Coupledcluster theory provides the exact solution to the timeindependent Schrödinger equation
H

Ψ
⟩
=
E

Ψ
⟩
where
H
is the Hamiltonian of the system,

Ψ
⟩
the exact wavefunction, and E the exact energy of the ground state. Coupledcluster theory can also be used to obtain solutions for excited states using, for example, linearresponse, equationofmotion, stateuniversal multireference coupled cluster, or valenceuniversal multireference coupled cluster approaches.
The wavefunction of the coupledcluster theory is written as an exponential ansatz:

Ψ
⟩
=
e
T

Φ
0
⟩
,
where

Φ
0
⟩
, the reference wave function, which is typically a Slater determinant constructed from Hartree–Fock molecular orbitals, though other wave functions such as configuration interaction, multiconfigurational selfconsistent field, or Brueckner orbitals can also be used.
T
is the cluster operator which, when acting on

Φ
0
⟩
, produces a linear combination of excited determinants from the reference wave function (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, also unlike other theories, does not depend on the size consistency of the reference wave function. This is easily seen, for example, in the single bond breaking of F
2
when using a restricted HartreeFock (RHF) reference, which is not size consistent, at the CCSDT level of theory which provides an almost exact, full CIquality, potential energy surface and does not dissociate the molecule into F
−
and F
+
ions, like the RHF wave function, but rather into two neutral F atoms. If one were to use, for example, the CCSD, CCSD[T], or CCSD(T) levels of theory, they would not provide reasonable results for the bond breaking of F
2
, with the latter two approaches providing unphysical potential energy surfaces, though this is for reasons other than just size consistency.
A criticism of the method is that the conventional implementation employing the similaritytransformed Hamiltonian (see below) is not variational, though there are bivariational and quasivariational approaches that have been developed since the first implementations of the theory. While the above ansatz for the wave function itself has no natural truncation, however, for other properties, such as energy, there is a natural truncation when examining expectation values, which has its basis in the linked and connectedcluster theorems, and thus does not suffer from issues such as lack of size extensivity, like the variational configuration interaction approach.
The cluster operator is written in the form,
T
=
T
1
+
T
2
+
T
3
+
⋯
,
where
T
1
is the operator of all single excitations,
T
2
is the operator of all double excitations and so forth. In the formalism of second quantization these excitation operators are expressed as
T
1
=
∑
i
∑
a
t
a
i
a
^
a
a
^
i
,
T
2
=
1
4
∑
i
,
j
∑
a
,
b
t
a
b
i
j
a
^
a
a
^
b
a
^
j
a
^
i
,
and for the general nfold cluster operator
T
n
=
1
(
n
!
)
2
∑
i
1
,
i
2
,
…
,
i
n
∑
a
1
,
a
2
,
…
,
a
n
t
a
1
,
a
2
,
…
,
a
n
i
1
,
i
2
,
…
,
i
n
a
^
a
1
a
^
a
2
…
a
^
a
n
a
^
i
n
…
a
^
i
2
a
^
i
1
.
In the above formulae
(
a
^
a
†
=
)
a
^
a
and
a
^
i
denote the creation and annihilation operators, respectively, and i, j stand for occupied (hole) and a, b for unoccupied (particle) orbitals (states). The creation and annihilation operators in the coupled cluster terms above are written in canonical form, where each term is in the normal order form, with respect to the Fermi vacuum,

Φ
0
⟩
. Being the oneparticle cluster operator and the twoparticle cluster operator,
T
1
and
T
2
convert the reference function

Φ
0
⟩
into a linear combination of the singly and doubly excited Slater determinants, respectively, if applied without the exponential (such as in CI where a linear excitation operator is applied to the wave function). Applying the exponential cluster operator to the wave function, one can then generate more than doubly excited determinants due to the various powers of
T
1
and
T
2
that appear in the resulting expressions (see below). Solving for the unknown coefficients
t
a
i
and
t
a
b
i
j
is necessary for finding the approximate solution

