Coupled cluster
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Coupled cluster (CC) is a numerical technique used for describing many-body systems. The method was initially developed by Fritz Coester and Hermann Kümmel in the 1950s for studying nuclear physics phenomena, but became more frequently used after Jiři Čížek and Josef Paldus reformulated the method for electronic correlation in atoms and molecules in the 1960s. It is now one of the most prevalent methods in quantum chemistry that include electronic correlation. It is important to note that coupled cluster is applicable specifically to fermionic systems, and in computational chemistry it is specifically applicable to electronic systems.
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A historical account
In the first reference below, Kümmel comments that
- 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 Jiri Cizek published his first paper on a quantum chemistry problem. He had looked into the 1957 and 1960 papers publised 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 Jiri'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 interaction
- Coupled-pair many-electron theory or CPMET, also known as the coupled cluster approximation
Background and derivation
There are two important theorems which are foundational to the coupled cluster approach. They are called the linked cluster theorem and the connected cluster theorem.
Wavefunction ansatz
To solve the time independent Schrödinger equation, one solves a system of the form
- <math>\hat{H} \vert{\Psi}\rangle = E \vert{\Psi}\rangle </math>
where <math>\hat{H}</math> denotes the Hamiltonian operator for the system, and E the energy levels of the system. The wavefunctions are denoted by <math>\vert{\Psi}\rangle</math>.
Coupled cluster theory proceeds by imposing an ansatz on the wavefunction, where an assumption is made that
- <math> \vert{\Psi}\rangle = e^\hat{T} \vert{\Phi_0}\rangle </math>,
where <math>\vert{\Phi_0}\rangle</math> is an appropriate reference function of some sort. For example, <math>\vert{\Phi_0}\rangle</math> can be a state function in Hartree-Fock space.
When this is substituted back into the original Schrödinger equation, the result is
- <math>\hat{H} e^\hat{T} \vert{\Psi_0}\rangle = E e^\hat{T} \vert{\Psi_0}\rangle </math>, from which we obtain
- <math>\hat{G} \vert{\Psi_0}\rangle = e^\hat{-T} \hat{H} e^\hat{T} \vert{\Psi_0}\rangle = E \vert{\Psi_0}\rangle </math>,
where <math>\hat{G} = e^\hat{-T} \hat{H} e^\hat{T} </math> is now a transformed operator.
This equation shows that the ansatz preserves the energies of the original Hamiltonian, while at the same time transforming the original Hamiltonian operator to the new operator G. Coupled cluster theory then suggests that there is an appropriate cluster operator <math>\hat{T}</math>, for which the resulting transformed Hamiltonian G is greatly simplified. This means that the matrix representation of G is a diagonal matrix or very nearly diagonal matrix.
The form of the wavefunction is an ansatz because there is no experimental rationale for suggesting that the wavefunction should be of this form - rather it is used because the form of the equation has particular properties which permit tractability of complicated quantum systems. In particular, because the cluster operator is exponentiated, an operator inside the cluster operator which corresponds to a single excitation of a state simultaneously excites other higher order states. This is because
- <math> e^\hat{T} = 1 + \hat{T} + \frac{\hat{T}^2}{2!} + \cdots = \sum_{n=0}^\infty \frac{\hat{T}^n}{n!} </math>
and from this Taylor series, it is apparent that higher order states are excited due to the compounding effect of repeated applications of the cluster operator. It is also noteworthy to remark that the excitation decreases as well. Finally, because the reference state <math>\Psi_0</math> usually represents a system of fermions, the Taylor series for the cluster operator terminates at a finite length, despite the fact that its mathematical representation is an infinite series. If the reference state is specific to, for example, electrons (which are spin-1/2 particles), there are only two spin states for each electron, and repeated application of the cluster operator will excite them out of the system. Hence, the series is finite due to the physical interpretation and relevance of the system of interest.
Hermiticity preservation
The transformation process, in general, does not preserve the hermitian nature of the original Hamiltonian - that is to say that
- <math>\hat{G}^\dagger = \left ( e^\hat{-T} \hat{H} e^\hat{T} \right )^\dagger
= e^{\hat{T}^\dagger} \hat{H} e^{\hat{-T}^\dagger}
\ne e^\hat{-T} \hat{H} e^\hat{T} = \hat{G} </math>
However, if provisions are made so that the cluster operator is antihermitian, where
- <math>\hat{T}^\dagger = -\hat{T} </math>
the transformation process preserves the hermiticity of the system.
Varieties of coupled cluster methods
Coupled cluster methods are able to give robust numerical calculations for systems of interest in quantum chemistry. Over the years, a number of minor modifications have been made or added to the original framework so that certain special cases of molecules and their energies can be computed. These modifications include
- Imposing the operator <math>e^{\hat{T}}</math> to be unitary;
- Adding higher order terms to the cluster operator <math>\hat{T}</math>
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 (shortened to doubles in coupled cluster terminology)
- T - for triple excitations (shortened to triples in coupled cluster terminology)
- Q - for quadruple excitations (shortened to quadruples in coupled cluster terminology)
Bracketed terms indicate that the higher order terms are calculated based on pertubation theory. For example, a CCSDT(Q) approach simply means:
- A Coupled cluster method
- It includes singles, doubles, and triples
- Quadruples are calculated with pertubation theory.
