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Knot-Isomers of Mo¨bius Cyclacene: How Does the Number of Knots Influence the Structure and First Hyperpolarizability? Hong-Liang Xu,† Zhi-Ru Li,‡ Zhong-Min Su,*,†,‡ Shabbir Muhammad,† Feng Long Gu,*,§ and Kikuo Harigaya| Institute of Functional Material Chemistry, Faculty of Chemistry, Northeast Normal UniVersity, Changchun 130024, Jilin, People’s Republic of China, State Key Laboratory of Theoretical and Computational Chemistry, Institute of Theoretical Chemistry, Jilin UniVersity, Changchun, 130023, China, Department of Molecular and Material Sciences, Faculty of Engineering Sciences, Kyushu UniVersity, 6-1 Kasuga-Park, Fukuoka, 816-8580, Japan, and Nanotechnology Research Institute, AIST, Umezono 1-1-1, Tsukuba, Ibaraki 305-8568, Japan ReceiVed: February 13, 2009; ReVised Manuscript ReceiVed: July 3, 2009
Four knot-isomers of Mo¨bius cyclacene are composed of 15 nitrogen-substituted benzo rings. They are nonMo¨bius cyclacenes without a knot (0), Mo¨bius cyclacenes with a knot (1), non-Mo¨bius cyclacenes with two knots (2), and Mo¨bius cyclacenes with three knots (3). Their structures and nonlinear optical properties are systematically studied. The order of first hyperpolarizability (β0) is 4693 (0) < 10 484 (2) < 25 419 (3) < 60 846 au (1). The β0 values of the knot-isomers with knot(s) are larger than that of the knot-isomer without a knot. It shows that the β0 value can be dramatically increased (13 times) by twisting the knot(s) to the cyclacene. Two noticeable relationships between the number of knots and the first hyperpolarizability have been observed: (i) the β0 values of one surface Mo¨bius cyclacene (1 and 3) with an odd number of knots are larger than that of two-surface non-Mo¨bius cyclacenes (0 and 2) with an even number of knots. (ii) For the one-surface Mo¨bius cyclacenes, the β0 value for 1 with one knot is larger than that for 3 with three knots. On the other hand, the largest component of β0 is alternated for the four knot-isomers. The largest components are βz for the 0 and βy for the 1 and 2. The largest component turns back to the βz for the 3. Introduction 1
Since the famous one-surface Mo¨bius strip was discovered by German mathematician Mo¨bius in 1858, the special structural2 and curious chemical and physical3 characteristics of one surface Mo¨bius strip with a knot have drawn extensive attention of the scientists. For example, for the aromaticity, the Hu¨ckel rules for aromaticity (4n + 2 electrons) are no longer valid for Mo¨bius annulenes, but the Mo¨bius ring with 4n π electrons is aromatic. For the magnetism, the unusual ring currents in Mo¨bius annulenes are also particularly interesting. Nonlinear optics4 (NLO) has developed very quickly in the past two decades. Much effort has been devoted to find the important influencing factors which can lead to a significant increase in the first hyperpolarizability and to design a new type of NLO materials. Theoretical investigations5 play an important role for the new high-performance NLO materials’ discovery, but the NLO properties for the Mo¨bius systems are seldom studied. The framework shape effect on the first hyperpolarizability has been investigated by comparing non-Mo¨bius cyclacenes and Mo¨bius cyclacenes which are composed by only seven nitrogensubstituted benzo rings.6 It was shown that by twisting a knot of non-Mo¨bius (normal) cyclacenes to form Mo¨bius cyclacenes the first hyperpolarizability is decreased from 1049 to 393 au. Can the first hyperpolarizability be increased when twisting a knot into non-Mo¨bius cyclacenes to form Mo¨bius cyclacenes? * Corresponding authors. E-mail: cube.kyushu-u.ac.jp. † Northeast Normal University. ‡ Jilin University. § Kyushu University. | Nanotechnology Research Institute.