Ψ
⟩
.
The exponential operator
e
T
may be expanded as a Taylor series and if we consider only the
T
1
and
T
2
cluster operators of
T
, we can write:
e
T
=
1
+
T
+
1
2
!
T
2
+
⋯
=
1
+
T
1
+
T
2
+
1
2
T
1
2
+
T
1
T
2
+
1
2
T
2
2
+
⋯
Though this series is finite in practice because the number of occupied molecular orbitals is finite, as is the number of excitations, it is still very large, to the extent that even modern day massively parallel computers are inadequate, except for problems of a dozen or so electrons and very small basis sets, when considering all contributions to the cluster operator and not just
T
1
and
T
2
. Often, as was done above, the cluster operator includes only singles and doubles (see CCSD below) as this offers a computationally affordable method that performs better than MP2 and CISD, but is not very accurate usually. For accurate results some form of triples (approximate or full) are needed, even near the equilibrium geometry (in the FranckCondon region), and especially when breaking singlebonds or describing diradical species (these latter examples are often what is referred to as multireference problems, since more than one determinant has a significant contribution to the resulting wave function). For double bond breaking, and more complicated problems in chemistry, quadruple excitations often become important as well, though usually they are small for most problems, and as such, the contribution of
T
5
,
T
6
etc. to the operator
T
is typically small. Furthermore, if the highest excitation level in the
T
operator is n,
T
=
T
1
+
.
.
.
+
T
n
then Slater determinants for an Nelectron system excited more than n (< N) times may still contribute to the coupled cluster wave function

Ψ
⟩
because of the nonlinear nature of the exponential ansatz, and therefore, coupled cluster terminated at
T
n
usually recovers more correlation energy than CI with maximum n excitations.
The Schrödinger equation can be written, using the coupledcluster wave function, as
H

Ψ
0
⟩
=
H
e
T

Φ
0
⟩
=
E
e
T

Φ
0
⟩
where there are a total of q coefficients (tamplitudes) to solve for. To obtain the q equations, first, we multiply the above Schrödinger equation on the left by
e
−
T
and then project onto the entire set of up to mtuply excited determinants, where m is the highest order excitation included in
T
, that can be constructed from the reference wave function

Φ
0
⟩
, denoted by

Φ
∗
⟩
, and individually,

Φ
i
a
⟩
are singly excited determinants where the electron in orbital i has been excited to orbital a;

Φ
i
j
a
b
⟩
are doubly excited determinants where the electron in orbital i has been excited to orbital a and the electron in orbital j has been excited to orbital b, etc. In this way we generate a set of coupled energyindependent nonlinear algebraic equations needed to determine the tamplitudes.
⟨
Φ
0

e
−
T
H
e
T

Φ
0
⟩
=
E
⟨
Φ
0

Φ
0
⟩
=
E
⟨
Φ
∗

e
−
T
H
e
T

Φ
0
⟩
=
E
⟨
Φ
∗

Φ
0
⟩
=
0
,
(note, we have made use of
e
−
T
e
T
=
1
, the identity operator, and we are also assuming that we are using orthogonal orbitals, though this does not necessarily have to be true, e.g., valence bond orbitals, and in such cases the last set of equations are not necessarily equal to zero) the latter being the equations to be solved and the former the equation for the evaluation of the energy.
Considering the basic CCSD method:
⟨
Φ
0

e
−
(
T
1
+
T
2
)
H
e
(
T
1
+
T
2
)

Φ
0
⟩
=
E
,
⟨
Φ
i
a

e
−
(
T
1
+
T
2
)
H
e
(
T
1
+
T
2
)

Φ
0
⟩
=
0
,
⟨
Φ
i
j
a
b

e
−
(
T
1
+
T
2
)
H
e
(
T
1
+
T
2
)

Φ
0
⟩
=
0
,
in which the similarity transformed Hamiltonian,
H
¯
, can be explicitly written down using Hadamard's formula in Lie algebra, also called Hadamard's lemma (see also Baker–Campbell–Hausdorff formula (BCH formula), though note they are different, in that Hadamard's formula is a lemma of the BCH formula):
H
¯
=
e
−
T
H
e
T
=
H
+
[
H
,
T
]
+
1
2
!
[
[
H
,
T
]
,
T
]
+
.
.
.
=
(
H
e
T
)
C
.
The subscript C designates the connected part of the corresponding operator expression.
The resulting similarity transformed Hamiltonian is nonHermitian, resulting in different left and righthanded vectors (wave functions) for the same state of interest (this is what is often referred to in coupled cluster theory as the biorthogonality of the solution, or wave function, though it also applies to other nonHermitian theories as well). The resulting equations are a set of nonlinear equations which are solved in an iterative manner. Standard quantum chemistry packages (GAMESS (US), NWChem, ACES II, etc.) solve the coupled cluster equations using the Jacobi method and direct inversion of the iterative subspace (DIIS) extrapolation of the tamplitudes to accelerate convergence.
The classification of traditional coupledcluster methods rests on the highest number of excitations allowed in the definition of
T
. The abbreviations for coupledcluster methods usually begin with the letters "CC" (for coupled cluster) followed by
 S – for single excitations (shortened to singles in coupledcluster terminology)
 D – for double excitations (doubles)
 T – for triple excitations (triples)
 Q – for quadruple excitations (quadruples)
Thus, the
T
operator in CCSDT has the form
T
=
T
1
+
T
2
+
T
3
.
Terms in round brackets indicate that these terms are calculated based on perturbation theory. For example, the CCSD(T) method means:
 Coupled cluster with a full treatment singles and doubles.
 An estimate to the connected triples contribution is calculated noniteratively using ManyBody Perturbation Theory arguments.
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 CCSD, while relatively inexpensive, does not provide sufficient accuracy except for the smallest systems (approximately 2 to 4 electrons), and often an approximate treatment of triples is needed. The most well known coupled cluster method that provides an estimate of connected triples is CCSD(T), which provides a good description of closedshell molecules near the equilibrium geometry, but breaks down in more complicated situations such as bond breaking and diradicals. Another popular method that makes up for the failings of the standard CCSD(T) approach is CRCC(2,3), where the triples contribution to the energy is computed from the difference between the exact solution and the CCSD energy, and is not based on perturbation theory arguments. More complicated coupledcluster methods such as CCSDT and CCSDTQ are used only for highaccuracy calculations of small molecules. The inclusion of all n levels of excitation for the nelectron system gives the exact solution of the Schrödinger equation within the given basis set, within the Born–Oppenheimer approximation (although schemes have also been drawn up to work without the BO approximation).
One possible improvement to the standard coupledcluster approach is to add terms linear in the interelectronic distances through methods such as CCSDR12. 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 coupledcluster method described above is also known as the singlereference (SR) coupledcluster method because the exponential ansatz involves only one reference function

