassembling of the C30 trefoils: (A) left-handed trefoil and (B) right-handed trefoil.
3000
VCD 2000
a arc (Right-handed trefoil)
Strength
1000
0 1700
1800
1900
2000
2100
-1000
a arc (Left-handed trefoil) -2000
-3000
Wavenumber, cm-1
Fig. 5. Comparison of the B3LYP /6-31G(d) [33] calculated VCD spectra of the a arcs of left- and right-handed C30 trefoils in the 1700–2100 cm1 regions. The calculated spectra were simulated with Lorentzian band shapes and 5 cm1 half-width in regions. The frequencies were un-scaled.
Fig. 6. The B3LYP/6-31G(d) [33] optimized structure of the right-handed [6]helicene.
right-handed trefoil can be distinguished by the helical characters of their arcs. Additionally, if the carbyne trefoil knots belong to other symmetries (for example C1 or C2), the orientation of the net helicity cannot be changed: right-handed trefoil has right-handed chirality, and the left-handed one possesses left-handed chirality, but differs in their optical intensities. 5. Conclusions
the rotation of the whole molecular knot, which is larger than the right-handed [6]helicene and reaches 10,000– 30,000°. In the same way, left-handed trefoil has the same helical parameters but opposite orientation. It is remarkable that the helix-cutting-reassembling method appears to work well: it is easy to identify the chirality of the trefoil knots from their corresponding helicity. The results also show that the helicities of the two carbyne trefoils present opposite property, that is, the left- and
In summary, on the basis of the calculated molecular structures, a helix-cutting-reassembling method has been developed for the helical chirality analysis of the carbyne trefoils. The results of the helicity analysis and VCD spectra show that the left- and right-handed carbyne trefoils are mirror-images, which are composed of three chiral helices, respectively. The results of this study provide a simple and direct method for the investigation and classification
W.-Y. Deng, W.-Y. Qiu / Journal of Molecular Structure 875 (2008) 515–519
left–right chirality in molecular knots. This approach may expand and enhance the helix theory. Furthermore, it is well known that the molecular knots are very convoluted and some molecular knots are chiral first due to the sheer topology and then to their chemical components. Thus, this model is believed to benefit the complex molecules, such as DNA knots, protein knots and other self-assembly architectures. Acknowledgments This work was supported by grants from The National Natural Science Foundation of China (Nos. 20173023 and 90203012) and Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20020730006). References [1] C.C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, W.H. Freeman & Company, New York, 1994. [2] D. Rolfsen, Knots and Links, Publish or Perish Inc., Berkely, CA, 1976. [3] E. Wasserman, J. Am. Chem. Soc. 82 (1960) 4433. [4] (a) C.O. Dietrich-Buchecker, J.-P. Sauvage, Angew. Chem. 101 (1989) 192; (b) C.O. Dietrich-Buchecker, J.-P. Sauvage, Angew. Chem. Int. Ed. Engl. 28 (1989) 189. [5] S.M. Du, B.D. Stollar, N.C. Seeman, J. Am. Chem. Soc. 117 (1995) 1194. [6] O. Lukin, F. Vo¨gtle, Angew. Chem. Int. Ed. 44 (2005) 1456. [7] S.A. Wasserman, N.R. Cozzarelli, Proc. Natl. Acad. Sci. USA 82 (1985) 1079. [8] J.D. Griffith, H.A. Nash, Proc. Natl. Acad. Sci. USA 82 (1985) 3124. [9] C. Liang, K. Mislow, J. Am. Chem. Soc. 116 (1994) 11189. [10] W.R. Taylor, Nature 406 (2000) 916. [11] W.R. Taylor, K. Lin, Nature 421 (2003) 25. [12] H.-X. Zhou, J. Am. Chem. Soc. 125 (2003) 9280. [13] Y.-Y. Yin, C.-Y. Liu, The Helix theory on the Optical Activity of the Organic Compounds, Chemical Industry Press, Beijing, 2000. [14] A. Fresnel, Mem. Acad. Sci. 7 (1827) 45.
