Editing Coupled cluster (section)

== Relation to other theories ==

=== Configuration interaction ===
The ''C<sub>j</sub>'' excitation operators defining the CI expansion of an ''N''-electron system for the wave function <math>|\Psi_0\rangle</math>,

: <math>|\Psi_0\rangle = (1 + C) |\Phi_0\rangle,</math>

: <math>C = \sum_{j=1}^N C_j,</math>

are related to the cluster operators <math>T</math>, since in the limit of including up to <math>T_N</math> in the cluster operator the CC theory must be equal to full CI, we obtain the following relationships<ref>{{cite book | last1 = Paldus | first1 = J. | title = Diagrammatic Methods for Many-Fermion Systems | year = 1981 | edition = Lecture Notes | location = University of Nijmegen, Njimegen, The Netherlands}}</ref><ref>{{cite book | last1 = Bartlett | first1 = R. J. | last2 = Dykstra | first2 = C. E. | last3 = Paldus | first3 = J. | title = Advanced Theories and Computational Approaches to the Electronic Structure of Molecules | editor-last = Dykstra | editor-first = C. E. | year = 1984 | pages = 127}}</ref>

: <math>C_1 = T_1,</math>

: <math>C_2 = T_2 + \frac{1}{2} (T_1)^2,</math>

: <math>C_3 = T_3 + T_1 T_2 + \frac{1}{6} (T_1)^3,</math>

: <math>C_4 = T_4 + \frac{1}{2} (T_2)^2 + T_1 T_3 + \frac{1}{2} (T_1)^2 T_2 + \frac{1}{24} (T_1)^4,</math>

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.&nbsp;99–194.

=== Symmetry-adapted cluster ===
The symmetry-adapted cluster (SAC)<ref>{{cite journal | last1 = Nakatsuji | first1 = H. | last2 = Hirao | first2 = K. | journal = Chemical Physics Letters | volume = 47 | issue = 3 | pages = 569 | year = 1977 | title = Cluster expansion of the wavefunction. Pseudo-orbital theory applied to spin correlation | doi = 10.1016/0009-2614(77)85042-2 |bibcode = 1977CPL....47..569N }}</ref><ref>{{cite journal | last1 = Nakatsuji | first1 = H. | last2 = Hirao | first2 = K. | journal = Journal of Chemical Physics | volume = 68 | issue = 5 | pages = 2053 | year = 1978 | title = Cluster expansion of the wavefunction. Symmetry-adapted-cluster expansion, its variational determination, and extension of open-shell orbital theory | doi = 10.1063/1.436028 |bibcode = 1978JChPh..68.2053N }}</ref> approach determines the (spin- and) symmetry-adapted cluster operator
: <math>S = \sum_I S_I</math>
by solving the following system of energy-dependent equations:

: <math>\langle\Phi| (H - E_0) e^S |\Phi\rangle = 0,</math>

: <math>\langle\Phi_{i_1 \ldots i_n}^{a_1 \ldots a_n}| (H - E_0) e^S |\Phi\rangle = 0,</math>

: <math>i_1 < \cdots < i_n, \quad a_1 < \cdots <a_n, \quad n = 1, \dots, M_s,</math>

where <math>|\Phi_{i_1 \ldots i_n}^{a_1 \ldots a_n}\rangle</math> are the {{not a typo|''n''-tuply}} excited determinants relative to <math>|\Phi\rangle</math> (usually, in practical implementations, they are the spin- and symmetry-adapted configuration state functions), and <math>M_s</math> is the highest order of excitation included in the SAC operator. If all of the nonlinear terms in <math>e^S</math> are included, then the SAC equations become equivalent to the standard coupled-cluster equations of Jiří Čížek. This is due to the cancellation of the energy-dependent terms with the disconnected terms contributing to the product of <math>H e^S</math>, resulting in the same set of nonlinear energy-independent equations. Typically, all nonlinear terms, except <math>\tfrac{1}{2} S_2^2</math> are dropped, as higher-order nonlinear terms are usually small.<ref>{{cite journal | last1 = Ohtsuka | first1 = Y. | last2 = Piecuch | first2 = P. | last3 = Gour | first3 = J. R. | last4 = Ehara | first4 = M. | last5 = Nakatsuji | first5 = H. | journal = Journal of Chemical Physics | volume = 126 | issue = 16 | pages = 164111 | year = 2007 | title = Active-space symmetry-adapted-cluster configuration-interaction and equation-of-motion coupled-cluster methods for high accuracy calculations of potential energy surfaces of radicals | doi = 10.1063/1.2723121 |bibcode = 2007JChPh.126p4111O | pmid = 17477593 | hdl = 2433/50108 | hdl-access = free }}</ref>