Knot-isomers of Moebius Cyclacene: How Does the Number of Knots Influence the Structure and First Hyperpolarizability?
Hong-Liang Xu, Zhi-Ru Li, Zhong-Min Su, Feng Long Gu, Kikuo Harigaya
KKnot-isomers of Möbius Cyclacene: How Does the Number of Knots Influence the Structure and First Hyperpolarizability?
Hong-Liang Xu, Zhi-Ru Li, Zhong-Min Su,
Feng Long Gu, and Kikuo Harigaya Institute of Functional Material Chemistry, Faculty of Chemistry, Northeast Normal Uni v ersity, Changchun 130024, Jilin, People’s Republic of China, [email protected] 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; E-mail: [email protected] Nanotechnology Research Institute, AIST, Umezono 1-1-1, Tsukuba, Ibaraki 305-8568, Japan; E-mail: [email protected]
Abstract
Four large ring molecules composed by 15 nitrogen-substituted benzene rings, named as “knot-isomers of Möbius cyclacene”, i.e. non-Möbius cyclacenes without a knot ( ), Möbius cyclacenes with a knot ( ), non-Möbius cyclacenes with two knots ( ), and Möbius cyclacenes with three knots ( ), are systematically studied for their structures and nonlinear optical properties. The first hyperpolarizability ( β ) values of these four knot-isomers structures are 4693 ( ) < 10484 ( ) < 25419 (
3) < he β values (60846 for , 10484 for and 25419 au for ) of the knot-isomers with knot(s) are larger than that (4693 au for ) of the knot-isomer without a knot. It shows that the β value can be dramatically increases (13 times) by introducing the knot(s) to the cyclacenes structures. It is found that introducing knots to cyclacenes is a new means to enhance the first hyperpolarizability. Two noticeable relationships between the number of knots and the first hyperpolarizability have een observed. i). the β values (60846 for and 25419 au for ) of one surface Möbius cyclacene ( and ) with odd number of knots are larger than that (4693 for and 10484 for ) of two surfaces non-Möbius cyclacenes ( and ) with even number of knots. ii). For the one surface Möbius cyclacenes, the β value (60846) for with one knot is larger than that (25419 au) for with three knots. On the other hand, the largest component of β for the four knot-isomer of Möbius cyclacene is alternated among x, y and z for different number of knots. The largest component is β z for the , after twisting the with the first knot and the second knots, the largest component turns to β y for the and . The largest component turns back to the β z for the . Introduction
Since the famous one surface Möbius strip was discovered by German mathematician Möbius in 1858, the special structural and curious chemical and physical characteristics of one surface Möbius strip with a knot have drawn extensive attention of the scientists. For example, for the aromaticity, the Hückel rules for aromaticity (4n+2 electrons) are no longer valid for Möbius annulenes, but the Möbius ring with 4n π electrons is aromatic. For the magnetism, the unusual ring currents in Möbius annulenes are also particularly interesting. Nonlinear optics (NLO) develops 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 new type NLO materials. Theoretical investigations play an important role for the new high-performance NLO materials’ discovery. But the NLO properties for the Möbius systems are seldom studied. The framework shape effect on the first hyperpolarizability, however, has been investigated by omparing non-Möbius cyclacenes and Möbius cyclacenes. It was shown that twisting a knot of non-Möbius (normal) cyclacenes to form
Möbius cyclacenes the first hyperpolarizability is decreased from 1049 to 393 au. Whether the first hyperpolarizability can be increased when adding a knot into non-Möbius cyclacenes to form
Möbius cyclacenes? This work is trying to answer this question.
