耦合簇理論 (Coupled Cluster, CC)指的是一種用於求解多體問題的理論方法。該理論首先由Fritz Coester和Hermann Kümmel 於1950年代提出，當時是為了研究核物理中的一些現象，但是後來由Jiři Čížek和Josef Paldus重新改善後，從1960年代開始，被廣泛的運用到研究原子和分子中的電子相關效應。耦合簇理論是目前最流行的包括電子相關的量子化學方法之一。值得一提的是，耦合簇主要套用於費密子體系，而在計算化學中則最主要套用於電子體系。
其他相關理論
組態相互作用
耦合對多電子理論 (Coupled-pair many-electron theory，簡稱CPMET)，有時稱為耦合簇近似。
[編輯]背景知識及推導
為求解時間無關的薛丁格方程, 我們要求解體系的這種形式的方程
在這裡 表示體系的哈密頓算符, E 表示體系的能量. 波函式用來表示.
耦合簇理論通過對波函施加一個ansatz 而改進對體系的解, 在這裡我們使用了一個假定[如下所示]
,
在這裡， 是一個合適的參考波函，可以有多種形式.如, 可以是Hartree-Fock 空間中的一個態函式.
把改進後的波函帶回到最初的薛丁格方程中，得到如下結果
, 根據這個式子，可以得到：
,
是一個轉換後的算符.
這個等式表明ansatz 保留了最初哈密頓量所對應的能量, 同時把初始哈密頓算符轉換成新的算符. 耦合簇理論提出存在一個合適的簇算符 , 它可以使轉換後的哈密頓函式G極大的被簡化，也就是說G 的矩陣表示是對角化的或者是接近對角化的[如可以以分塊對角化的形式存在].
波函式的這種形式是先驗的[ansatz]，沒有試驗結果表明波函式是以這種形式存在。選用這種波函的原因是它使複雜的量子體系變得易於處理. 特別是由於簇算符是取冪的,簇算符中對應著一個態的單激發的算符同時也激發[或理解為相關]其它的高階態. 下式對此給出了很好的解釋
根據這個泰勒級數Taylor series, 我們可以清楚地看出更高階的態由於簇算符重複使用的混合效果而被激發. 同時也可以注意到，隨著激發度的增加，該激發態所占比重也在減小。[？原文：It is also noteworthy to remark that the excitation decreases as well]. 最終,由於參考態Ψ0 所表示的通常都是fermion體系, 所以簇算符泰勒級數的長度是有限的, 儘管從數學角度來說它應該被表示為一個無限的級數. 下面一個例子解釋為什麼簇算符泰勒級數的長度應該有限。如果參考態被指定,比如說是電子(自旋為-1/2的粒子 ), 則對每個電子就只有兩個自旋態,簇算符的重複使用將把它們激發到體系之外[and repeated application of the cluster operator will excite them out of the system]. 因此簇算符的級數由於所研究體系的物理解釋和適用性而應是有限的.
[編輯]Hermiticity preservation
It is worthwhile to note that the transformation process in general, does not preserve the hermiticity of the original Hamiltonian - that is to say that
However, if provisions are made so that the cluster operater is antihermitian, where the transformation process preserves the hermiticity of the system.
[編輯]厄米性的保守
有必要指出在一個通常的轉換過程中，初始哈密頓算符不再能保持它的厄米性，也就是說：
不過，如果指定簇算符是反厄米算符,也就是，則在轉換過程中可以保持體系的厄米性質
[編輯]Varieties of coupled cluster methods
Coupled cluster methods are able to give robust numerical calculations for systems of interest in quantum chemistry. Over the years, a number of minor modifications have been made or added to the original framework so that certain special cases of molecules and their energies can be computed. These modifications include
Imposing the operator to be unitary;
Adding higher order terms to the cluster operator
The abbreviations for coupled cluster methods usually begin with the letters "CC" (for coupled cluster) followed by
S - for single excitations (shortened to singles in coupled cluster terminology)
D - for double excitations (shortened to doubles in coupled cluster terminology)
T - for triple excitations (shortened to TripleS in coupled cluster terminology)
Q - for quadruple excitations (shortened to quadruples in coupled cluster terminology)
Bracketed terms indicate that the higher order terms are calculated based on pertubation theory. For example, a CCSDT(Q) approach simply means:
A Coupled cluster method
It includes singles, doubles, and triples
Quadruples are calculated with pertubation theory.
