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'''耦合簇方法'''（'''coupled cluster''', '''CC'''）是[[量子化學]][[全始計算法]]中對[[多電子相關能]]的其中一種高精確計算方法。它从[[哈特里－福克]]分子轨道出发，通过指数形式的耦合算符运算得到真实体系的波函数。一些小分子和中等大小的分子精度最高的计算结果是通过 CC 方法得到的。<ref>
{{cite book | first1 = H. G. |last1 =  Kümmel |chapter = A biography of the coupled cluster method | editor1-first=R. F. | editor1-last = Bishop  | editor2-first= T.  | editor2-last = Brandes
 | editor3-first =  K. A. | editor3-last =  Gernoth | editor4-first =  N. R. |editor4-last = Walet |editor5-first = Y.
| title = Recent progress in many-body theories Proceedings of the 11th international conference| publisher =  World Scientific Publishing | location =  Singapore | year = 2002
| pages =334–348 | isbn= 978-981-02-4888-8 | editor-last =Xian }}
</ref><ref>{{cite book
  | last = Cramer  | first = Christopher J.
  | title = Essentials of Computational Chemistry
  | url = https://archive.org/details/essentialscomput00cram_296  | publisher = John Wiley & Sons, Ltd.  | year = 2002  | location = Chichester
  | pages = [https://archive.org/details/essentialscomput00cram_296/page/n202 191]–232  | isbn = 0-471-48552-7}}</ref><ref>
{{Cite book|isbn=978-0-521-81832-2|publisher=Cambridge University Press |title=Many-Body Methods in Chemistry and Physics: MBPT and Coupled-Cluster Theory|year=2009| first1=Isaiah |last1=Shavitt | first2=Rodney J. | last2=Bartlett}}</ref>
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{{Electronic structure methods}}

The method was initially developed by [[Fritz Coester]] and [[Herman Kummel|Hermann Kümmel]] in the 1950s for studying [[nuclear physics]] phenomena, but became more frequently used when in 1966 [[Jiri Cizek|Jiři Čížek]] (and later together with [[Josef Paldus]]) reformulated the method for electron correlation in [[atoms]] and [[molecules]]. It is now one of the most prevalent methods in [[quantum chemistry]] that includes electronic correlation.
CC theory is simply the perturbative variant of the Many Electron Theory (MET) of [[Oktay Sinanoğlu]], which is the exact (and variational) solution of the many electron problem, so it was also called "Coupled Pair MET (CPMET)". J. Čížek used the correlation function of MET and used Goldstone type perturbation theory to get the energy expression while original MET was completely variational. Čížek first developed the Linear-CPMET and then generalized it to full CPMET in the same paper in 1966. He then also performed an application of it on benzene molecule with O. Sinanoğlu in the same year. Because MET is somewhat difficult to perform computationally, CC is simpler and thus, in today's computational chemistry, CC is the best variant of MET and gives highly accurate results in comparison to experiments.<ref>
{{cite journal | doi = 10.1063/1.1727484 | title = On the Correlation Problem in Atomic and Molecular Systems. Calculation of Wavefunction Components in Ursell-Type Expansion Using Quantum-Field Theoretical Methods | year = 1966 | last1 = Čížek | first1 = Jiří | journal = The Journal of Chemical Physics | volume = 45 | pages = 4256 |bibcode = 1966JChPh..45.4256C }}</ref><ref>
{{cite book | last1 = Sinanoğlu | first1= O. | last2 = Brueckner | first2 =  K. | title = Three approaches to electron correlation in atoms 
| publisher =  Yale Univ. Press | year = 1971| isbn=0-300-01147-4}} and references therein</ref><ref>
{{cite journal | doi = 10.1063/1.1732596 | title = Many-Electron Theory of Atoms and Molecules. I. Shells, Electron Pairs vs Many-Electron Correlations | year = 1962 | last1 = Si̇nanoğlu | first1 = Oktay | journal = The Journal of Chemical Physics | volume = 36 | pages = 706|bibcode = 1962JChPh..36..706S }}</ref>
-->