Computational Details
The optimized geometric structures of four large knot-isomers of Möbius 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 sophisticated the single, double, and perturbative triple excitation coupled-cluster [CCSD(T)]. For the Möbius cyclacenes with seven nitrogen-substituted benzene rings and relative systems, the satisfying results of BhandHlyp β value are obtained (see Supporting Information). Thus, the first (hyper)polarizabilities are evaluated for the four large knot-isomers of Möbius cyclacene in the present work at BhandHlyp/6-31+G(d) level. The magnitude of the applied electric field is chosen as 0.001 au for the calculation of the (hyper)polarizabilities. The polarizability ( α ) is defined as follows: )(31 zzyyxx αααα ++= (1) The static first hyperpolarizability is noted as: )( zyx ββββ ++= (2) where zyxkji ikkijjiiii ,,,,),(53 =++= ββββ . All of the calculations were performed with the GAUSSIAN 03 program package. The imensional plots of molecular orbitals were generated with the GaussView program. Results and Discussions
A. Equilibrium Geometries
The optimized geometric structures of four knot-isomers of Möbius cyclacene with all real frequencies are shown in Figure 1. The four knot-isomers of Möbius cyclacene are named by the knot number (
0, 1, 2 and ), and the dihedral angles C n -C-C-C (n=1, 2, 3… 15) are denoted in Figure 1a. In Figure 1, the fifteen nitrogen substituted [15]cyclacene is two surface non-Möbius cyclacene without a knot. Twisting the first knot, the one surface Möbius cyclacene with one knot is formed. Further twisting second knot, we obtained the two surfaces non-Möbius cyclacene with two knots. Further twisting third knot, the new one surface Möbius cyclacene with three knots is obtained. How does the number of knots influence the structure? It is found that the number of the dihedral angle C n -C-C-C peaks consists with the knot number (see Figure 2). From Table I and Figure 2, the has the same the dihedral angles C n -C-C-C (0.005~0.035°). Twisting first knot to form one surface Möbius cyclacenes ( ), the dihedral angle peak is formed on the C -C-C-C dihedral angle (31.693°). Twisting second knot form two surfaces cyclacenes ( ), two peaks are formed on the C -C-C-C (31.061°) and C -C-C-C dihedral angle (22.635°). Twisting third knot to form another Möbius cyclacenes ( ), three peaks are formed on the C -C-C-C (29.962°), C -C-C-C (32.207°) and C -C-C-C dihedral angle (38.134°). B. The Static First Hyperpolarizabilities
The electric properties of
0, 1, 2 and calculated at the BHandHLYP/6-31+G(d) level are given in Table II. From Table II, the order of polarizability ( α ) is 927.17 ( ) < 960.48 ( ) < 1077.67 ( ) < ) au. The knot number effect on the polarizability for knot-isomers of Möbius cyclacene is shown: the α values (1088.85 for and 1077.67 au for ) of one surface Möbius cyclacene ( and ) with odd number of knots are larger than that (960.48 for and 927.17 au for ) of two surfaces non-Möbius cyclacenes ( and ) with even number of knots. Especially, the relationships between the first hyperpolarizability and knot number are investigated. The order of β values is 4693 ( ) < 10484 ( ) < 25419 ( ) < ) . Comparing these β values, we find that the β values (60846 for , 10484 for and 25419 au for ) of the structures with knot(s) are larger than that (4693 au for ) of the structure without knot (see Figure 3). It shows that the β value can be increased by introducing the knot(s) to the cyclacene, which is new factor to enhance the first hyperpolarizability. Two noticeable relationships between the knot number and the first hyperpolarizability have been observed. i). The β values (60846 and 25419 au) of one surface Möbius cyclacenes ( and ) with odd number of knots are large that (4693 and 10484 au) of two surfaces non-Möbius cyclacenes ( and ) with even number of knots. ii). For the one surface Möbius cyclacenes, the β value (60846) for with one knot is larger than that (25419 au) for with three knots, which shows that the structure with small knot number of has large β value. Among the four knot-isomers of Möbius cyclacenes, the one surface Möbius cyclacene with a knot has the largest β values (60846 au), which can compare with that of other high NLO systems. For example, the known electrides (HCN) n Li , Li@calix[4]pyrrole (the range of the β values is 3385 ~ 15682 a.u.), and the large donor-acceptor polyenes systems (the range of the β values is 8818 ~ cis -[RuII(NH ) (2-PymQ + ) ][PF ] (34487 a.u). The first hyperpolarizability can be estimated by the two-state approximation, gegege E μμβ Δ∝ , ggeege μμμ −= (3) where the subscript “g” indicates the ground state and the subscript “e” indicates the charge-transfer excited state. Δ μ ge is the dipole moment difference, μ ge is the transition dipole moment, and E ge is the transition energy. From the two-level expression (eq. 3), it is obvious that the transition energy is the decisive factor in the first hyperpolarizability. A simple approximation is to represent the ground and excited states using the HOMO (H) and LUMO (L) orbital energies, respectively, so the E ge is denoted ε gap in the present work. From Eq 3, the β is