Coupled cluster
Encyclopedia
Coupled cluster is a numerical technique used for describing many-body systems. Its most common use is as one of several quantum chemical
post-Hartree–Fock ab initio quantum chemistry methods
in the field of computational chemistry
. It essentially takes the basic Hartree–Fock molecular orbital
method and constructs multi-electron wavefunctions using the exponential cluster operator to account for electron correlation
. 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 Jiři Čížek (and later together with Josef Paldus
) reformulated the method for electron correlation in atoms and molecules. It is now one of the most prevalent methods in quantum chemistry
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 paper in 1966. He then also performed an application of it on benzene molecule with O. 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.
where
is the Hamiltonian
of the system. The wavefunction and the energy of the lowest-energy state are denoted by
and E, respectively. Other variants of the coupled-cluster theory, such as equation-of-motion coupled cluster and multi-reference coupled cluster may also produce approximate solutions for the excited state
s (and sometimes ground state
s) of the system.
The wavefunction of the coupled-cluster theory is written as an exponential ansatz
:
,
where
is a Slater determinant
usually constructed from Hartree–Fock molecular orbital
s.
is an excitation operator which, when acting on 
, produces a linear combination of excited Slater determinants (see section below for greater detail).
The choice of the exponential ansatz is opportune because (unlike other ansätze, for example, configuration interaction
) it guarantees the size extensivity of the solution. Size consistency
in CC theory, however, depends on the size consistency of the reference wave function. A drawback of the method is that it is not variational
.
,
where
is the operator of all single excitations, 
is the operator of all double excitations and so forth. In the formalism of second quantization these excitation operators are conveniently expressed as
and so forth.
In the above formulae
and 
denote the creation and annihilation operators respectively and i, j stand for occupied and a, b for unoccupied orbitals. The creation and annihilation operators in the coupled cluster terms above are written in canonical form, where each term is in normal order
. Being the one-particle excitation operator and the two-particle excitation operator,
and 
convert the reference function 
into a linear combination of the singly and doubly excited Slater determinants, respectively. Solving for the unknown coefficients 
and 
is necessary for finding the approximate solution 
.
Taking into consideration the structure of
, the exponential operator 
may be expanded into Taylor series
:
This series is finite in practice because the number of occupied molecular orbitals is finite, as is the number of excitations. In order to simplify the task for finding the coefficients t, the expansion of
into individual excitation operators is terminated at the second or slightly higher level of excitation (rarely exceeding four). This approach is warranted by the fact that even if the system admits more than four excitations, the contribution of 
, 
etc. to the operator 
is small. Furthermore, if the highest excitation level in the 
operator is n,
then Slater determinants excited more than n times may (and usually do) still contribute to the wave function
because of the non-linear
nature of the exponential ansatz. Therefore, coupled cluster terminated at
usually recovers more correlation energy than configuration interaction with maximum n excitations.
Suppose there are q coefficients t to solve for. Therefore, we need q equations. It is easy to notice that each t-coefficient may be put in correspondence with a certain excited determinant:
corresponds to the determinant obtained from 
by substituting the occupied orbitals i,j,k,... with the virtual orbitals a,b,c,... Projecting the Schrödinger equation above by q such different determinants from the left, we obtain the sought-for q equations:
where by
we understand the whole set of the appropriate excited determinants. To manifest the connectivity of these equations, we can reformulate the above equation in a more convenient form. We apply
to both sides of the coupled-cluster Schroedinger equations. After this we project the Schroedinger equation to 
and 
, and obtain:
,
the latter being the equations to be solved and the former the equation for the evaluation of the energy. Consider the standard CCSD method:
,
,
,
in which the similarity transformed Hamiltonian (defined as
) can be explicitly written down with the BCH formula:
.
The resulting similarity transformed Hamiltonian is not hermitian. Standard quantum chemistry packages (ACES II
, NWChem
, etc.) solve the coupled-equations iteratively using the Jacobi updates and the DIIS extrapolations of the t amplitudes.
. The abbreviations for coupled-cluster methods usually begin with the letters "CC" (for coupled cluster) followed by
Thus, the
operator in CCSDT has the form
Terms in round brackets indicate that these terms are calculated based on perturbation theory
. For example, a CCSD(T) approach simply means:
of quantum chemistry" for its excellent compromise between the accuracy and the cost for the molecules near equilibrium geometries. More complicated coupled-cluster methods such as CCSDT and CCSDTQ are used only for high-accuracy calculations of small molecules. The inclusion of all n levels of excitation for the n-electron system gives the exact solution of the Schrödinger equation
within the given basis set
, within the Born–Oppenheimer approximation (although schemes could also be drawn up to work without the BO approximation with great cost).
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.
The coupled-cluster method described above is also known as the single-reference (SR) coupled-cluster method because the exponential ansatz involves only one reference function
. The standard generalizations of the SR-CC method are the multi-reference (MR) approaches: state-universal coupled cluster
(also known as Hilbert space
coupled cluster), valence-universal coupled cluster (or Fock space
coupled cluster) and state-selective coupled cluster (or state-specific coupled cluster).
Quantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
post-Hartree–Fock ab initio quantum chemistry methods
Ab initio quantum chemistry methods
Ab initio quantum chemistry methods are computational chemistry methods based on quantum chemistry. The term ab initiowas first used in quantum chemistry by Robert Parr and coworkers, including David Craig in a semiempirical study on the excited states of benzene.The background is described by Parr...
in the field of computational chemistry
Computational chemistry
Computational chemistry is a branch of chemistry that uses principles of computer science to assist in solving chemical problems. It uses the results of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids...
. It essentially takes the basic Hartree–Fock molecular orbital
Molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The term "orbital" was first...
method and constructs multi-electron wavefunctions using the exponential cluster operator to account for electron correlation
Electronic correlation
Electronic correlation is the interaction between electrons in the electronic structure of a quantum system.- Atomic and molecular systems :...
. 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
Nuclear physics
Nuclear physics is the field of physics that studies the building blocks and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those...
phenomena, but became more frequently used when in 1966 Jiři Čížek (and later together with Josef Paldus
Josef Paldus
Josef Paldus is a Distinguished Emeritus of Applied Mathematics at the University of Waterloo, Ontario, Canada. His research is mainly in the field of quantum chemistry. Among other things, he is known for the introduction of coupled cluster theory in quantum chemistry and the adaptation of the...
) reformulated the method for electron correlation in atoms and molecules. It is now one of the most prevalent methods in quantum chemistry
Quantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
that includes electronic correlation.
CC theory is simply the perturbative variant of the Many Electron Theory (MET) of Oktay Sinanoğlu
Oktay Sinanoglu
Oktay Sinanoğlu is a Turkish scientist specializing in theoretical chemistry and molecular biology. In May 1963 at the age of 29 , he was full professor at Yale University...
, 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 paper in 1966. He then also performed an application of it on benzene molecule with O. 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 an approximate solution to the time-independent Schrödinger equationwhere
is the HamiltonianMolecular Hamiltonian
In atomic, molecular, and optical physics as well as in quantum chemistry, molecular Hamiltonian is the name given to the Hamiltonian representing the energy of the electrons and nuclei in a molecule...
of the system. The wavefunction and the energy of the lowest-energy state are denoted by
and E, respectively. Other variants of the coupled-cluster theory, such as equation-of-motion coupled cluster and multi-reference coupled cluster may also produce approximate solutions for the excited stateExcited state
Excitation is an elevation in energy level above an arbitrary baseline energy state. In physics there is a specific technical definition for energy level which is often associated with an atom being excited to an excited state....
s (and sometimes ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...
s) of the system.
The wavefunction of the coupled-cluster theory is written as an exponential ansatz
Ansatz
Ansatz is a German noun with several meanings in the English language.It is widely encountered in physics and mathematics literature.Since ansatz is a noun, in German texts the initial a of this word is always capitalised.-Definition:...
:
,where
is a Slater determinantSlater determinant
In quantum mechanics, a Slater determinant is an expression that describes the wavefunction of a multi-fermionic system that satisfies anti-symmetry requirements and consequently the Pauli exclusion principle by changing sign upon exchange of fermions . It is named for its discoverer, John C...
usually constructed from Hartree–Fock molecular orbital
Molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The term "orbital" was first...
s.
is an excitation operator which, when acting on , produces a linear combination of excited Slater determinants (see section below for greater detail).The choice of the exponential ansatz is opportune because (unlike other ansätze, for example, configuration interaction
Configuration interaction
Configuration interaction 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. Mathematically, configuration simply describes the linear combination...
) it guarantees the size extensivity of the solution. Size consistency
Size consistency
In quantum chemistry, size consistency is a property that guarantees the consistency of the energy behavior when interaction between the involved molecular system is nullified ....
in CC theory, however, depends on the size consistency of the reference wave function. A drawback of the method is that it is not variational
Variational principle
A variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which minimize or maximize the value of quantities that depend upon those functions...
.
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 these excitation operators are conveniently expressed asand so forth.
In the above formulae
and denote the creation and annihilation operators respectively and i, j stand for occupied and a, b for unoccupied orbitals. The creation and annihilation operators in the coupled cluster terms above are written in canonical form, where each term is in normal orderNormal order
In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered when all creation operators are to the left of all annihilation operators in the product. The process of putting a product into normal order is...
