Editing Coupled cluster (section)

== Cluster operator ==

The cluster operator is written in the form

: <math>T = T_1 + T_2 + T_3 + \cdots,</math>

where <math>T_1</math> is the operator of all single excitations, <math>T_2</math> is the operator of all double excitations, and so forth. In the formalism of [[second quantization]] these excitation operators are expressed as

: <math>
 T_1 = \sum_i \sum_a t_a^i \hat{a}^a \hat{a}_i,
</math>

: <math>
 T_2 = \frac{1}{4} \sum_{i,j} \sum_{a,b} t_{ab}^{ij} \hat{a}^a \hat{a}^b \hat{a}_j \hat{a}_i,
</math>

and for the general ''n''-fold cluster operator

: <math>
 T_n = \frac{1}{(n!)^2} \sum_{i_1,i_2,\ldots,i_n} \sum_{a_1,a_2,\ldots,a_n} t_{a_1,a_2,\ldots,a_n}^{i_1,i_2,\ldots,i_n} \hat{a}^{a_1} \hat{a}^{a_2} \ldots \hat{a}^{a_n} \hat{a}_{i_n} \ldots \hat{a}_{i_2} \hat{a}_{i_1}.
</math>

In the above formulae <math>\hat{a}^a = \hat{a}^\dagger_a</math> and <math>\hat{a}_i</math> denote the [[creation and annihilation operator]]s respectively, while ''i'',&nbsp;''j'' stand for occupied (hole) and ''a'',&nbsp;''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 <math>|\Phi_0\rangle</math>. Being the one-particle cluster operator and the two-particle cluster operator, <math>T_1</math> and <math>T_2</math> convert the reference function <math>|\Phi_0\rangle</math> into a linear combination of the singly and doubly excited Slater determinants respectively, if applied without the exponential (such as in [[Configuration interaction|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 <math>T_1</math> and  <math>T_2</math> that appear in the resulting expressions (see below). Solving for the unknown coefficients <math>t_a^i</math> and <math>t_{ab}^{ij}</math> is necessary for finding the approximate solution <math>|\Psi\rangle</math>.

The exponential operator <math>e^T</math> may be expanded as a [[Taylor series]], and if we consider only the <math>T_1</math> and <math>T_2</math> cluster operators of <math>T</math>, we can write

: <math>e^T = 1 + T + \frac{1}{2!} T^2 + \cdots = 1 + T_1 + T_2 + \frac{1}{2} T_1^2 + \frac{1}{2} T_1 T_2 + \frac{1}{2} T_2 T_1 + \frac{1}{2} T_2^2 + \cdots</math>

Though in practice this series is finite 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 <math>T_1</math> and <math>T_2</math>. 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 [[Møller–Plesset perturbation theory|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 [[Franck–Condon principle|Franck–Condon]] region), and especially when breaking single bonds or describing [[diradical]] species (these latter examples are often what is referred to as multi-reference 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 have small contributions for most problems, and as such, the contribution of <math>T_5</math>, <math>T_6</math> etc. to the operator <math>T</math> is typically small. Furthermore, if the highest excitation level in the <math>T</math> operator is ''n'',

: <math>T = T_1 + ... + T_n,</math>

then Slater determinants for an ''N''-electron system excited more than <math>n</math> (<math><N</math>) times may still contribute to the coupled-cluster wave function <math>|\Psi\rangle</math> because of the [[nonlinearity|non-linear]] nature of the exponential ansatz, and therefore, coupled cluster terminated at <math>T_n</math> usually recovers more correlation energy than CI with maximum ''n'' excitations.