耦合簇方法

（重定向自偶合簇

波函数拟设

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

${\displaystyle \vert {\Psi }\rangle =e^{\hat {T}}\vert {\Phi _{0}}\rangle }$

簇算符

${\displaystyle {\hat {T}}={\hat {T}}_{1}+{\hat {T}}_{2}+{\hat {T}}_{3}+\cdots }$

${\displaystyle {\hat {T}}_{1}=\sum _{i}\sum _{a}t_{i}^{a}{\hat {a}}_{a}^{\dagger }{\hat {a}}_{i},}$
${\displaystyle {\hat {T}}_{2}={\frac {1}{4}}\sum _{i,j}\sum _{a,b}t_{ij}^{ab}{\hat {a}}_{a}^{\dagger }{\hat {a}}_{b}^{\dagger }{\hat {a}}_{j}{\hat {a}}_{i},}$

${\displaystyle e^{\hat {T}}=1+{\hat {T}}+{\frac {{\hat {T}}^{2}}{2!}}+\cdots =1+{\hat {T}}_{1}+{\hat {T}}_{2}+{\frac {{\hat {T}}_{1}^{2}}{2}}+{\hat {T}}_{1}{\hat {T}}_{2}+{\frac {{\hat {T}}_{2}^{2}}{2}}+\cdots }$

${\displaystyle {\hat {T}}={\hat {T}}_{1}+...+{\hat {T}}_{n}}$

耦合簇方程

${\displaystyle {\hat {H}}e^{\hat {T}}\vert {\Psi _{0}}\rangle =Ee^{\hat {T}}\vert {\Psi _{0}}\rangle }$

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

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

${\displaystyle \langle {\Psi _{0}}\vert e^{-({\hat {T}}_{1}+{\hat {T}}_{2})}{\hat {H}}e^{({\hat {T}}_{1}+{\hat {T}}_{2})}\vert {\Psi _{0}}\rangle =E}$
${\displaystyle \langle {\Psi _{S}}\vert e^{-({\hat {T}}_{1}+{\hat {T}}_{2})}{\hat {H}}e^{({\hat {T}}_{1}+{\hat {T}}_{2})}\vert {\Psi _{0}}\rangle =0}$
${\displaystyle \langle {\Psi _{D}}\vert e^{-({\hat {T}}_{1}+{\hat {T}}_{2})}{\hat {H}}e^{({\hat {T}}_{1}+{\hat {T}}_{2})}\vert {\Psi _{0}}\rangle =0}$

${\displaystyle {\bar {H}}=e^{-{\hat {T}}}{\hat {H}}e^{\hat {T}}={\hat {H}}+\left[{\hat {H}},{\hat {T}}\right]+{\frac {1}{2}}\left[\left[{\hat {H}},{\hat {T}}\right],{\hat {T}}\right]+\cdots }$

${\displaystyle {\bar {H}}}$  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)

${\displaystyle T={\hat {T}}_{1}+{\hat {T}}_{2}+{\hat {T}}_{3}.}$

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. Xian, R. F.; Brandes, T.; Gernoth, K. A.; Walet, N. R. (编). 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); Editors list列表中的|first5=缺少|last5= (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]. （原始内容存档于2012-06-16）.