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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 Projector augmented wave method ： ウィキペディア英語版 Projector augmented wave method The projector augmented wave method (PAW) is a technique used in ab initio electronic structure calculations. It is a generalization of the pseudopotential and linear augmented-plane-wave methods, and allows for density functional theory calculations to be performed with greater computational efficiency. Valence wavefunctions tend to have rapid oscillations near ion cores due to the requirement that they be orthogonal to core states; this situation is problematic because it requires many Fourier components (or in the case of grid-based methods, a very fine mesh) to describe the wavefunctions accurately. The PAW approach addresses this issue by transforming these rapidly oscillating wavefunctions into smooth wavefunctions which are more computationally convenient, and provides a way to calculate all-electron properties from these smooth wavefunctions. This approach is somewhat reminiscent of a change from the Schrödinger picture to the Heisenberg picture. == Transforming the wavefunction == The linear transformation $\backslash mathcal$ transforms the fictitious pseudo wavefunction $|\backslash tilde\backslash rangle$ to the all-electron wavefunction $|\backslash Psi\backslash rangle$: :$|\backslash Psi\backslash rangle=\backslash mathcal|\backslash tilde\backslash rangle$ Note that the "all-electron" wavefunction is a Kohn-Sham single particle wavefunction, and should not be confused with the many-body wavefunction. In order to have $|\backslash tilde\backslash rangle$ and $|\backslash Psi\backslash rangle$ differ only in the regions near the ion cores, we write :$\backslash mathcal=1+\backslash sum\_R\backslash hat\}\_R$ is non-zero only within some spherical augmentation region $\backslash Omega\_R$ enclosing atom $R$. Around each atom, it is useful to expand the pseudo wavefunction into pseudo partial waves: :$|\backslash tilde\backslash rangle=\backslash sum\_i|\backslash tilde\_i\backslash rangle\; c\_i$ within $\backslash Omega\_R$. Because the operator $\backslash mathcal$ is linear, the coefficients $c\_i$ can be written as an inner product with a set of so-called projector functions, $|p\_i\backslash rangle$: :$c\_i=\backslash langle\; p\_i|\backslash tilde\backslash rangle$ where $\backslash langle\; p\_i|\backslash tilde\_j\backslash rangle=\backslash delta\_$. The all-electron partial waves, $|\backslash phi\_i\backslash rangle=\backslash mathcal|\backslash tilde\_i\backslash rangle$, are typically chosen to be solutions to the Kohn-Sham Schrödinger equation for an isolated atom. The transformation $\backslash mathcal$ is thus specified by three quantities: # a set of all-electron partial waves $|\backslash phi\_i\backslash rangle$ # a set of pseudo partial waves $|\backslash tilde\_i\backslash rangle$ # a set of projector functions $|p\_i\backslash rangle$ and we can explicitly write it down as :$\backslash mathcal\; =\; 1\; +\; \backslash sum\_i\; \backslash left(\; |\; \backslash phi\_i\; \backslash rangle\; -\; |\; \backslash tilde\_i\; \backslash rangle\; \backslash right)\; \backslash langle\; p\_i\; |$ Outside the augmentation regions, the pseudo partial waves are equal to the all-electron partial waves. Inside the spheres, they can be any smooth continuation, such as a linear combination of polynomials or Bessel functions. The PAW method is typically combined with the frozen core approximation, in which the core states are assumed to be unaffected by the ion's environment. There are several online repositories of pre-computed atomic PAW data.〔(【引用サイトリンク】title=PAW atomic data for ABINIT code )〕〔(【引用サイトリンク】title=Periodic Table of the Elements for PAW Functions )〕〔(【引用サイトリンク】title=Atomic PAW Setups )〕 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「Projector augmented wave method」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.