519
[15] I. Tinoco, M.P. Freeman, J. Phys. Chem. 61 (1957) 1196. [16] J.H. Brewster, J. Am. Chem. Soc. 81 (1959) 5475, 5483, 5493. [17] (a) D.Z. Wang, Mendeleev Commun. 14 (2004) 244; (b) D.Z. Wang, Tetrahedron 61 (2005) 7125; (c) D.Z. Wang, Tetrahedron 61 (2005) 7134; (d) D.Z. Wang, Chirality 17 (2005) S177. [18] E. Flapan, When Topology Meets Chemistry – A Topological Look at Molecular Chirality, Cambridge University Press, Cambridge, 2000. [19] W.-Y. Qiu, in: D. Bonchev, D.H. Rouvray (Eds.), Chemical Topology – Applications and Techniques, Mathematical Chemistry Series, vol. 6, Gordon and Breach Science Publishers, Amsterdam, 2000 (Chapter 3). [20] W.-Y. Qiu, H.-W. Xin, J. Mol. Struct. (Theochem) 401 (1997) 151. [21] W.-Y. Qiu, H.-W. Xin, J. Mol. Struct. (Theochem) 397 (1997) 33. [22] W.-Y. Qiu, H.-W. Xin, J. Mol. Struct. (Theochem) 428 (1998) 35. [23] W.-Y. Qiu, H.-W. Xin, J. Mol. Struct. (Theochem) 429 (1998) 81. [24] C. Liang, K. Mislow, J. Am. Chem. Soc. 116 (1994) 11189. [25] C. Liang, K. Mislow, J. Am. Chem. Soc. 117 (1995) 4201. [26] J.Cz. Dobrowolski, A.P. Mazurek, Int. J. Quant. Chem. 70 (1998) 1009. [27] J.Cz. Dobrowolski, A.P. Mazurek, J. Chem. Inf. Comput. Sci. 43 (2003) 861. [28] J.Cz. Dobrowolski, A.P. Mazurek, J. Mol. Struct. 563–564 (2001) 309. [29] W.-Y. Deng, W.-Y. Qiu, Acta Chim. Sin. 64 (2006) 2322 (in Chinese). [30] G. Holtzwarth, E.C. Hsu, H.S. Mosher, T.R. Faulkner, A. Moscowitz, J. Am. Chem. Soc. 96 (1974) 251. [31] M. Quack, Faraday Discuss. 99 (1994) 87. [32] M. Dehn, Math. Ann. 75 (1914) 402. [33] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, V.G. Zakrzewski, J.A. Montgomery Jr., R.E. Stratmann, J.C. Burant, S. Dapprich, J.M. Millam, A.D. Daniels, K.N. Kudin, M.C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G.A. Petersson, P.Y. Ayala, Q. Cui, K. Morokuma, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J. Cioslowski, J.V. Ortiz, A.G. Baboul, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R.L. Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, J.L. Andres, C. Gonzalez, M. Head-Gordon, E.S. Replogle, J.A. Pople, Gaussian 98, Revision A.9, Gaussian, Inc., Pittsburgh, PA, 1998. [34] K.P. Meurer, F. Vo¨gtle, Top. Curr. Chem. Springer-Verlag, Berlin, 127 (1985) 1.