==波函数拟设==
耦合簇方法提供了一种近似求解不含时[[薛定谔方程]]的方法：
<center><math>\hat{H} \vert{\Psi}\rangle = E \vert{\Psi}\rangle </math></center>

这里 <math>\hat{H}</math> 表示体系的哈密顿量。体系的基态波函数与基态能量分别用 <math>\vert{\Psi}\rangle</math> 和 ''E'' 来表示。耦合簇理论的其它变体，如{{link-en|运动方程耦合簇方法|equation-of-motion coupled cluster}} 和{{link-en|多参考态耦合簇方法|multi-reference coupled cluster}}，则提供了求解体系激发态的方法。<ref>{{cite journal | last1 =Koch | first1 =Henrik | last2 =Jo̸rgensen | first2 =Poul | title =Coupled cluster response functions | journal =The Journal of Chemical Physics | volume =93 | pages =3333 | year =1990 | doi =10.1063/1.458814|bibcode = 1990JChPh..93.3333K }}</ref><ref>{{cite journal |last1 = Stanton |first1 = John F. |last2 = Bartlett |first2 = Rodney J. |title = The equation of motion coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties |journal = The Journal of Chemical Physics |volume = 98 |pages = 7029 |year = 1993 |doi = 10.1063/1.464746|bibcode = 1993JChPh..98.7029S }}</ref>

体系的基态波函数可以用下面的[[拟设]]来表出：

<center><math> \vert{\Psi}\rangle = e^{\hat{T}} \vert{\Phi_0}\rangle  </math></center>

式中 <math>\vert{\Phi_0}\rangle</math> 为[[哈特里－福克]]基态波函数，<math>\hat{T}</math> 是一个激发算符，称为簇算符，当它作用在 <math>\vert{\Phi_0}\rangle</math> 上时，得到一组[[斯莱特行列式]]的线性组合。（详情见下文）

在拟设的选取上，CC 方法比起其它的方法例如[[组态相互作用方法]]（CI）有优势。这是因为这一拟设具有[[大小广延性问题|大小广延性]]。CC 方法的大小一致性取决于参考波函数的大小一致性。CC 方法的一个主要缺陷是，它不是[[变分法|变分]]的。

== 簇算符 ==

簇算符由下式给出：

<center><math>  \hat{T}=\hat{T}_1 + \hat{T}_2 + \hat{T}_3 + \cdots </math></center>

其中 <math>\hat{T}_1</math> 是包含所有单激发的算符，<math>\hat{T}_2</math> 是包含所有双激发的算符，余类推。这些算符可以通过[[正则量子化]]表达为下列形式<ref>{{cite web|url=http://www.ua.es/cuantica/docencia/otros/cc/node5.html|accessdate=2012-06-24|title=The Cluster Operator|archive-url=https://web.archive.org/web/20120616125557/http://www.ua.es/cuantica/docencia/otros/cc/node5.html|archive-date=2012-06-16|dead-url=yes}}</ref>：

<center><math> 
\hat{T}_1=\sum_{i}\sum_{a} t_{i}^{a} \hat{a}^{\dagger}_{a}\hat{a}_{i},
</math></center>

<center><math>
\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},
</math></center>

余类推。

在上面的式子中，<math>\hat{a}^{\dagger}</math> 和 <math>\hat{a}</math> 分别是电子的[[产生及湮没算符]]。下标 ''i, j'' 表示占据轨道，而 ''a, b'' 表示空轨道。在耦合簇算符中的产生和湮没算符按照[[正规序]]排列。单粒子激发算符 <math>\hat{T}_1</math> 和双粒子激发算符 <math>\hat{T}_2</math> 分别把 <math>\vert{\Phi_0}\rangle</math> 变为单激发和双激发斯莱特行列式的线性组合。为了最终得到体系的波函数，需要求解拟设中的待定系数 <math>t_{i}^{a}</math>，<math>t_{ij}^{ab}</math> 等。