There is generally no accepted standard for higher order excitations, as their implementation and computational complexity are significantly demanding to perform. Additionally, the accuracy gained from using such methods is relatively minimal when compared to lower order methods. In principle, one would follow the conventions of appending extra letters for the ordinals, or use any appropriate notational method which conveys similar information.
[編輯]常見耦合簇方法
對於量子化學所感興趣的體系，耦合簇方法涉及到大量的數值計算。在這些年中，對最初的理論骨架做了各種各樣的修正以計算特定的分子以及它們的能量，具體的包括如下這些：
強制性的使算符 成為單位算符
對簇算符添加更高階的項
各種耦合簇方法的縮寫通常以字母"CC" (coupled cluster)開頭，後跟以下的字母
S 單激發 [是singles的縮寫]
D 雙激發 [是doubles的縮寫]
T 三激發 [是triples的縮寫]
Q 四激發 [是quadruples的縮寫]
括弧里的項表示更高階的項是在pertubation theory的基礎上計算得到. 如CCSDT(Q)方法的意思是
一個耦合簇方法
包含了單、雙和三重激發態
四重激發態用微擾理論計算
對於更高階的激發還沒有普遍被接受的標準，因為包含更高階激發的計算不是很容易實現，此外，以這種方法獲得的準確度比低一階的方法只有略微的改進。原則上來說，對這些方法的描述要么遵循根據序數添加額外字母的慣例，要么使用可以傳遞簡潔信息的合適的符號方法
[編輯]Cluster operator
,
where in the formalism of second quantization:
In the above formulae and denote the creation and annihilation operators respectively and i,j stand for occupied and a,b for unoccupied orbitals. T1 and T2 are called the one-particle excitation operator, and the two-particle excitation operator, because they effectively convert the reference function into a linear combination of singly- and doubly-excited Slater determinants. Solving for the coefficients and , in order to satisfy the definition of the cluster operator, constitutes a coupled cluster calculation. The operators in the coupled cluster term are normally written in canonical form, where each term is in normal order. Similar operators also appear in canonical pertubation theory.
The cluster operator can be represented in a vector space which spans the sequence of creation/annihilation operators which are in the cluster operator itself.
[編輯]Coupled cluster with doubles (CCD)
In the simplest version one considers only operator (double excitations). This method is called coupled cluster with doubles (CCD in short).
[編輯]Coupled cluster with singles and doubles (CCSD)
This version, as the name suggests, considers both and operators, accounting for both double and single excitations. The approximation is that = + .
[編輯]包含單重態和雙重態的耦合簇理論
如名所示，包含單激發和雙激發的耦合簇理論 (CCSD)通過使用單激發 跟雙激發 算符，同時考慮到單激發跟雙激發態的影響。其近似可記為 = + 。
[編輯]Description of the theory
The method gives exact non-relativistic solution of the Schrödinger equation of the n-body problem if one includes up to the cluster operator. However, the computational effort of solving the equations grows steeply with the order of the cluster operator and in practical applications the method is limited to the first few orders.
Most frequently, one solves the CC equation using the operator , which produces all Slater determinants which differ from the reference determinant by one or two spin-orbitals. This approach, called coupled-cluster singles and doubles (CCSD), has the effect of describing coupled two-body electron correlation effects and orbital relaxation effects. Because the operator is exponentiated in coupled-cluster theory, higher-order "disconnected" electron correlations are also accounted for in an approximate way. It is also fairly common (although also more computationally expensive) to include an approximate, pertubative correction accounting for three-body electron correlations in a method designated CCSD(T). For ground electronic states near their equilibrium geometries, CCSD(T) is often called a "gold standard" of quantum chemistry because it provides results very close to those of full configuration interaction (full CI), which solves the non-relativistic electronic Schrödinger equation exactly within the given one-particle basis set. More recently, coupled cluster methods have been developed which use the operator , or even the operator . These methods are called CCSDT and CCSDTQ. A simplification of the later method is CCSDT(Q) where the four-body terms are introducted by a perturbative correction to CCSDT.
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 valid.
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 . 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).
The coupled cluster equations are usually derived using diagrammatic technique and result in nonlinear equations which can be solved in an iterative way. Converged solution requires usually a few dozens of iterations.
