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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 Eckart conditions ： ウィキペディア英語版 Eckart conditions The Eckart conditions, named after Carl Eckart, sometimes referred to as Sayvetz conditions, simplify the nuclear motion (rovibrational) Schrödinger equation that arises in the second step of the Born–Oppenheimer approximation. The Eckart conditions allow to a large extent the separation of the external (rotation and translation) motions from the internal (vibration) motions. Although the rotational and vibrational motions of the nuclei in a molecule cannot be fully separated, the Eckart conditions minimize the coupling between these two. ==Definition of Eckart conditions== The Eckart conditions can only be formulated for a semi-rigid molecule, which is a molecule with a potential energy surface ''V''(R1, R2,..R''N'') that has a well-defined minimum for R''A''0 ($A=1,\backslash ldots,\; N$). These equilibrium coordinates of the nuclei—with masses ''M''''A''—are expressed with respect to a fixed orthonormal principal axes frame and hence satisfy the relations :$$ \sum_^N M_A\,\big(\delta_|\mathbf_A^0|^2 - R^0_ R^0_\big) = \lambda^0_i \delta_ \quad\mathrm\quad \sum_^N M_A \mathbf_A^0 = \mathbf. Here λi0 is a principal inertia moment of the equilibrium molecule. The triplets R''A''0 = (''R''''A''10, ''R''''A''20, ''R''''A''30) satisfying these conditions, enter the theory as a given set of real constants. Following Biedenharn and Louck we introduce an orthonormal body-fixed frame, the ''Eckart frame'', :$\backslash vec\_1,\; \backslash vec\_2,\; \backslash vec\_3\backslash \}$. If we were tied to the Eckart frame, which—following the molecule—rotates and translates in space, we would observe the molecule in its equilibrium geometry when we would draw the nuclei at the points, :$$ \vec_A^0 \equiv \vec_A^0 =\sum_^3 \vec_i\, R^0_,\quad A=1,\ldots,N . Let the elements of R''A'' be the coordinates with respect to the Eckart frame of the position vector of nucleus ''A'' ($A=1,\backslash ldots,\; N$). Since we take the origin of the Eckart frame in the instantaneous center of mass, the following relation :$$ \sum_A M_A \mathbf_A = \mathbf holds. We define ''displacement coordinates'' :$\backslash mathbf\_A\backslash equiv\backslash mathbf\_A-\backslash mathbf^0\_A$. Clearly the displacement coordinates satisfy the translational Eckart conditions, :$$ \sum_^N M_A \mathbf_A = 0 . The rotational Eckart conditions for the displacements are: :$$ \sum_^N M_A \mathbf^0_A \times \mathbf_A = 0, where $\backslash times$ indicates a vector product. These rotational conditions follow from the specific construction of the Eckart frame, see Biedenharn and Louck, ''loc. cit.'', page 538. Finally, for a better understanding of the Eckart frame it may be useful to remark that it becomes a principal axes frame in the case that the molecule is a rigid rotor, that is, when all ''N'' displacement vectors are zero. 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「Eckart conditions」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.