[email protected];
gu@
TABLE 1: Dihedral Angles Cn-C-C-C (n ) 1, 2, 3 ... 15) for Knot-Isomers of Mo¨bius Cyclacene 0, 1, 2, and 3 Cn-C-C-Ca
0
1
2
3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.008 0.006 0.010 0.007 0.005 0.030 0.006 0.035 0.010 0.012 0.018 0.026 0.015 0.008 0.009
8.408 2.458 0.058 0.643 0.455 3.093 3.029 0.476 5.388 14.573 24.322 31.106 31.693 26.000 17.129
8.237 10.312 18.487 27.37 31.061 27.367 18.48 10.314 8.292 12.665 18.884 22.635 22.604 18.806 12.564
28.206 27.513 28.323 28.47 29.962 26.258 27.259 30.783 32.207 29.790 27.824 30.608 36.177 38.134 33.382
a
See Figure 1b.
To answer this question, four knot-isomers of large Mo¨bius cyclacene are chosen. The single, double, and triple twisted Mo¨bius [15]cyclacenes are reported by Henry S. Rzepa.7 Computational Details The optimized geometric structures of four large knot-isomers of Mo¨bius cyclacene with all real frequencies are obtained by using the density functional theory (DFT) B3LYP/6-31G(d) level. Champagne and Nakano pointed out that for a medium-size system, p-quinodimethane, the BHandHLYP method can also reproduce the (hyper)polarizability values from the more
10.1021/jp901358f CCC: $40.75 2009 American Chemical Society Published on Web 08/05/2009
Knot-Isomers of Mo¨bius Cyclacene
J. Phys. Chem. C, Vol. 113, No. 34, 2009 15381
Figure 1. Optimized structures of the non-Mo¨bius cyclacenes without a knot (0), the Mo¨bius cyclacenes with a knot (1), new non-Mo¨bius cyclacenes with two knots (2), and new Mo¨bius cyclacenes with three knots (3) nitrogen-substituted polyacenes (a). The dihedral angles Cn-C-C-C (n ) 1, 2, 3 ...15) (azury arrowhead) (b).
sophisticated single, double, and perturbative triple excitation coupled cluster [CCSD(T)].8 For the Mo¨bius cyclacenes with seven nitrogen-substituted benzo rings and relative systems,6 the satisfying results of BHandHLYP β0 value are obtained (see Supporting Information). Thus, the first (hyper)polarizabilities are evaluated for the four large knot-isomers of Mo¨bius cyclacene in the present work at the BHandHLYP/6-31+G(d) level. The transition energies (∆E) of 0, 1, 2, and 3 are estimated by the CIS method with the sto-3g basis set. The magnitude of the applied electric field is chosen as 0.001 au for the calculation of the (hyper)polarizabilities. The polarizability (R0) is defined as follows
1 R0 ) (Rxx + Ryy + Rzz) 3
(1)
The static first hyperpolarizability is noted as
β0 ) (β2x + β2y + βz2)1/2
(2)
where βi ) (3/5)(βiii + βijj + βikk), i, j, k ) x, y, z. All of the calculations were performed with the GAUSSIAN 03 program package.9 The dimensional plots of molecular orbitals were generated with the GaussView program.10 Results and Discussions A. Equilibrium Geometries. The optimized geometric structures of four knot-isomers of Mo¨bius cyclacene with all real frequencies are shown in Figure 1. The four knot-isomers of Mo¨bius cyclacene are named by the knot number (0, 1, 2, and 3), and the dihedral angles Cn-C-C-C (n ) 1, 2, 3 ... 15) are denoted in Figure 1b. In Figure 1, the nitrogen-substituted [15]cyclacene is the twosurface non-Mo¨bius cyclacene 0 without a knot. Twisting the
Figure 2. Dihedral angles Cn-C-C-C (n ) 1, 2, 3 ... 15) for Knotisomers of Mo¨bius cyclacene 0, 1, 2, and 3.
