Editing Coupled cluster (section)

== Wavefunction ansatz ==
Coupled-cluster theory provides the exact solution to the time-independent Schrödinger equation

: <math>H |\Psi\rangle = E |\Psi\rangle,</math>

where <math>H</math> is the [[Molecular Hamiltonian|Hamiltonian]] of the system, <math>|\Psi\rangle</math> is the exact wavefunction, and ''E'' is the exact energy of the ground state. Coupled-cluster theory can also be used to obtain solutions for [[excited state]]s using, for example, [[linear response coupled-cluster|linear-response]],<ref>{{cite journal | last1 = Monkhorst | first1 = H. J. | title = Calculation of properties with the coupled-cluster method | journal = International Journal of Quantum Chemistry | volume = 12, S11 | pages = 421–432 | year = 1977 | doi =10.1002/qua.560120850}}</ref> [[equation-of-motion coupled cluster|equation-of-motion]],<ref>{{cite journal |last1 = Stanton |first1 = John F. |last2 = Bartlett |first2 = Rodney J. |title = The equation of motion coupled-cluster method. A systematic biorthogonal approach to molecular excitation energies, transition probabilities, and excited state properties |journal = The Journal of Chemical Physics |volume = 98 |pages = 7029 |year = 1993 |doi = 10.1063/1.464746 |bibcode = 1993JChPh..98.7029S |issue = 9 }}</ref> [[State-universal coupled cluster|state-universal multi-reference]],<ref>{{cite journal | last1 = Jeziorski | first1 = B. | last2 = Monkhorst | first2 = H. | title = Coupled-cluster method for multideterminantal reference states | journal = Physical Review A | volume = 24 | pages = 1668 | year = 1981 | doi = 10.1103/PhysRevA.24.1668 |bibcode = 1981PhRvA..24.1668J | issue = 4 }}</ref> or [[valence-universal multi-reference coupled cluster]]<ref>{{cite journal | last1 = Lindgren | first1 = D. | last2 = Mukherjee | title = On the connectivity criteria in the open-shell coupled-cluster theory for general model spaces | journal = Physics Reports | volume = 151 | issue = 2| year = 1987 | doi = 10.1016/0370-1573(87)90073-1 |bibcode = 1987PhR...151...93L | first2 = Debashis | pages = 93 }}</ref> approaches.

The wavefunction of the coupled-cluster theory is written as an exponential [[ansatz]]:

: <math>|\Psi\rangle = e^T |\Phi_0\rangle,</math>

where <math>|\Phi_0\rangle</math> is the reference wave function, which is typically a [[Slater determinant]] constructed from [[Hartree–Fock]] [[molecular orbital]]s, though other wave functions such as [[configuration interaction]], [[multi-configurational self-consistent field]], or [[Brueckner orbitals]] can also be used. <math>T</math> is the cluster operator, which, when acting on <math>|\Phi_0\rangle</math>, 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<sub>2</sub> when using a restricted Hartree–Fock (RHF) reference, which is not size-consistent, at the CCSDT (coupled cluster single-double-triple) level of theory, which provides an almost exact, full-CI-quality, potential-energy surface and does not dissociate the molecule into F<sup>−</sup> and F<sup>+</sup> ions, like the RHF wave function, but rather into two neutral F atoms.<ref>{{cite journal | last1 = Kowalski | first1 = K. | last2= Piecuch |first2 = P. | title = A comparison of the renormalized and active-space coupled-cluster methods: Potential energy curves of BH and F2 | journal = Chemical Physics Letters | volume = 344 | issue = 1–2 | pages = 165–175 | year = 2001 | doi=10.1016/s0009-2614(01)00730-8 |bibcode = 2001CPL...344..165K }}</ref> If one were to use, for example, the CCSD, or CCSD(T) levels of theory, they would not provide reasonable results for the bond breaking of F<sub>2</sub>, with the latter one approaches unphysical potential energy surfaces,<ref>{{cite journal | last1 =Ghose | first1 =K. B. | last2= Piecuch | first2=P. | last3=Adamowicz | first3= L. |title = Improved computational strategy for the state-selective coupled-cluster theory with semi-internal triexcited clusters: Potential energy surface of the HF molecule | journal= Journal of Chemical Physics |volume= 103 | issue =21 | pages= 9331 | year=1995 | doi=10.1063/1.469993 |bibcode = 1995JChPh.103.9331G }}</ref> though this is for reasons other than just size consistency.

A criticism of the method is that the conventional implementation employing the similarity-transformed Hamiltonian (see below) is not [[Variational principle|variational]], though there are bi-variational and quasi-variational 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 connected-cluster theorems, and thus does not suffer from issues such as lack of size extensivity, like the variational configuration-interaction approach.