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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Möbius aromaticity ： ウィキペディア英語版 Möbius aromaticity In organic chemistry, Möbius aromaticity is a special type of aromaticity believed to exist in a number of organic molecules. In terms of molecular orbital theory these compounds have in common a monocyclic array of molecular orbitals in which there is an odd number of out-of-phase overlaps, the opposite pattern compared to the aromatic character to Hückel systems. The spatial configuration of the orbitals is reminiscent of a Möbius strip, hence the name. The smallest member of this class of compounds would be trans-benzene. Möbius molecular systems were considered in 1964 by Edgar Heilbronner by application of the Hückel method,〔''Hückel molecular orbitals of Möbius-type conformations of annulenes'' Tetrahedron Letters, Volume 5, Issue 29, 1964, Pages 1923-1928 E. Heilbronner 〕 but the first such compound was not synthesized until 2003 by the group of Rainer Herges.〔''Synthesis of a Möbius aromatic hydrocarbon'' D. Ajami, O. Oeckler, A. Simon, R. Herges Nature 426, 819-821 (18 December 2003) PMID 14685233〕 ==Ansatz Wavefunction & Hückel-Möbius Energy== For the Mobius geometry, the boundary conditions differ from the standard particle in a ring problem. Supposing to have a strip of length $L\_x$ and $L\_z$, we can see that ''general'' Mobius boundary conditions for the $\backslash psi$ wavefunction are: * $\backslash psi(x,0)=\backslash psi(x,L\_z)$ * $\backslash psi(0,z)=\backslash psi(L\_x,-z)$ or using the spherical azimuthal angle $\backslash phi$: :$\backslash psi(\backslash phi)=-\backslash psi(\backslash phi+2\backslash pi)$ For an $N$-carbons, the proposed ansatz LCAO wavefunction is: :$|\backslash rangle=\backslash sum\_^\backslash rangle=\backslash sum\_^\}\backslash rangle=\backslash sum\_^^\backslash lambda=-c\_j^\backslash lambda$ Using this equation and the Euler rule we can find the right $\backslash lambda$ value satisfying previous boundary conditions: :$e^=-e^$ :$e^=-1$ :$\backslash lambda\_k=\backslash frac\backslash ;\backslash ;\backslash ;\; k=0,1,2,\backslash ldots,(N-1)$ From the last equation we see that to fulfil the general boundary conditions, $\backslash lambda$ must be a half-integer number. The coefficients of the ansatz become: :$$ c_j^=e^ From figure above, it can also be seen that the overlap between two consecutive $p\_z$ AOs is at a constant angle $\backslash omega=\backslash pi/N$, and for this reason resonance integral $\backslash beta^\backslash prime$ it's considered as a constant into the Huckel matrix we will write later. It could be simply written as: :$\backslash beta^\backslash prime=\backslash beta\backslash cos(\backslash pi/N)$ where $\backslash beta$ is the standard Huckel’s resonance integral value (the one with $\backslash omega=0$). Nevertheless, the presence of a $C\_2$ axis as the only symmetry element brings to a full phase change at the end of the ring, e.i. between the first and the $N$-th carbon atoms. For this reason, in the Huckel matrix the resonance integral between carbon $1$ and $N$ is $-\backslash beta^\backslash prime$. For the generic $N$ carbons Mobius system, the Huckel matrix $\backslash mathbf$ is: :$$ \mathbf= \begin \alpha& \beta & 0 &\cdots& -\beta \\ \beta & \alpha& \beta & \cdots & 0 \\ 0 & \beta & \alpha & \cdots & 0 \\ \vdots &\vdots &\vdots &\ddots &\vdots \\ -\beta &0& 0 & \cdots & \alpha \end Eigenvalues equation can now be solved. Since $\backslash mathbf$ is a $N\backslash times\; N$ matrix, we will have $N$ eigenvalues $E\_k$ and $N$ MOs. Defining the variable :$x\_k=\backslash frac$ we have: :$$ \begin x_k& 1 & 0 &\cdots& -1 \\ 1 & x_k& 1 & \cdots & 0 \\ 0 & 1 & x_k & \cdots & 0 \\ \vdots &\vdots &\vdots &\ddots &\vdots \\ -1 &0& 0 & \cdots & x_k \end \cdot \begin c_1^ \\ c_2^ \\ c_3^ \\ \vdots\\ c_N^ \\ \end=0 Hence we obtain a system of $N$ equations, in which the first one ($k=0$) and the last one ($k=N-1$) have a $-1$ coefficient: :$$ \begin x_0c_1^+c_2^-c_N^=0\\ \vdots\\ c_^+x_kc_j^+c_^=0\\ \vdots\\ c_^+x_c_N^-c_1^=0 \end All these equations can be easily solved using Euler's rule, leading to :$$ x_k=-2\cos} hence :$$ E_k=\alpha+2\beta^\prime\cos}=\alpha+2\beta^\prime\cos} 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Möbius aromaticity」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.