first knot, the one-surface Mo¨bius cyclacene 1 with one knot is formed. Further twisting the second knot, we obtained the twosurface non-Mo¨bius cyclacene 2 with two knots. Further twisting
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TABLE 2: Values of the r, β0, ∆E, f0, ∆µ, LUMO, and HOMO at the BHandHLYP/6-31+G(d) Level for Knot-Isomers of Mo¨bius Cyclacenes 0, 1, 2, and 3 R (au) βx (au) βy (au) βz (au) β0 (DFT, au) β0 (HF, au) f0 ∆E (eV) ∆µ (au)a HOMO (eV) LUMO (eV) εgap (eV) a
0
1
2
3
960 –29 –11 4693 4693 10082 0.4068 1.7149 0.537 –6.081 –3.944 2.137
1089 –7885 59949 6756 60845 22297 0.1218 1.2542 3.037 –5.709 –4.250 1.459
927 –37 10484 –2 10484 11302 0.1082 1.6535 2.965 –6.086 –3.994 2.092
1078 9886 4966 22885 25419 12298 0.2655 1.4230 1.206 –6.313 –4.299 2.014
1 au ) 2.542 D.
the third knot, the new one-surface Mo¨bius cyclacene 3 with three knots is obtained. How does the number of knots influence the structure? It is found that the knot number consists of the number of the dihedral angle peaks (see Figure 2). From Table 1 and Figure 2, the 0 has the same the dihedral angles (Cn-C-C-C to be 0.005-0.035°). Twisting the first knot to form one-surface Mo¨bius cyclacenes (1), a clear maximum (31.693°) is observed. Interestingly, a small bump (about 3°) of dihedral angle has been observed, which relates to the two N atoms in trans position (see molecule 1 in Figure 2). Twisting the second knot forms two-surface cyclacenes (2) without symmetry, and its two unequal maxima of dihedral angle, C5-C-C-C (31.061°) and C12-C-C-C (22.635°), are shown in Figure 2. The two related N-substituted benzo rings are localized in different areas of the molecule (see molecule 2 in Figure 2). Twisting the third knot forms another Mo¨bius cyclacene (3) without symmetry, and three dihedral angle peaks are unequal to be C4-C-C-C (29.962°) < C8-C-C-C (32.207°) < C13-C-C-C (38.134°). The N atom near the 38.134° is located outside of 3, and the N atoms near 32.207° and 29.962° are located inside of 3. While the 32.207° is larger than 29.962°, it may be due to 32.207° near two N atoms in the trans position (see molecule 3 in Figure 2). B. Static First Hyperpolarizability. The electric properties of 0, 1, 2, and 3 calculated at the BHandHLYP/6-31+G(d) level are given in Table 2. From Table 2, the order of polarizability (R0) is 927.17 (2) < 960.48 (0) < 1077.67 (3) < 1088.85 au (1). The knot number effect on the polarizability for knot-isomers of Mo¨bius cyclacene has shown that the R0 values (1088.85-
1077.67 au) of one-surface Mo¨bius cyclacenes (1 and 3) with an odd number of knots are larger than that (960.48-927.17 au) of two-surface non-Mo¨bius cyclacenes (0 and 2) with an even number of knots. Especially the relationships between the first hyperpolarizability and knot number are investigated. The order of β0 values is 4693 (0) < 10 484 (2) < 25 419 (3) < 60 846 au (1). Comparing these β0 values, we find that the β0 values (60 846 for 1, 10 484 for 2, and 25 419 au for 3) of the structures with knot(s) are larger than that (4693 au for 0) of the structure without knot(s) (see Figure 3a). It shows that the β0 value can be increased by twisting the knot(s) to the cyclacene, which is a new approach to enhance the first hyperpolarizability. Two noticeable relationships between the knot number and the first hyperpolarizability have been observed. (i) The β0 values (60 846 and 25 419 au) of one-surface Mo¨bius cyclacenes (1 and 3) with an odd number of knots are larger than those (4693 and 10 484 au) of two-surface nonMo¨bius cyclacenes (0 and 2) with an even number of knots. (ii) For the one-surface Mo¨bius cyclacenes, the β0 value (60 846) for 1 with one knot is larger than that (25 419 au) for 3 with three knots. Among the four knot-isomers of Mo¨bius cyclacenes, the onesurface Mo¨bius cyclacene with a knot has the largest β0 values (60 846 au). This β0 value is considerable, compared with reported values such as, for example, the known electrides (HCN)nLi,11a Li@calix[4]pyrrole11b (the range of the β0 values is 3385-15 682 au), and the large donor-acceptor polyenes systems12a (the range of the β0 values is 8818-152 502 au) as well as the organometallic system cis-[RuII(NH3)4(2-PymQ+)2][PF6]4 (34 487 au).12b To understand the interesting knot effect on the first hyperpolarizability, the two-state approximation13 is used.
