耦合簇理论

用于求解多体问题的理论方法

耦合簇理论（Coupled Cluster）是一种用于求解多体问题的理论方法。

耦合簇理论 (Coupled Cluster, CC)指的是一种用于求解多体问题的理论方法。该理论首先由Fritz Coester和Hermann Kümmel 于1950年代提出，当时是为了研究核物理中的一些现象，但是后来由Jiři Čížek和Josef Paldus重新改善后，从1960年代开始，被广泛的运用到研究原子和分子中的电子相关效应。耦合簇理论是最流行的包括电子相关的量子化学方法之一。值得一提的是，耦合簇主要应用于费米子体系，而在计算化学中则最主要应用于电子体系。

其他相关理论组态相互作用耦合对多电子理论 (Coupled-pair many-electron theory，简称CPMET)，有时称为耦合簇近似。 [编辑]背景知识及推导为求解时间无关的薛定谔方程, 我们要求解体系的这种形式的方程

在这里表示体系的哈密顿算符, E 表示体系的能量. 波函数用来表示.

耦合簇理论通过对波函施加一个ansatz 而改进对体系的解, 在这里我们使用了一个假定[如下所示]

, 在这里，是一个合适的参考波函，可以有多种形式.如, 可以是Hartree-Fock 空间中的一个态函数.

把改进后的波函带回到最初的薛定谔方程中，得到如下结果

, 根据这个式子，可以得到： , 是一个转换后的算符.这个等式表明ansatz 保留了最初哈密顿量所对应的能量, 同时把初始哈密顿算符转换成新的算符. 耦合簇理论提出存在一个合适的簇算符 , 它可以使转换后的哈密顿函数G极大的被简化，也就是说G 的矩阵表示是对角化的或者是接近对角化的[如可以以分块对角化的形式存在].

波函数的这种形式是先验的[ansatz]，没有试验结果表明波函数是以这种形式存在。选用这种波函的原因是它使复杂的量子体系变得易于处理. 特别是由于簇算符是取幂的,簇算符中对应着一个态的单激发的算符同时也激发[或理解为相关]其它的高阶态. 下式对此给出了很好的解释

根据这个泰勒级数Taylor series, 我们可以清楚地看出更高阶的态由于簇算符重复使用的混合效果而被激发. 同时也可以注意到，随着激发度的增加，该激发态所占比重也在减小。[？原文：It is also noteworthy to remark that the excitation decreases as well]. 最终,由于参考态Ψ0 所表示的通常都是fermion体系, 所以簇算符泰勒级数的长度是有限的, 尽管从数学角度来说它应该被表示为一个无限的级数. 下面一个例子解释为什么簇算符泰勒级数的长度应该有限。如果参考态被指定,比如说是电子(自旋为-1/2的粒子 ), 则对每个电子就只有两个自旋态,簇算符的重复使用将把它们激发到体系之外[and repeated application of the cluster operator will excite them out of the system]. 因此簇算符的级数由于所研究体系的物理解释和适用性而应是有限的.

[编辑]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.

[编辑]厄米性的保守有必要指出在一个通常的转换过程中，初始哈密顿算符不再能保持它的厄米性，也就是说：

不过，如果指定簇算符是反厄米算符,也就是，则在转换过程中可以保持体系的厄米性质

[编辑]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

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.

[编辑]常见耦合簇方法对于量子化学所感兴趣的体系，耦合簇方法涉及到大量的数值计算。在这些年中，对最初的理论骨架做了各种各样的修正以计算特定的分子以及它们的能量，具体的包括如下这些：

S 单激发 [是singles的缩写] D 双激发 [是doubles的缩写] T 三激发 [是triples的缩写] Q 四激发 [是quadruples的缩写] 括号里的项表示更高阶的项是在perturbation theory的基础上计算得到. 如CCSDT(Q)方法的意思是

一个耦合簇方法包含了单、双和三重激发态四重激发态用微扰理论计算对于更高阶的激发还没有普遍被接受的标准，因为包含更高阶激发的计算不是很容易实现，此外，以这种方法获得的准确度比低一阶的方法只有略微的改进。原则上来说，对这些方法的描述要么遵循根据序数添加额外字母的惯例，要么使用可以传递简洁信息的合适的符号方法

[编辑]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.

[编辑]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).

[编辑]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 = + .

[编辑]包含单重态和双重态的耦合簇理论如名所示，包含单激发和双激发的耦合簇理论 (CCSD)通过使用单激发跟双激发算符，同时考虑到单激发跟双激发态的影响。其近似可记为 = + 。

[编辑]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.

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.

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