Journal of Molecular Structure 875 (2008) 515–519 www.elsevier.com/locate/molstruc
Helical chirality in model mirror-imaged carbyne trefoil knots Wen-Ye Deng, Wen-Yuan Qiu
*
Department of Chemistry, State Key Laboratory of Applied Organic Chemistry, Lanzhou University, Lanzhou 730000, PR China Received 4 March 2007; received in revised form 22 May 2007; accepted 22 May 2007 Available online 2 June 2007
Abstract A new method for understanding the helical chirality of molecular knots has been developed on the basis of the helix theory. Our results show that the model right-handed carbyne trefoil knot is composed of three small right-handed helices with right-handed chirality, and the left-handed one has three small left-handed helices with left-handed chirality. These two trefoils present mirror-imaged symmetry. Moreover, the rotation strengths of the two carbyne trefoils have been suggested by an analogy to helicene. Helical chirality analysis leads us to infer that helical structure is the essence of molecular knot’s chirality, which is also a possible measure for the chirality of a molecular knot. Ó 2007 Elsevier B.V. All rights reserved. Keywords: Trefoil knot; Carbyne; Chirality; Helix theory; Mirror-image
1. Introduction Originally, knot theory began as a result of a hypothesis made by Lord Kelvin that atoms were rings knotted in different ways to produce different elements. Although chemists lost interest in knots after the hypothesis of Kelvin turned out to be wrong, mathematicians became interested in the study of knots. There are two main applications of knot theory. One is in the study of statistical mechanics and quantum mechanics, the other is the study of the determination of chirality [1,2]. The first topological catenane was synthesized in 1960 [3]. From the time being, the domain of knot theory is fully appreciated by researchers in chemistry and biochemistry. So far, many exciting advances have been made in the discoveries and syntheses of diverse molecular knots [4–12]. Thus, novel challenge and opportunity are raised in the field of chemistry, particularly, the chirality problem of molecular knots. Helix theory [13–17] was developed in the use of molecular chirality. This ideology that a microstructure (possibly helical in nature) is the essence of optical activity was first *
Corresponding author. Tel.: +86 931 8911356; fax: +86 931 8912582. E-mail address: [email protected] (W.-Y. Qiu).
0022-2860/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.molstruc.2007.05.035
suggested by Fresnel [14] in 1827, which was then proved by Tinoco and Freeman [15] and Brewster [16] by experiment. A recent achievement that diverse molecular chiralities generalized on the basis of their inherent helicities has been made by Wang [17]. However, the molecules in the helix theory only include the rigid molecules having the classical stereogenic units (points, axes, helices, and planes), but not the nonplanar molecules, especially the molecular knots. Although some prominent results have been obtained [18–25], the investigation of molecular knot’s chirality still presents enormous difficulties because of the convoluted structures of molecular knots. Fortunately, the circular dichroism (CD) spectrum and X-ray results [4–6] show that helical structures can be found everywhere in the molecular knots, which are concerned to the molecular knots’ chirality and synthesis. Despite this knowledge, the basic factors between the helical structures and molecular knot’s chirality are little understood. As Dobrowolski and Mazurek put, the carbyne knot is intertwined by a sp-hybridized (C„CA)n chain which has been used to the study of the properties of DNA knots [26–28]. Among them, the carbyne trefoil is regarded as one of the simplest and most ideal models in the molecular knot field. A novel research field on the structure and
W.-Y. Deng, W.-Y. Qiu / Journal of Molecular Structure 875 (2008) 515–519
stability of the C30 trefoil has been explored, on which the inherently helical structure and nonplanar conjugated character are shown [29]. The main goal of this paper is to present a method for investigating the relations between the helical structures and chiralities of carbyne knots. The research firstly shows that helical chirality exists in the model left- and righthanded carbyne trefoils by the helix-cutting-reassembling method on the basis of their helical characters and vibrational circular dichroism (VCD) spectra [28–31]; the left–right classification and rotation strengths are also discussed. The significance of this method makes an important contribution to exploring and measuring the chiralities of molecular knots. 2. C30 carbyne trefoil knots In a knot projection, an arc is a segment between two adjacent (without break) under-crossings, and a knot can be viewed as an interrelated collection of arcs [1,18]. Trefoil is the simplest nontrivial knot with three arcs (a, b, and c) as shown in Fig. 1, bottom diagram. In 1914, Dehn [32] firstly proved that the trefoil cannot superimpose upon its mirror image, which is topologically chiral. The left- and right-handed C30 trefoils of our study both belong to D3 point-group of symmetry. Three arcs a, b, and c (Fig. 1, top diagram) in each of them are the same. The structures of the two carbyne trefoils possess three primary features: C„C nonplanar conjugated structure, inherently helical structure, and weak crossing bonds [29]. Although the electron structures of the two carbyne trefoils possess neither asymmetric carbon atom nor axis of chirality, the analysis of the C„C stretching vibrations region of the calculated VCD spectra (Fig. 2) show that these two trefoils are chiral. The most intense VCD m(C„C) bands of the two knots both appear at the 1916 cm1 wavenumber. Obviously, the two spectra exhibit opposite signs but the same values at every wavenumber, while one appears a positive rotatory strength the other has a negative one, vice versa. Globally, the spectral fea-
1500
VCD 1000 Right-handed trefoil 500 Strength
516
0
1800
1900
2000
2100
-500 Left-handed trefoil -1000
-1500
Wavenumbers, cm-1
Fig. 2. Comparison of the B3LYP /6-31G(d) [33] calculated VCD spectra of the left- and right-handed C30 trefoils in the 1700–2100 cm1 regions. The calculated spectra were simulated with Lorentzian band shapes and 5 cm1 half-width in regions. The frequencies were un-scaled.