β0 ∝
∆µ · f0 ∆E3
(3)
In the above expression, β0 is proportional to the difference of the dipole moment between the ground state and the crucial excited state (∆µ) and the oscillator strength (f0) but inversely proportional to the third power of the transition energy (∆E). The order of the ∆E is 1.7149 (0) > 1.2542 (1) < 1.6535 (2) > 1.4230 (3) eV. This order is related to the order of β0 values (see Figure 3b). Among the four knot-isomers of Mo¨bius cyclacene, the ∆E value (1.2542 for 1, 1.6535 for 2, and 1.4230 eV for 3) of structures with knot(s) is smaller than that (1.7149
Figure 3. Relationship (a) between the first hyperpolarizability and knot number and the relationship (b) between the crucial transition energy (∆E) and knot number.
Knot-Isomers of Mo¨bius Cyclacene
J. Phys. Chem. C, Vol. 113, No. 34, 2009 15383 Acknowledgment. This work was supported by the National Natural Science Foundation of China (No. 20773046 and 20703008), and Chang Jiang Scholars Program (2006), Program for Changjiang Scholars and Innovative Research Team in University (IRT0714), snd Science Foundation of Young Teachers of Northeast Normal University (20090402). Supporting Information Available: Complete ref 8 and optimized Cartesian coordinates for four big knot-isomers of Mo¨bius cyclacene. This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes
Figure 4. Crucial transitions of non-Mo¨bius cyclacenes (blue lines) and Mo¨bius cyclacenes (red lines).
eV for 0) of structures without a knot, and the ∆E of one-surface Mo¨bius cyclacenes (1 and 3) with an odd number of knots is smaller than that of two-surface cyclacene (0 and 2) with an even number of knots. For the one-surface Mo¨bius cyclacenes, the ∆E (1.2542) for 1 (one knot) is smaller than that (1.4230 eV) for 3 (three knots). From eq 3, the transition energies ∆E can well explain the order of the β0 values for 0, 1, 2, and 3. On the other hand, it is observed that the largest component of β0 is alternated for the four knot-isomers. The largest components are βz for the 0 and βy for the 1 and 2. The largest component turns back to the βz for the 3. It is found that the largest component of β0 relates to the direction of charge transfer in the crucial transition. The crucial transitions for knot-isomers of Mo¨bius cyclacenes are shown in Figure 4. From Figure 4, the directions of charge transfer in crucial transitions are in the z direction, and the largest components of β0 are βz for 0 and 3. The directions of charge transfer in crucial transitions are in the y direction, and the largest components of β0 are βy for 1 and 2. Conclusions In the present work, we have obtained a valuable description of the knot effect on the first hyperpolarizabilities for the four knot-isomers of big Mo¨bius cyclacene, twisting a knot into nonMo¨bius cyclacenes, and the β0 values are dramatically increased (13 times). It is a new approach to enhance the first hyperpolarizability. Two noticeable relationships between the knot number and the first hyperpolarizability have been observed. (i) The β0 values of Mo¨bius cyclacenes (1 and 3) with an odd number of knots are larger than those of non-Mo¨bius cyclacenes (0 and 2) with an even number of knots. (ii) For the Mo¨bius cyclacenes, the β0 value for 1 with one knot is larger than that for 3 with three knots. On the other hand, the largest component of β0 is alternated for the four knot-isomers. The largest components are βz for the 0 and βy for the 1 and 2. The largest component turns back to the βz for the 3.
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