考虑到簇算符 <math>\hat{T}</math> 的结构后，指数耦合算符 <math>e^{\hat{T}}</math> 可以展开成[[泰勒级数]]：

<center><math> 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 </math></center>

事实上，这一级数是有限的，因为分子轨道的数目与激发的数目都是有限的。为了简化求解系数 <math>t</math> 的过程，<math>\hat{T}</math>的展开式中一般在双激发或略高一点的激发处截断，很少有超过四激发的。这是因为是否包含五激发以上的算符 <math>\hat{T}_5</math>、<math>\hat{T}_6</math> 等，对最终计算结果的影响很小。而且，即使只在簇算符的表达式中取前 <math>n</math> 项：

<center><math> \hat{T} = \hat{T}_1 + ... + \hat{T}_n </math></center>

那么由于耦合算符具有指数形式，高于 <math>n</math> 激发的斯莱特行列式仍然会对最终的波函数有贡献。因此，在 <math>\hat{T}_n</math> 处截断的 CC 方法通常能比激发数最高为 <math>n</math> 的 CI 方法获得更多的电子相关能修正。

== 耦合簇方程 ==
耦合簇方程就是展开系数 <math>t</math> 所满足的方程。有多种方法来书写这一方程，其中标准的做法是会得到一个可以迭代求解的方程组。耦合簇方法的薛定谔方程可以写成：

:<math> \hat{H} e^{\hat{T}} \vert{\Psi_0}\rangle = E e^{\hat{T}} \vert {\Psi_0}\rangle </math>

假设现在共有 <math>q</math> 个 <math>t</math> 系数需要求解。于是我们需要 <math>q</math> 个方程。注意到每一个 <math>t</math> 系数都与唯一的一个激发斯莱特行列式相关联：<math>t_{ijk...}^{abc...}</math> 对应的是 <math>\vert{\Phi_0}\rangle</math> 中处于 <math>i, j, k, \cdots</math> 轨道上的电子分别被激发到 <math>a, b, c, \cdots</math> 轨道上所得的行列式。上式两边向对应的行列式投影，就得到了我们所要的 <math>q</math> 个方程。

<center><math>\langle {\Psi^{*}}\vert \hat{H} e^{\hat{T}} \vert{\Psi_0}\rangle = E \langle {\Psi^{*}} \vert e^{\hat{T}} \vert {\Psi_0}\rangle </math></center>

式中 <math>\vert{\Psi^{*}}\rangle</math> 表示任意一个与待求的 <math>t</math> 系数相关联的激发行列式。为了更好地利用这些方程之间的联系，我们可以把上面的方程改写成一种更方便的形式，将 <math> e^{-\hat{T}} </math> 乘到耦合簇薛定谔方程两端，然后分别向 <math> \Psi_0 </math> 和 <math> \Psi^* </math> 投影，我们得到：

<center><math> \langle {\Psi_0}\vert e^{-\hat{T}} \hat{H} e^{\hat{T}} \vert{\Psi_0}\rangle = E </math></center>

<center><math> \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 </math></center>

第一式提供了求解 CC 能量的方法，第二式则是用来求解 <math>t</math> 系数的方程。以标准的 CCSD 方法为例，方程组中包括下面三组方程：

<center><math>\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 </math></center>
<center><math>\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</math></center>
<center><math>\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</math></center>

上式中经相似变换后的哈密顿量（用 <math>\bar{H}</math> 表示）可以通过{{link-en|BCH 公式|BCH formula}} 求出：
 
<center><math> \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 </math></center>

<math>\bar{H}</math> 不是厄米的。
<!-- Standard quantum chemistry packages ([[ACES (computational chemistry)|ACES II]], [[NWChem]], etc.) solve the coupled-equations iteratively using the Jacobi updates and the DIIS extrapolations of the ''t'' amplitudes. -->

== 耦合簇方法的种类 ==
传统上耦合簇方法依照 <math>\hat{T}</math> 中包含哪些 <math>\hat{T}_n</math> 算符来进行分类。相应的方法名称则由 CC 后面加上相应的字母构成：