. Being the one-particle excitation operator and the two-particle excitation operator,
and convert the reference function into a linear combination of the singly and doubly excited Slater determinants, respectively. Solving for the unknown coefficients and is necessary for finding the approximate solution .Taking into consideration the structure of
, the exponential operator may be expanded into Taylor seriesTaylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
:
This series is finite in practice because the number of occupied molecular orbitals is finite, as is the number of excitations. In order to simplify the task for finding the coefficients t, the expansion of
into individual excitation operators is terminated at the second or slightly higher level of excitation (rarely exceeding four). This approach is warranted by the fact that even if the system admits more than four excitations, the contribution of , etc. to the operator is small. Furthermore, if the highest excitation level in the operator is n,then Slater determinants excited more than n times may (and usually do) still contribute to the wave function
because of the non-linearNonlinearity
In mathematics, a nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input; a linear system fulfills these conditions. In other words, a nonlinear system is any problem where the variable to be solved for cannot be...
nature of the exponential ansatz. Therefore, coupled cluster terminated at
usually recovers more correlation energy than configuration interaction with maximum n excitations.Coupled-cluster equations
Coupled-cluster equations are equations whose solution is the set of coefficients t. There are several ways of writing such equations but the standard formalism results in a terminating set of equations which may be solved iteratively. The naive variational approach does not take advantage of the connected nature of the cluster amplitudes and results in a non-terminating set of equations. The coupled cluster Schrödinger equation is formally:Suppose there are q coefficients t to solve for. Therefore, we need q equations. It is easy to notice that each t-coefficient may be put in correspondence with a certain excited determinant:
corresponds to the determinant obtained from by substituting the occupied orbitals i,j,k,... with the virtual orbitals a,b,c,... Projecting the Schrödinger equation above by q such different determinants from the left, we obtain the sought-for q equations:where by
we understand the whole set of the appropriate excited determinants. To manifest the connectivity of these equations, we can reformulate the above equation in a more convenient form. We apply to both sides of the coupled-cluster Schroedinger equations. After this we project the Schroedinger equation to and , and obtain:,the latter being the equations to be solved and the former the equation for the evaluation of the energy. Consider the standard CCSD method:
,,,in which the similarity transformed Hamiltonian (defined as
) can be explicitly written down with the BCH formula:.The resulting similarity transformed Hamiltonian is not hermitian. Standard quantum chemistry packages (ACES II
ACES (computational chemistry)
Aces II is an ab initio computational chemistry package for performing high-level quantum chemical ab initio calculations...
, NWChem
NWChem
NWChem is an ab initio computational chemistry software package which also includes quantum chemical and molecular dynamics functionality.It was designed to run on high-performance parallel supercomputers as well as conventional workstation clusters. It aims to be scalable both in its ability to...
, etc.) solve the coupled-equations iteratively using the Jacobi updates and the DIIS extrapolations of the t amplitudes.
Types of coupled-cluster methods
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- S - for single excitations (shortened to singles in coupled-cluster terminology)
- D - for double excitations (doubles)
- T - for triple excitations (triples)
- Q - for quadruple excitations (quadruples)
Thus, the
operator in CCSDT has the formTerms in round brackets indicate that these terms are calculated based on perturbation theory
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
. For example, a CCSD(T) approach simply means:
- A coupled-cluster method
- It includes singles and doubles fully
- Triples are calculated with perturbation theory.
General description of the theory
The complexity of equations and the corresponding computer codes, as well as the cost of the computation increases sharply with the highest level of excitation. For many applications the sufficient accuracy may be obtained with CCSD, and the more accurate (and more expensive) CCSD(T) is often called "the gold standardGold standard
The gold standard is a monetary system in which the standard economic unit of account is a fixed mass of gold. There are distinct kinds of gold standard...
of quantum chemistry" for its excellent compromise between the accuracy and the cost for the molecules near equilibrium geometries. More complicated coupled-cluster methods such as CCSDT and CCSDTQ are used only for high-accuracy calculations of small molecules. The inclusion of all n levels of excitation for the n-electron system gives the exact solution of the Schrödinger equation
Schrödinger equation
The Schrödinger equation was formulated in 1926 by Austrian physicist Erwin Schrödinger. Used in physics , it is an equation that describes how the quantum state of a physical system changes in time....
within the given basis set
Basis set
Basis set can refer to:* Basis * Basis set...
, within the Born–Oppenheimer approximation (although schemes could also be drawn up to work without the BO approximation with great cost).
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.
The coupled-cluster method described above is also known as the single-reference (SR) coupled-cluster method because the exponential ansatz involves only one reference function
. The standard generalizations of the SR-CC method are the multi-reference (MR) approaches: state-universal coupled clusterState-universal coupled cluster
State-universal coupled cluster method is one of several multi-reference coupled-cluster generalizations of single-reference coupled cluster method. It was first formulated by Bogumił Jeziorski and Hendrik Monkhorst in their work published in Physical Review A in 1981. State-universal coupled...
(also known as Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
coupled cluster), valence-universal coupled cluster (or Fock space
Fock space
The Fock space is an algebraic system used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named after V. A...
coupled cluster) and state-selective coupled cluster (or state-specific coupled cluster).
A historical account
In the first reference below, Kümmel comments:- 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 Jiři Čížek published his first paper on a quantum chemistry problem. He had looked into the 1957 and 1960 papers published in Nuclear Physics by Fritz and myself. 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 gave 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 Jiři'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.
Relation to other theories
- Configuration interactionConfiguration interactionConfiguration interaction 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. Mathematically, configuration simply describes the linear combination...
- Coupled-electron pair approximation (CEPA)