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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 Diffusion Monte Carlo ： ウィキペディア英語版 Diffusion Monte Carlo Diffusion Monte Carlo (DMC) is a quantum Monte Carlo method that uses a Green's function to solve the Schrödinger equation. DMC is potentially numerically exact, meaning that it can find the exact ground state energy within a given error for any quantum system. When actually attempting the calculation, one finds that for bosons, the algorithm scales as a polynomial with the system size, but for fermions, DMC scales exponentially with the system size. This makes exact large-scale DMC simulations for fermions impossible; however, DMC employing a clever approximation known as the fixed-node approximation can still yield very accurate results.〔() J. Anderson, J. Chem. Phys. 65, 4121 (1976)〕 What follows is an explanation of the basic algorithm, how it works, why fermions cause a problem, and how the fixed-node approximation resolves this problem. == The Projector Method == To motivate the algorithm, let's look at the Schrödinger equation for a particle in some potential in one dimension: :$i\backslash frac=-\backslash frac\backslash frac\; +\; V(x)\backslash Psi(x,t).$ We can condense the notation a bit by writing it in terms of an ''operator'' equation, with :$H=-\backslash frac\backslash frac\; +\; V(x)$. So then we have :$i\backslash frac=H\backslash Psi(x,t),$ where we have to keep in mind that H is an operator, not a simple number or function. There are special functions, called eigenfunctions, for which $H\backslash Psi(x)=E\backslash Psi(x)$, where E is a number. These functions are special because no matter where we evaluate the action of the H operator on the wave function, we always get the same number E. These functions are called stationary states, because the time derivative at any point x is always the same, so the amplitude of the wave function never changes in time. Since the overall phase of a wave function is not measurable, the system does not change in time. We are usually interested in the wave function with the lowest energy eigenvalue, the ground state. We're going to write a slightly different version of the Schrödinger equation that will have the same energy eigenvalue, but, instead of being oscillatory, it will be convergent. Here it is: :$-\backslash frac=(H-E\_0)\backslash Psi(x,t)$. We've removed the imaginary number from the time derivative and added in a constant offset of $E\_0$, which is the ground state energy. We don't actually know the ground state energy, but there will be a way to determine it self-consistently which we'll introduce later. Our modified equation(some people call it the imaginary-time Schrödinger equation) has some nice properties. The first thing to notice is that if we happen to guess the ground state wave function, then $H\backslash Phi\_0(x)=E\_0\backslash Phi\_0(x)$ and the time derivative is zero. Now suppose that we start with another wave function($\backslash Psi$), which is not the ground state but is not orthogonal to it. Then we can write it as a linear sum of eigenfunctions: :$\backslash Psi=c\_0\backslash Phi\_0+\backslash sum\_^\backslash infty\; c\_i\backslash Phi\_i$ Since this is a linear differential equation, we can look at the action of each part separately. We already determined that $\backslash Phi\_0$ is stationary. Suppose we take $\backslash Phi\_1$. Since $\backslash Phi\_0$ is the lowest-energy eigenfunction, the associate eigenvalue of $\backslash Phi\_1$ satisfies the property $E\_1\; >\; E\_0$. Thus the time derivative of $c\_1$ is negative, and will eventually go to zero, leaving us with only the ground state. This observation also gives us a way to determine $E\_0$. We watch the amplitude of the wave function as we propagate through time. If it increases, then decrease the estimation of the offset energy. If the amplitude decreases, then increase the estimate of the offset energy. 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「Diffusion Monte Carlo」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.