Coupled cluster (CC) is a numerical technique used for describing
many-body system
The many-body problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of many interacting particles. ''Microscopic'' here implies that quantum mechanics has to be used to provid ...
s. Its most common use is as one of several
post-Hartree–Fock
In computational chemistry, post-Hartree–Fock methods are the set of methods developed to improve on the Hartree–Fock (HF), or self-consistent field (SCF) method. They add electron correlation which is a more accurate way of including the rep ...
ab initio quantum chemistry methods
''Ab initio'' quantum chemistry methods are computational chemistry methods based on quantum chemistry. The term was first used in quantum chemistry by Robert Parr and coworkers, including David Craig in a semiempirical study on the excit ...
in the field of
computational chemistry
Computational chemistry is a branch of chemistry that uses computer simulation to assist in solving chemical problems. It uses methods of theoretical chemistry, incorporated into computer programs, to calculate the structures and properties of m ...
, but it is also used in
nuclear physics. Coupled cluster essentially takes the basic
Hartree–Fock molecular orbital method and constructs multi-electron wavefunctions using the exponential cluster operator to account for
electron correlation
Electronic correlation is the interaction between electrons in the electronic structure of a quantum system. The correlation energy is a measure of how much the movement of one electron is influenced by the presence of all other electrons.
Atom ...
. Some of the most accurate calculations for small to medium-sized molecules use this method.
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
Josef Paldus, (born November 25, 1935 in Bzí, Czech Republic, died January 15, 2023 in Kitchener, Canada) is a Distinguished Professor Emeritus of Applied Mathematics at the University of Waterloo, Ontario, Canada.
Josef Paldus became associ ...
) reformulated the method for electron correlation in
atoms
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
and
molecules
A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioche ...
. It is now one of the most prevalent methods in
quantum chemistry
Quantum chemistry, also called molecular quantum mechanics, is a branch of physical chemistry focused on the application of quantum mechanics to chemical systems, particularly towards the quantum-mechanical calculation of electronic contributions ...
that includes electronic correlation.
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)". J. Čížek used the correlation function of MET and used Goldstone-type perturbation theory to get the energy expression, while original MET was completely variational. Čížek first developed the linear CPMET and then generalized it to full CPMET in the same work in 1966. He then also performed an application of it on the benzene molecule with Sinanoğlu in the same year. Because MET is somewhat difficult to perform computationally, CC is simpler and thus, in today's computational chemistry, CC is the best variant of MET and gives highly accurate results in comparison to experiments.
Wavefunction ansatz
Coupled-cluster theory provides the exact solution to the time-independent Schrödinger equation
:
where
is the
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
of the system,
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 states using, for example,
linear-response,
equation-of-motion,
state-universal multi-reference, or
valence-universal multi-reference coupled cluster approaches.
The wavefunction of the coupled-cluster theory is written as an exponential
ansatz
In physics and mathematics, an ansatz (; , meaning: "initial placement of a tool at a work piece", plural Ansätze ; ) is an educated guess or an additional assumption made to help solve a problem, and which may later be verified to be part of th ...
:
:
where
is the reference wave function, which is typically a
Slater determinant
In quantum mechanics, a Slater determinant is an expression that describes the wave function of a multi-fermionic system. It satisfies anti-symmetry requirements, and consequently the Pauli principle, by changing sign upon exchange of two electro ...
constructed from
Hartree–Fock molecular orbitals, though other wave functions such as
configuration interaction
Configuration interaction (CI) is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematical ...
,
multi-configurational self-consistent field
Multi-configurational self-consistent field (MCSCF) is a method in quantum chemistry used to generate qualitatively correct reference states of molecules in cases where Hartree–Fock and density functional theory are not adequate (e.g., for mole ...
, or
Brueckner orbitals Brueckner is a surname of German origin. Notable people with the surname include:
* Benedikt Brueckner (born 1990), German professional ice hockey player
* Georg F Brueckner (1930–1992), German martial arts pioneer and inventor
* Guenter Brueckne ...
can also be used.
is the cluster operator, which, when acting on
, 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
Configuration interaction (CI) is a post-Hartree–Fock linear variational method for solving the nonrelativistic Schrödinger equation within the Born–Oppenheimer approximation for a quantum chemical multi-electron system. Mathematical ...
) it guarantees the
size extensivity of the solution.
Size consistency In quantum chemistry, size consistency and size extensivity are concepts relating to how the behaviour of quantum chemistry calculations changes with size. Size consistency (or strict separability) is a property that guarantees the consistency of th ...
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 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. 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
2, with the latter one approaches 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 similarity-transformed Hamiltonian (see below) is not
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.
Cluster operator
The cluster operator is written in the form
:
where
is the operator of all single excitations,
is the operator of all double excitations, and so forth. In the formalism of
second quantization
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as t ...
these excitation operators are expressed as
:
:
and for the general ''n''-fold cluster operator
:
In the above formulae
and
denote the
creation and annihilation operator
Creation operators and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator (usually d ...
s respectively, while ''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
. Being the one-particle cluster operator and the two-particle cluster operator,
and
convert the reference function
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
and
that appear in the resulting expressions (see below). Solving for the unknown coefficients
and
is necessary for finding the approximate solution
.
The exponential operator
may be expanded as a
Taylor series, and if we consider only the
and
cluster operators 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
and
. 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
In chemistry, a diradical is a molecular species with two electrons occupying molecular orbitals (MOs) which are degenerate. The term "diradical" is mainly used to describe organic compounds, where most diradicals are extremely reactive and i ...
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
,
etc. to the operator
is typically small. Furthermore, if the highest excitation level in the
operator is ''n'',
:
then Slater determinants for an ''N''-electron system excited more than
(