# S - 单激发 (在英语的 CC 术语里面简称 ''singles'')
# D - 双激发 (''doubles'')
# T - 三激发 (''triples'')
# Q - 四激发 (''quadruples'')

例如，CCSDT 方法里面簇算符 <math>\hat{T}</math> 的表达式如下： 

<center><math> T = \hat{T}_1 + \hat{T}_2 + \hat{T}_3.</math></center>

在圆括号里面的项则表示它们是通过[[微扰理论]]求得的。例如 CCSD(T) 表示： 

# 耦合簇方法
# 包含完整的单激发和双激发
# 三激发则采用微扰理论而不是迭代求解
<!--
== General description of the theory ==

The complexity of equations and the corresponding computer codes, as well as the cost of the computation increases sharply with the highest level of excitation. For many applications the sufficient accuracy may be obtained with CCSD, and the more accurate (and more expensive) CCSD(T) is often called "the [[gold standard]] of quantum chemistry" for its excellent compromise between the accuracy and the cost for the molecules near equilibrium geometries. More complicated coupled-cluster methods such as CCSDT and CCSDTQ are used only for high-accuracy calculations of small molecules. The inclusion of all ''n'' levels of excitation for the ''n''-electron system gives the exact solution of the [[Schrödinger equation]] within the given [[basis set]], within the Born&ndash;Oppenheimer approximation (although schemes could also be drawn up to work without the BO approximation with great cost).

One possible improvement to the standard coupled-cluster approach is to add terms linear in the interelectronic distances through methods such as CCSD-R12.  This improves the treatment of dynamical electron correlation by satisfying the [[Kato cusp]] condition and accelerates convergence with respect to the orbital basis set.  Unfortunately, R12 methods invoke the [[resolution of the identity]] which requires a relatively large basis set in order to be a good approximation. 

The coupled-cluster method described above is also known as the ''[[single-reference]]'' (SR) coupled-cluster method because the exponential ansatz involves only one reference function <math>\vert{\Phi_0}\rangle</math>. The standard generalizations of the SR-CC method are the ''[[multi-reference]]'' (MR) approaches: [[state-universal coupled cluster]] (also known as [[Hilbert space]] coupled cluster), [[valence-universal coupled cluster]] (or [[Fock space]] coupled cluster) and [[state-selective coupled cluster]] (or state-specific coupled cluster).

== A historical account ==
In the first reference below, Kümmel comments:

:''Considering the fact that the CC method was well understood around the late fifties it looks strange that nothing happened with it until 1966, as Jiři Čížek published his first paper on a quantum chemistry problem. He had looked into the 1957 and 1960 papers published in ''Nuclear Physics'' by Fritz and myself. I always found it quite remarkable that a quantum chemist would open an issue of a nuclear physics journal. I myself at the time had almost gave up the CC method as not tractable and, of course, I never looked into the quantum chemistry journals. The result was that I learnt about Jiři's work as late as in the early seventies, when he sent me a big parcel with reprints of the many papers he and Joe Paldus had written until then.
-->

==参见==
*[[组态相互作用方法]]
<!-- * [[Coupled-electron pair approximation]] (CEPA) -->
== 參考文獻 ==
{{reflist}}

== 扩展阅读 ==
* [https://web.archive.org/web/20060219064754/http://zopyros.ccqc.uga.edu/lec_top/cc/html/review.html A theoretical review and introduction to coupled cluster theory]
* [https://web.archive.org/web/20120703030217/http://www.ua.es/cuantica/docencia/otros/cc/cc.html An Introduction to Coupled-Cluster Theory]
* [https://web.archive.org/web/20060511015415/http://student.ccbcmd.edu/%7Ebhoffm30/instructional/ci/node14.html The Coupled Cluster (CC) Approach]
* Cramer, C. J. Essentials of Computational Chemistry: Theories and Models. Wiley

[[Category:量子化学]]
[[Category:電子結構方法]]