# 耦合簇方法

## 波函数拟设

$\hat{H} \vert{\Psi}\rangle = E \vert{\Psi}\rangle$

$\vert{\Psi}\rangle = e^{\hat{T}} \vert{\Phi_0}\rangle$

## 簇算符

$\hat{T}=\hat{T}_1 + \hat{T}_2 + \hat{T}_3 + \cdots$

$\hat{T}_1=\sum_{i}\sum_{a} t_{i}^{a} \hat{a}^{\dagger}_{a}\hat{a}_{i},$
$\hat{T}_2=\frac{1}{4}\sum_{i,j}\sum_{a,b} t_{ij}^{ab} \hat{a}^{\dagger}_{a}\hat{a}^{\dagger}_{b}\hat{a}_j\hat{a}_{i},$

$e^{\hat{T}} = 1 + \hat{T} + \frac{\hat{T}^2}{2!} + \cdots = 1 + \hat{T}_1 + \hat{T}_2 + \frac{\hat{T}_1^2}{2} + \hat{T}_1\hat{T}_2 + \frac{\hat{T}_2^2}{2} + \cdots$

$\hat{T} = \hat{T}_1 + ... + \hat{T}_n$

## 耦合簇方程

$\hat{H} e^{\hat{T}} \vert{\Psi_0}\rangle = E e^{\hat{T}} \vert {\Psi_0}\rangle$

$\langle {\Psi^{*}}\vert \hat{H} e^{\hat{T}} \vert{\Psi_0}\rangle = E \langle {\Psi^{*}} \vert e^{\hat{T}} \vert {\Psi_0}\rangle$

$\langle {\Psi_0}\vert e^{-\hat{T}} \hat{H} e^{\hat{T}} \vert{\Psi_0}\rangle = E$
$\langle {\Psi^{*}}\vert e^{-\hat{T}} \hat{H} e^{\hat{T}} \vert{\Psi_0}\rangle = E \langle {\Psi^{*}}\vert e^{-\hat{T}} e^{\hat{T}} \vert{\Psi_0}\rangle = 0$

$\langle {\Psi_0}\vert e^{-(\hat{T}_1+\hat{T}_2)} \hat{H} e^{(\hat{T}_1+\hat{T}_2)} \vert{\Psi_0}\rangle = E$
$\langle {\Psi_{S}}\vert e^{-(\hat{T}_1+\hat{T}_2)} \hat{H} e^{(\hat{T}_1+\hat{T}_2)} \vert{\Psi_0}\rangle =0$
$\langle {\Psi_{D}}\vert e^{-(\hat{T}_1+\hat{T}_2)} \hat{H} e^{(\hat{T}_1+\hat{T}_2)} \vert{\Psi_0}\rangle =0$

$\bar{H} = e^{-\hat{T}} \hat{H} e^{\hat{T}} = \hat H + \left[\hat H, \hat T\right] + \frac12\left[\left[\hat H,\hat T\right],\hat T\right] + \cdots$

$\bar{H}$ 不是厄米的。

## 耦合簇方法的种类

1. S - 单激发 (在英语的 CC 术语里面简称 singles)
2. D - 双激发 (doubles)
3. T - 三激发 (triples)

$T = \hat{T}_1 + \hat{T}_2 + \hat{T}_3.$

1. 耦合簇方法
2. 包含完整的单激发和双激发
3. 三激发则采用微扰理论而不是迭代求解

## 參考文獻

1. ^ Kümmel, H. G. A biography of the coupled cluster method. (编) Xian, R. F.; Brandes, T.; Gernoth, K. A. et al. Recent progress in many-body theories Proceedings of the 11th international conference. Singapore: World Scientific Publishing. 2002: 334–348. ISBN 978-981-02-4888-8. |editor1-last=|editor-last=只需其一 (帮助);
2. ^ Cramer, Christopher J. Essentials of Computational Chemistry. Chichester: John Wiley & Sons, Ltd. 2002: 191–232. ISBN 0-471-48552-7.
3. ^ Shavitt, Isaiah; Bartlett, Rodney J. Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory. Cambridge University Press. 2009. ISBN 978-0-521-81832-2.
4. ^ Koch, Henrik; Jo̸rgensen, Poul. Coupled cluster response functions. The Journal of Chemical Physics. 1990, 93: 3333. Bibcode:1990JChPh..93.3333K. doi:10.1063/1.458814.
5. ^ Stanton, John F.; Bartlett, Rodney J. The equation of motion coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties. The Journal of Chemical Physics. 1993, 98: 7029. Bibcode:1993JChPh..98.7029S. doi:10.1063/1.464746.
6. ^ The Cluster Operator. [2012-06-24].