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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 Ewald summation ： ウィキペディア英語版 Ewald summation Ewald summation, named after Paul Peter Ewald, is a method for computing long-range interactions (e.g., Coulombic interactions) in periodic systems. It was first developed as the method for calculating electrostatic energies of ionic crystals, and is now commonly used for calculating long-range interactions in computational chemistry. Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space. In this method, the long-range interaction is divided into two parts: a short-range contribution, and a long-range contribution which does not have a singularity. The short-range contribution is calculated in real space, whereas the long-range contribution is calculated using a Fourier transform. The advantage of this method is the rapid convergence of the energy compared with that of a direct summation. This means that the method has high accuracy and reasonable speed when computing long-range interactions, and it is thus the de facto standard method for calculating long-range interactions in periodic systems. The method requires charge neutrality of the molecular system in order to calculate accurately the total Coulombic interaction. A study of the truncation errors introduced in the energy and force calculations of disordered point-charge systems is provided by Kolafa and Perram.〔 〕 ==Derivation== Ewald summation rewrites the interaction potential as the sum of two terms, :$\backslash varphi(\backslash mathbf)\; \backslash \; \backslash stackrel\backslash \; \backslash varphi\_(\backslash mathbf)\; +\; \backslash varphi\_(\backslash mathbf)$, where $\backslash varphi\_(\backslash mathbf)$ represents the short-range term whose sum quickly converges in real space and $\backslash varphi\_(\backslash mathbf)$ represents the long-range term whose sum quickly converges in Fourier (reciprocal) space. The long-ranged part should be finite for all arguments (most notably ''r'' = 0) but may have any convenient mathematical form, most typically a Gaussian distribution. The method assumes that the short-range part can be summed easily; hence, the problem becomes the summation of the long-range term. Due to the use of the Fourier sum, the method implicitly assumes that the system under study is infinitely periodic (a sensible assumption for the interiors of crystals). One repeating unit of this hypothetical periodic system is called a ''unit cell''. One such cell is chosen as the "central cell" for reference and the remaining cells are called ''images''. The long-range interaction energy is the sum of interaction energies between the charges of a central unit cell and all the charges of the lattice. Hence, it can be represented as a ''double'' integral over two charge density fields representing the fields of the unit cell and the crystal lattice :$$ E_ = \iint d\mathbf\, d\mathbf^\prime\, \rho_\text(\mathbf) \rho_(\mathbf^\prime) \ \varphi_(\mathbf - \mathbf^\prime) where the unit-cell charge density field $\backslash rho\_(\backslash mathbf)$ is a sum over the positions $\backslash mathbf\_k$ of the charges $q\_k$ in the central unit cell :$$ \rho_(\mathbf) \ \stackrel\ \sum_ q_k \delta(\mathbf - \mathbf_k) and the ''total'' charge density field $\backslash rho\_\backslash text(\backslash mathbf)$ is the same sum over the unit-cell charges $q\_$ and their periodic images :$$ \rho_\text(\mathbf) \ \stackrel\ \sum_ \sum_ q_k \delta(\mathbf - \mathbf_k - n_1 \mathbf_1 - n_2 \mathbf_2 - n_3 \mathbf_3) Here, $\backslash delta(\backslash mathbf)$ is the Dirac delta function, $\backslash mathbf\_1$, $\backslash mathbf\_2$ and $\backslash mathbf\_3$ are the lattice vectors and $n\_1$, $n\_2$ and $n\_3$ range over all integers. The total field $\backslash rho\_\backslash text(\backslash mathbf)$ can be represented as a convolution of $\backslash rho\_(\backslash mathbf)$ with a ''lattice function'' $L(\backslash mathbf)$ :$$ L(\mathbf) \ \stackrel\ \sum_ \delta(\mathbf - n_1 \mathbf_ - n_ \mathbf_2 - n_3 \mathbf_3) Since this is a convolution, the Fourier transformation of $\backslash rho\_\backslash text(\backslash mathbf)$ is a product :$$ \tilde_\text(\mathbf) = \tilde(\mathbf) \tilde_(\mathbf) where the Fourier transform of the lattice function is another sum over delta functions :$$ \tilde(\mathbf) = \frac \sum_ \delta(\mathbf - m_1 \mathbf_1 - m_2 \mathbf_2 - m_3 \mathbf_3) where the reciprocal space vectors are defined $\backslash mathbf\_\; \backslash \; \backslash stackrel\backslash \; \backslash frac\; \backslash times\; \backslash mathbf\_\}$ (and cyclic permutations) where $\backslash Omega\; \backslash \; \backslash stackrel\backslash \; \backslash mathbf\_\; \backslash cdot\; \backslash left(\; \backslash mathbf\_\; \backslash times\; \backslash mathbf\_\; \backslash right)$ is the volume of the central unit cell (if it is geometrically a parallelepiped, which is often but not necessarily the case). Note that both $L(\backslash mathbf)$ and $\backslash tilde(\backslash mathbf)$ are real, even functions. For brevity, define an effective single-particle potential :$$ v(\mathbf) \ \stackrel\ \int d\mathbf^\, \rho_(\mathbf^\prime) \ \varphi_(\mathbf - \mathbf^\prime) Since this is also a convolution, the Fourier transformation of the same equation is a product :$$ \tilde(\mathbf) \ \stackrel\ \tilde_(\mathbf) \tilde(\mathbf) where the Fourier transform is defined :$$ \tilde(\mathbf) = \int d\mathbf \ v(\mathbf) \ e^} The energy can now be written as a ''single'' field integral :$$ E_ = \int d\mathbf \ \rho_\text(\mathbf) \ v(\mathbf) Using Parseval's theorem, the energy can also be summed in Fourier space :$$ E_ = \int \frac \ \tilde_\text^ *(\mathbf) \tilde(\mathbf) = \int \frac \tilde^ *(\mathbf) \left| \tilde_(\mathbf)\right|^2 \tilde(\mathbf) = \frac \sum_ \left| \tilde_(\mathbf)\right|^2 \tilde(\mathbf) where $\backslash mathbf\; =\; m\_1\; \backslash mathbf\_1\; +\; m\_2\; \backslash mathbf\_2\; +\; m\_3\; \backslash mathbf\_3$ in the final summation. This is the essential result. Once $\backslash tilde\_(\backslash mathbf)$ is calculated, the summation/integration over $\backslash mathbf$ is straightforward and should converge quickly. The most common reason for lack of convergence is a poorly defined unit cell, which must be charge neutral to avoid infinite sums. 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「Ewald summation」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.