tures present mirror-imaged symmetry, which suggests that the two C30 trefoils are enantiomers. In the absence of any chiral inducing agents, the synthesized trefoil knots always exist as a mixture of stereoisomers [4–6]. It is worth noting that [6]helicene [13,34], as one of the most successful demonstrations by helix theory, displays unique nonplanar conjugated and inherently helical chiral structure. These structural characters similar to the C30 trefoils [29] provide a critical insight with regard to the validity of helical chirality analysis for the carbyne trefoils. 3. Helical chirality According to the helix theory and the structures of C30 trefoils, a helix-cutting-reassembling method is applied to analyze the helical chiralities of the two trefoils. This approach can be mainly characterized by two operations: cutting and reassembling. Cutting involves the disconnection of the carbyne trefoil in the three trefoil arcs with the three crossings. The goal of the reassembling operation is to stick these three helical arcs together to form a net helicity. A detailed description of the approach for the right-handed trefoil can be found as follows. Step 1. Cutting (in Fig. 3B): Cutting the crossing bonds (C(8)AC(9),C(18)AC(19), and C(28)AC(29): marked with blue lines1), three equal 12-carbon (saving two end bonds) chains labeled with a, b, and c arcs (I) can be obtained, respectively. Then, rotating each arc clockwise about C2-axis of the
Fig. 1. The B3LYP/6-31G(d) [33] optimized structure (top) and projections (bottom) of the C30 trefoils and the arcs (a, b, and c): (A) left-handed trefoil and (B) right-handed trefoil.
1 For interpretation of colour, the reader is referred to the web version of this article.
W.-Y. Deng, W.-Y. Qiu / Journal of Molecular Structure 875 (2008) 515–519
517
Fig. 3. The process of helix-cutting of the C30 trefoils: (A) left-handed trefoil and (B) right-handed trefoil.
arc, the top view (II) and side view (III) can be observed. It is obvious that each arc twists into just a one-turn small helix, and there are three small right-handed helices. Note that if this process is reversed: sticking a, b, and c arcs in proper order, a right-handed trefoil can be rebuilt. Step 2. Reassembling (in Fig. 4B): Placing these arc chains in three cylinders, respectively, the helices could be more clearly viewed from IV. Finally, these helices can be reassembled into a big right-handed helix (V). Now, the right-handed trefoil has been transformed into a net right-handed helicity. This might account for its right-handed chirality. Similarly, the left-handed trefoil forms a net left-handed helicity, as illustrated in part A of Figs. 3 and 4, which has left-handed chirality. Analysis of the C„C stretching vibrations regions of the theoretical VCD spectra (Fig. 5) show that the chirality of the a arcs of the two C30 trefoils can be detected. At the same time, their VCD spectra exhibit mirror-imaged symmetry. Furthermore, the most intense VCD band of the arcs is nearly twice larger than that of the trefoils (mentioned above). It was reported that the VCD spectrum of trefoil molecule decreases with the increase of the molecular size [28]. It means that there are some quantitative relations between the chiralities of the carbyne trefoils and the chiralities of their arcs.