There is generally no accepted standard for higher order excitations, as their implementation and computational complexity are significantly demanding to perform. Additionally, the accuracy gained from using such methods is relatively minimal when compared to lower order methods. In principle, one would follow the conventions of appending extra letters for the ordinals, or use any appropriate notational method which conveys similar information.
Cluster operator
- <math> \hat{T}=\hat{T}_1 + \hat{T}_2 + \cdots </math>,
where in the formalism of second quantization:
- <math>
\hat{T}_1=\sum_{i}\sum_{a} t_{i}^{a} \hat{a}_{i}\hat{a}^{\dagger}_{a}, </math>
- <math>
\hat{T}_2=\frac{1}{4}\sum_{i,j}\sum_{a,b} t_{ij}^{ab} \hat{a}_{i}\hat{a}_j\hat{a}^{\dagger}_{a}\hat{a}^{\dagger}_{b}. </math>
In the above formulae <math>\hat{a}</math> and <math>\hat{a}^{\dagger}</math> denote the creation and annihilation operators respectively and i,j stand for occupied and a,b for unoccupied orbitals. T1 and T2 are called the one-particle excitation operator, and the two-particle excitation operator, because they effectively convert the reference function into a linear combination of singly- and doubly-excited Slater determinants. Solving for the coefficients <math>t_{i}^{a}</math> and <math>t_{ij}^{ab}</math>, in order to satisfy the definition of the cluster operator, constitutes a coupled cluster calculation. The operators in the coupled cluster term are normally written in canonical form, where each term is in normal order. Similar operators also appear in canonical pertubation theory.
The cluster operator can be represented in a vector space which spans the sequence of creation/annihilation operators which are in the cluster operator itself.
Coupled cluster with doubles (CCD)
In the simplest version one considers only <math>\hat{T}_2</math> operator (double excitations). This method is called coupled cluster with doubles (CCD in short).
Coupled cluster with singles and doubles (CCSD)
This version, as the name suggests, considers both <math>\hat{T}_2</math> and <math>\hat{T}_1</math> operators, accounting for both double and single excitations. The approximation is that <math>\hat{T}</math> = <math>\hat{T}_1</math> + <math>\hat{T}_2</math>.
Description of the theory
The method gives exact non-relativistic solution of the Schrödinger equation of the n-body problem if one includes up to the <math>\hat{T}_n</math> cluster operator. However, the computational effort of solving the equations grows steeply with the order of the cluster operator and in practical applications the method is limited to the first few orders.
Most frequently, one solves the CC equation using the operator <math>\hat{T} = \hat{T}_1 + \hat{T}_2</math>, which produces all Slater determinants which differ from the reference determinant by one or two spin-orbitals. This approach, called coupled-cluster singles and doubles (CCSD), has the effect of describing coupled two-body electron correlation effects and orbital relaxation effects. Because the operator <math>\hat{T}</math> is exponentiated in coupled-cluster theory, higher-order "disconnected" electron correlations are also accounted for in an approximate way. It is also fairly common (although also more computationally expensive) to include an approximate, pertubative correction accounting for three-body electron correlations in a method designated CCSD(T). For ground electronic states near their equilibrium geometries, CCSD(T) is often called a "gold standard" of quantum chemistry because it provides results very close to those of full configuration interaction (full CI), which solves the non-relativistic electronic Schrödinger equation exactly within the given one-particle basis set. More recently, coupled cluster methods have been developed which use the operator <math>\hat{T} = \hat{T}_1 + \hat{T}_2 + \hat{T}_3</math>, or even the operator <math>\hat{T} = \hat{T}_1 + \hat{T}_2 + \hat{T}_3 + \hat{T}_4</math>. These methods are called CCSDT and CCSDTQ. A simplification of the later method is CCSDT(Q) where the four-body terms are introducted by a perturbative correction to CCSDT.
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 valid.
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 <math>\vert{\Phi_0}\rangle</math>. 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).
The coupled cluster equations are usually derived using diagrammatic technique and result in nonlinear equations which can be solved in an iterative way. Converged solution requires usually a few dozens of iterations.
References
H.G. Kümmel, A biography of the coupled cluster method - found in R.F. Bishop, T. Brandes, K.A. Gernoth, N.R. Walet, Y. Xian (Eds.), Recent progress in many-body theories, Proceedings of the 11th international conference, World Scientific Publishing, Singapore, 2002, pp. 334-348.
External resources
- Cramer, C. J. Essentials of Computational Chemistry: Theories and Models. Wileyzh:偶合簇