inversely proportional to the second power of the transition energy ε gap . The order of the ε gap is 2.137 ( ) > 1.459 ( ) < 2.092 ( ) > 2.014 ( ) eV, which explains the relationship between the knot number and β . Among the four knot-isomers of Möbius cyclacene, the ε gap value (1.459 for , 2.092 for and 2.014 eV for ) of structures with knot(s) are smaller than that (2.137 eV for ) of structure without knot. And the ε gap of one surface Möbius cyclacene ( and ) with odd number of knots are smaller than that of two surfaces cyclacene ( and ) with even number of knots. For the one surface Möbius cyclacenes, the ε gap (1.459) for (one knot) smaller than that (2.014 eV) for (three knots). Interestingly, Figure 3 shows the shape of the ε curve is very similar to that of β curve. On the other hand, it is observed that the largest component of β for the four knot-isomer of Möbius cyclacene is alternated among x, y and z for different number of knots. The largest component is β z for the , after twisting the with the first knot and the second knots, the largest component turns to β y for the and . The largest component turns back to the β z for the . rom Eq 3, the largest component o f Δ μ ge is related to the largest component of β . From Figure 4, the HOMO of is mainly located on the un-substituted side, and LUMO of is mainly located on the N-substituted side. Thus, for the HOMO-LUMO transition of , the charge transfer and Δ μ ge along with the z-axis from un-substituted side to the N-substituted side. So the largest component of β is β z . While For the HOMO is mainly located on the side of positive y-axis, and LUMO is mainly located on the side of negative y-axis. The transition of , the charge transfer and Δ μ ge from side of positive y-axis to the side of negative y-axis, and thus the largest component of β is β y . Similar to , the largest component of β for is also β y . Analogically to , the largest component of β for is β z . Conclusions
In the present work, we have obtained a valuable description of the knot effect on the static (hyper)polarizabilities for the four big knot-isomers of Möbius cyclacene, the β values (60846 for , 10484 for and 25419 au for ) of isomer cyclacenes with knot(s) are larger than that (4693 au for ) of isomer cyclacenes without a knot. It shows that the β value can be increased by twisting the knot(s), which is a new factor to enhance the first hyperpolarizability. Two noticeable relationships between the knot number and the first hyperpolarizability have been observed. i). The β values (60846 for and 25419 au for ) of one surface Möbius cyclacene ( and ) with odd knot are large that (4693 for and 10484 for ) of two surfaces non-Möbius cyclacenes ( and ) with even knot. ii). For the one surface Möbius cyclacenes, the β value (60846) for with one knot larger than that (25419 au) for with three knots. On the other hand, the largest component of β for the four knot-isomer of Möbius cyclacene can be turning with changing of knot number: the largest component is β z for the , after twisting the with the first knot and the second knots, the largest component turns to β y for the and . The largest omponent turns back to the β z for the . As a result, our investigation may offer new way to design new NLO compounds. Acknowledgment
This work was supported by the National Natural Science Foundation of China (No. 20573043, 20773046 and 20703008), and
Chang Jiang Scholars Program (2006), Program for Changjiang Scholars and Innovative Research Team in University (IRT0714).
Supporting Information Available:
Complete ref 8 and optimized Cartesian coordinates for four big knot-isomers of Möbius cyclacene. This material is available free of charge via the Internet at http://pubs.acs.org.
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0, 1, 2 and C n -C-C-C a
0 1 2 3
1 0.008 8.408 8.237 28.2062 0.006 2.458 10.312 27.5133 0.010 0.061 18.487 28.3234 0.007 0.676 27.37 28.47 5 0.005 0.455 31.061 29.9626 0.030 3.093 27.367 27.9037 0.006 3.029 18.48 27.2598 0.035 0.476 10.314 30.7839 0.010 5.388 8.292 32.20710 0.012 14.573 12.665 29.79011 0.018 24.322 18.884 27.82412 0.026 31.106 22.635 30.60813 0.015 31.693 22.604 36.17714 0.008 26.000 18.806 38.13415 0.009 17.129 12.564 33.382 a see Figure 1a. able II. The values of the α , β , LUMO and HOMO at BhandhLYP/6-31+G(d) level for Knot-isomers of Möbius Cyclacene
0, 1, 2 and
0 1 2 3 α (au )
960 1089 927 1078 β x (au ) -29 -7885 -37 9886 β y (au ) -11 59949 10484 4966 β z (au ) β (au ) HOMO (eV) -6.081 -5.709 -6.086 -6.313
LUMO (eV) -3.944 -4.250 -3.994 -4.299 ε gap (eV) / ε gap (au ) a b Figure 1.
The dihedral angles C n -C-C-C (n=1, 2, 3… 15) (azury arrowhead) ( a ). The optimized structures of the the non-Möbius cyclacenes without a knot ( ), the Möbius cyclacenes with a knot ( ), new non-Möbius cyclacenes with two knots ( ) and new Möbius cyclacenes with three knots ( ) nitrogen-substituted polyacenes ( b ). Figure 2. Th e dihedral angles C n -C-C-C (n=1, 2, 3…15) for Knot-isomers of Möbius Cyclacene
0, 1, 2 and a b Figure 3.
The relationship ( a ) between the first hyperpolarizability and knot number, the relationship ( b ) between the reciprocal of ε gap and knot number. Figure 4.