COUPLED CLUSTER

'Coupled cluster' ('CC') is a numerical technique used for describing many-body systems. Its most common use is as one of several quantum chemical post-Hartree-Fock ab initio quantum chemistry methods in the field of computational chemistry. It starts from the Hartree-Fock molecular orbital method and adds a correction term to take into account electron correlation. Some of the most accurate calculations for small to medium sized molecules use this method.
The method was initially developed by Fritz Coester and Hermann Kümmel in the 1950s for studying nuclear physics phenomena, but became more frequently used after Jiři Čížek and Josef Paldus reformulated the method for electron correlation in atoms and molecules in the 1960s. It is now one of the most prevalent methods in quantum chemistry that includes electronic correlation.
=== Wavefunction ansatz ===
Coupled-cluster theory provides an approximate solution to the time-independent Schrödinger equation
:
angle = E ert{Psi}
angle
where $hat\{H\}$ is the Hamiltonian of the system. The wavefunction and the energy of the lowest-energy state are denoted by
angle and ''E'', respectively. Other variants of the coupled-cluster theory, such as equation-of-motion coupled cluster and multi-reference coupled cluster may also produce approximate solutions for the excited states of the system.
The wavefunction of the coupled-cluster theory is written as an exponential ansatz:
:
angle = e^{hat{T}} ert{Phi_0}
angle ,
where
angle is a Slater determinant usually constructed from Hartree-Fock molecular orbitals. $hat\{T\}$ is an excitation operator which, when acting on
angle, produces a linear combination of excited Slater determinants (see section below for greater detail).
The choice of the exponential ansatz is opportune because (unlike other ansatzes, for example, configuration interaction) it guarantees the size extensivity of the solution. Size consistency in CC theory, however, depends on the size consistency of the reference wave function.

Contents

Cluster operator

Coupled-cluster equations

Types of coupled-cluster methods

General description of the theory

A historical account

Relation to other theories

References

External resources

Cluster operator

The cluster operator is written in the form
:$hat\{T\}=hat\{T\}\_1\; +\; hat\{T\}\_2\; +\; hat\{T\}\_3\; +\; cdots$,
where $hat\{T\}\_1$ is the operator of all single excitations, $hat\{T\}\_2$ is the operator of all double excitations and so forth. In the formalism of second quantization these excitation operators are conveniently expressed as
:$$
hat{T}_1=sum_{i}sum_{a} t_{i}^{a} hat{a}_{i}hat{a}^{dagger}_{a},

:$$
hat{T}_2=rac{1}{4}sum_{i,j}sum_{a,b} t_{ij}^{ab} hat{a}_{i}hat{a}_jhat{a}^{dagger}_{a}hat{a}^{dagger}_{b}

etc
In the above formulae $hat\{a\}^\{dagger\}$ and $hat\{a\}$ denote the creation and annihilation operators respectively and ''i, j'' stand for occupied and ''a, b'' for unoccupied orbitals. The creation and annihilation operators in the coupled cluster terms above are written in canonical form, where each term is in normal order. Being the one-particle excitation operator and the two-particle excitation operator, $hat\{T\}\_1$ and $hat\{T\}\_2$ convert the reference function
angle into a linear combination of the singly- and doubly-excited Slater determinants, respectively. Solving for the unknown coefficients $t\_\{i\}^\{a\}$ and $t\_\{ij\}^\{ab\}$ is necessary for finding the approximate solution
angle.
Taking into consideration the structure of $hat\{T\}$, the exponential operator $e^\{hat\{T\}\}$ may be expanded into Taylor series:
:
This series is finite in practice because the number of occupied molecular orbitals is finite, as is the number of excitations. In order to simplify the task for finding the coefficients ''t'', the expansion of $hat\{T\}$ into individual excitation operators is terminated at the second or slightly higher level of excitation (rarely exceeding four). This approach is warranted by the fact that even if the system admits more than four excitations, the contribution of $hat\{T\}\_5$, $hat\{T\}\_6$ etc to the operator $hat\{T\}$ is small. Furthermore, if the highest excitation level in the $hat\{T\}$ operator is ''n'',
:$T\; =\; 1\; +\; hat\{T\}\_1\; +\; ...\; +\; hat\{T\}\_n$
then Slater determinants excited more than ''n'' times may (and usually do) still contribute to the wave function
angle because of the non-linear nature of the exponential ansatz. Therefore, coupled cluster terminated at $hat\{T\}\_n$ usually recovers more correlation energy than configuration interaction with maximum ''n'' excitations.

Coupled-cluster equations

Coupled-cluster equations are equations whose solution is the set of coefficients ''t''. There are several ways of writing such equations but the standard formalism results in a terminating set of equations which may be solved iteratively. The naive variational approach does not take advantage of the connected nature of the cluster amplitudes and results in a non-terminating set of equations.
Suppose there are ''q'' coefficients ''t'' to solve for. Therefore, we need ''q'' equations. It is easy to notice that each ''t''-coefficient may be put in correspondence with a certain excited determinant: $t\_\{ijk...\}^\{abc...\}$ corresponds to the determinant obtained from
angle by substituting the occupied orbitals ''i,j,k,...'' with the virtual orbitals ''a,b,c,...'' Projecting the Schrödinger equation with the exponential ansatz by ''q'' such different determinants from the left, we obtain the sought-for ''q'' equations:
:$langle\; \{Psi^\{$
★ }}ert hat{H} e^{hat{T}} ert{Psi_0}
angle = E langle {Psi^{
★ }} ert e^{hat{T}} ert {Psi_0}
angle
where by
★ }}
angle we understand the whole set of the appropriate excited determinants.
Unfortunately, $langle\; \{Psi^\{$
★ }} ert e^{hat{T}} ert {Psi_0}
angle is a non-terminating sequence. The coupled-cluster equations are reduced to a closed form in the similarity transformed representation:
:
angle = langle {Psi_0}ert ar{H}_N ert{Psi_0}
angle,
:$0\; =\; langle\; \{Psi^\{$
★ }}ert e^{hat{-T}} hat{H}_N e^{hat{T}} ert{Psi_0}
angle ,
the latter being the equations to be solved and the former the equation for the evaluation of the energy. Consider the standard CCSD method:
:
angle ,
:
angle ,
:
angle ,
which when reduced using the BCH formula for the similarity transformed Hamiltonian, $e^\{hat\{-T\}\}\; hat\{H\}\; e^\{hat\{T\}\}\; =\; H\; +\; [H,T]\; +\; (1/2)$