Φ
0
⟩
. The standard generalizations of the SRCC method are the multireference (MR) approaches: stateuniversal coupled cluster (also known as Hilbert space coupled cluster), valenceuniversal coupled cluster (or Fock space coupled cluster) and stateselective coupled cluster (or statespecific coupled cluster).
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 Jiří 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 given 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 Jiří'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.
Josef Paldus also wrote his firsthand account of the origins of coupledcluster theory, its implementation, and exploitation in electronic wave function determination; his account is primarily about the making of coupledcluster theory rather than about the theory itself.
The C_{j} excitation operators defining the CI expansion of an Nelectron system for the wave function

Ψ
0
⟩
,

Ψ
0
⟩
=
(
1
+
C
)

Φ
0
⟩
,
C
=
∑
j
=
1
N
C
j
,
are related to the cluster operators
T
, since in the limit of including up to
T
N
in the cluster operator the CC theory must be equal to full CI, we obtain the following relationships
C
1
=
T
1
,
C
2
=
T
2
+
1
2
(
T
1
)
2
,
C
3
=
T
3
+
T
1
T
2
+
1
6
(
T
1
)
3
,
C
4
=
T
4
+
1
2
(
T
2
)
2
+
T
1
T
3
+
1
2
(
T
1
)
2
T
2
+
1
24
(
T
1
)
4
,
etc. For general relationships see J. Paldus, in Methods in Computational Molecular Physics, Vol. 293 of Nato Advanced Study Institute Series B: Physics, edited by S. Wilson and G.H.F. Diercksen (Plenum, New York, 1992), pp. 99–194.
The Symmetry adapted cluster (SAC) approach determines the (spin and) symmetry adapted cluster operator
S
=
∑
I
S
I
by solving the following system of energy dependent equations,
⟨
Φ

(
H
−
E
0
)
e
S

Φ
⟩
=
0
,
⟨
Φ
i
1
…
i
n
a
1
…
a
n

(
H
−
E
0
)
e
S

Φ
⟩
=
0
,
i
1
<
⋯
<
i
n
,
a
1
<
⋯
<
a
n
,
n
=
1
,
…
,
M
s
,
where

Φ
i
1
…
i
n
a
1
…
a
n
⟩
are the ntuply excited determinants relative to

Φ
⟩
(usually they are the spin and symmetryadapted configuration state functions, in practical implementations), and
M
s
is the highestorder of excitation included in the SAC operator. If all of the nonlinear terms in
e
S
are included then the SAC equations become equivalent to the standard coupledcluster equations of Jiří Čížek. This is due to the cancellation of the energydependent terms with the disconnected terms contributing to the product of
H
e
S
, resulting in the same set of nonlinear energyindependent equations. Typically, all nonlinear terms, except
1
2
S
2
2
are dropped, as higherorder nonlinear terms are usually small.
In nuclear physics, coupled cluster saw significantly less use than in quantum chemistry during the 1980s and 1990s. More powerful computers as well as advances in theory (such as the inclusion of threenucleon interactions) have spawned renewed interest in the method since then, and it has been successfully applied to neutronrich and medium mass nuclei. Coupled cluster is one of several ab initio methods in nuclear physics, and is specifically suitable for nuclei having closed or nearly closed shells.