4. Rotation strength In this section, we will explore the totalizing characters of the chiralities of the carbyne trefoils, and predict their rotation strengths. As discussed previously [13–17,34], the optical intensities of the diverse helical structures are related merely to their helical parameters, such as radius (r), pitch (S), length (L), and polarizability. Here we will focus on the a helix of right-handed trefoil. In the view of structural property, the polarizability of the (C„CA)n is similar to the polycyclic aromatic system. Moreover, since the carbyne chain has an approximately curvilinear shape, and the helix length (L) is equal to the sum of the lengths of the carbon–carbon bonds in the arc, as shown in IV. With the following relationship: 2
L2 ¼ ð2prÞ þ S 2 ; The values of the parameters used are shown as below: ˚ , L = 14.69 A ˚ , and r = 2.31 A ˚. S = 2.27 A Then, the ratio S/L = 0.1546 < 0.5774, the optical intensity can be increased with the value of S/L [13]. Compared with the right-handed [6]helicene (Fig. 6) [13,33,34], a helix ˚ ), but they have the similar has a small pitch (2.27 < 3.81 A ˚ lengths (14.69 14.60 A) and approximate radius (2.31 ˚ ). It can be seen that a helix has a relatively smaller 2.24 A rotation than that of [6]helicene whose rotation is 12,000°. According to helix theory, totalizing a, b, and c helices is
518
W.-Y. Deng, W.-Y. Qiu / Journal of Molecular Structure 875 (2008) 515–519
Fig. 4. The process of helix-reassembling of the C30 trefoils: (A) left-handed trefoil and (B) right-handed trefoil.
3000
VCD 2000
a arc (Right-handed trefoil)
Strength
1000
0 1700
1800
1900
2000
2100
-1000
a arc (Left-handed trefoil) -2000
-3000
Wavenumber, cm-1
Fig. 5. Comparison of the B3LYP /6-31G(d) [33] calculated VCD spectra of the a arcs of left- and right-handed C30 trefoils in the 1700–2100 cm1 regions. The calculated spectra were simulated with Lorentzian band shapes and 5 cm1 half-width in regions. The frequencies were un-scaled.
Fig. 6. The B3LYP/6-31G(d) [33] optimized structure of the right-handed [6]helicene.
right-handed trefoil can be distinguished by the helical characters of their arcs. Additionally, if the carbyne trefoil knots belong to other symmetries (for example C1 or C2), the orientation of the net helicity cannot be changed: right-handed trefoil has right-handed chirality, and the left-handed one possesses left-handed chirality, but differs in their optical intensities. 5. Conclusions
the rotation of the whole molecular knot, which is larger than the right-handed [6]helicene and reaches 10,000– 30,000°. In the same way, left-handed trefoil has the same helical parameters but opposite orientation. It is remarkable that the helix-cutting-reassembling method appears to work well: it is easy to identify the chirality of the trefoil knots from their corresponding helicity. The results also show that the helicities of the two carbyne trefoils present opposite property, that is, the left- and
In summary, on the basis of the calculated molecular structures, a helix-cutting-reassembling method has been developed for the helical chirality analysis of the carbyne trefoils. The results of the helicity analysis and VCD spectra show that the left- and right-handed carbyne trefoils are mirror-images, which are composed of three chiral helices, respectively. The results of this study provide a simple and direct method for the investigation and classification
W.-Y. Deng, W.-Y. Qiu / Journal of Molecular Structure 875 (2008) 515–519
left–right chirality in molecular knots. This approach may expand and enhance the helix theory. Furthermore, it is well known that the molecular knots are very convoluted and some molecular knots are chiral first due to the sheer topology and then to their chemical components. Thus, this model is believed to benefit the complex molecules, such as DNA knots, protein knots and other self-assembly architectures. Acknowledgments This work was supported by grants from The National Natural Science Foundation of China (Nos. 20173023 and 90203012) and Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20020730006). References [1] C.C. Adams, The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, W.H. Freeman & Company, New York, 1994. [2] D. Rolfsen, Knots and Links, Publish or Perish Inc., Berkely, CA, 1976. [3] E. Wasserman, J. Am. Chem. Soc. 82 (1960) 4433. [4] (a) C.O. Dietrich-Buchecker, J.-P. Sauvage, Angew. Chem. 101 (1989) 192; (b) C.O. Dietrich-Buchecker, J.-P. Sauvage, Angew. Chem. Int. Ed. Engl. 28 (1989) 189. [5] S.M. Du, B.D. Stollar, N.C. Seeman, J. Am. Chem. Soc. 117 (1995) 1194. [6] O. Lukin, F. Vo¨gtle, Angew. Chem. Int. Ed. 44 (2005) 1456. [7] S.A. Wasserman, N.R. Cozzarelli, Proc. Natl. Acad. Sci. USA 82 (1985) 1079. [8] J.D. Griffith, H.A. Nash, Proc. Natl. Acad. Sci. USA 82 (1985) 3124. [9] C. Liang, K. Mislow, J. Am. Chem. Soc. 116 (1994) 11189. [10] W.R. Taylor, Nature 406 (2000) 916. [11] W.R. Taylor, K. Lin, Nature 421 (2003) 25. [12] H.-X. Zhou, J. Am. Chem. Soc. 125 (2003) 9280. [13] Y.-Y. Yin, C.-Y. Liu, The Helix theory on the Optical Activity of the Organic Compounds, Chemical Industry Press, Beijing, 2000. [14] A. Fresnel, Mem. Acad. Sci. 7 (1827) 45.
