Coupled cluster (CC) is a numerical technique used for describing many-body systems.
[1][2][3] The method was initially developed by Fritz Coester and Hermann Kümmel in the 1950s for studying nuclear-physics phenomena, but became more frequently used when in 1966 Jiří Čížek (and later together with Josef Paldus) reformulated the method for electron correlation in atoms and molecules.
CC theory is simply the perturbative variant of the many-electron theory (MET) of Oktay Sinanoğlu, which is the exact (and variational) solution of the many-electron problem, so it was also called "coupled-pair MET (CPMET)".
, produces a linear combination of excited determinants from the reference wave function (see section below for greater detail).
This is easily seen, for example, in the single bond breaking of F2 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− and F+ ions, like the RHF wave function, but rather into two neutral F atoms.
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.
denote the creation and annihilation operators respectively, while i, j stand for occupied (hole) and a, b for unoccupied (particle) orbitals (states).
Applying the exponential cluster operator to the wave function, one can then generate more than doubly excited determinants due to the various powers of
, we can write 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
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 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).
The Schrödinger equation can be written, using the coupled-cluster wave function, as where there are a total of q coefficients (t-amplitudes) to solve for.
The resulting similarity-transformed Hamiltonian is non-Hermitian, resulting in different left and right 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 non-Hermitian theories as well).
solve the coupled-cluster equations using the Jacobi method and direct inversion of the iterative subspace (DIIS) extrapolation of the t-amplitudes to accelerate convergence.
The classification of traditional coupled-cluster methods rests on the highest number of excitations allowed in the definition of
The abbreviations for coupled-cluster methods usually begin with the letters "CC" (for "coupled cluster") followed by Thus, the
The most well known coupled-cluster method that provides an estimate of connected triples is CCSD(T), which provides a good description of closed-shell 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 CR-CC(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 coupled-cluster methods such as CCSDT and CCSDTQ are used only for high-accuracy calculations of small molecules.
One possible improvement to the standard coupled-cluster approach is to add terms linear in the interelectronic distances through methods such as CCSD-R12.
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.
Kümmel comments:[1] 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.
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.
[15] The Cj excitation operators defining the CI expansion of an N-electron system for the wave function
in the cluster operator the CC theory must be equal to full CI, we obtain the following relationships[16][17] etc.
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.
More powerful computers, as well as advances in theory (such as the inclusion of three-nucleon interactions), have spawned renewed interest in the method since then, and it has been successfully applied to neutron-rich and medium-mass nuclei.