519
[15] I. Tinoco, M.P. Freeman, J. Phys. Chem. 61 (1957) 1196. [16] J.H. Brewster, J. Am. Chem. Soc. 81 (1959) 5475, 5483, 5493. [17] (a) D.Z. Wang, Mendeleev Commun. 14 (2004) 244; (b) D.Z. Wang, Tetrahedron 61 (2005) 7125; (c) D.Z. Wang, Tetrahedron 61 (2005) 7134; (d) D.Z. Wang, Chirality 17 (2005) S177. [18] E. Flapan, When Topology Meets Chemistry – A Topological Look at Molecular Chirality, Cambridge University Press, Cambridge, 2000. [19] W.-Y. Qiu, in: D. Bonchev, D.H. Rouvray (Eds.), Chemical Topology – Applications and Techniques, Mathematical Chemistry Series, vol. 6, Gordon and Breach Science Publishers, Amsterdam, 2000 (Chapter 3). [20] W.-Y. Qiu, H.-W. Xin, J. Mol. Struct. (Theochem) 401 (1997) 151. [21] W.-Y. Qiu, H.-W. Xin, J. Mol. Struct. (Theochem) 397 (1997) 33. [22] W.-Y. Qiu, H.-W. Xin, J. Mol. Struct. (Theochem) 428 (1998) 35. [23] W.-Y. Qiu, H.-W. Xin, J. Mol. Struct. (Theochem) 429 (1998) 81. [24] C. Liang, K. Mislow, J. Am. Chem. Soc. 116 (1994) 11189. [25] C. Liang, K. Mislow, J. Am. Chem. Soc. 117 (1995) 4201. [26] J.Cz. Dobrowolski, A.P. Mazurek, Int. J. Quant. Chem. 70 (1998) 1009. [27] J.Cz. Dobrowolski, A.P. Mazurek, J. Chem. Inf. Comput. Sci. 43 (2003) 861. [28] J.Cz. Dobrowolski, A.P. Mazurek, J. Mol. Struct. 563–564 (2001) 309. [29] W.-Y. Deng, W.-Y. Qiu, Acta Chim. Sin. 64 (2006) 2322 (in Chinese). [30] G. Holtzwarth, E.C. Hsu, H.S. Mosher, T.R. Faulkner, A. Moscowitz, J. Am. Chem. Soc. 96 (1974) 251. [31] M. Quack, Faraday Discuss. 99 (1994) 87. [32] M. Dehn, Math. Ann. 75 (1914) 402. [33] M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, V.G. Zakrzewski, J.A. Montgomery Jr., R.E. Stratmann, J.C. Burant, S. Dapprich, J.M. Millam, A.D. Daniels, K.N. Kudin, M.C. Strain, O. Farkas, J. Tomasi, V. Barone, M. Cossi, R. Cammi, B. Mennucci, C. Pomelli, C. Adamo, S. Clifford, J. Ochterski, G.A. Petersson, P.Y. Ayala, Q. Cui, K. Morokuma, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J. Cioslowski, J.V. Ortiz, A.G. Baboul, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R. Gomperts, R.L. Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, J.L. Andres, C. Gonzalez, M. Head-Gordon, E.S. Replogle, J.A. Pople, Gaussian 98, Revision A.9, Gaussian, Inc., Pittsburgh, PA, 1998. [34] K.P. Meurer, F. Vo¨gtle, Top. Curr. Chem. Springer-Verlag, Berlin, 127 (1985) 1.