Coupled cluster 2 method

Perturbation methods add all types of corrections (S, D, T, Q etc.) to the reference wave 

The T operator acting on a HF reference wave function generates all iih excited Slater determinants. 

It is customary to use the term amplitudes for the expansion coefficients t, which are 

The first term generates the reference HF and the second all singly excited states. The first parenthesis generates all doubly excited states, which may be considered as connected (T2) or disconnected (Tj). The second parenthesis generates all triply excited states, which again may be either true (T3) or product triples (T2T1, Tj). The quadruply excited states can similarly be viewed as composed of five terms, a true quadruple and four product terms. Physically a connected type such as T4 corresponds to four electrons interacting simultaneously, while a disconnected term such as T2 corresponds to two 

With the coupled cluster wave function (4.46) the Schrddinger equation becomes 

Perturbation methods add all types of corrections (S, D, T, Q etc.) to the reference wave function to a given order (2, 3, 4 etc.). The idea in Coupled Cluster (CC) methods is to include all corrections of a given type to infinite order. The (intermediate normalized) coupled cluster wave function is written as 

It is customary to use the term amplitudes for the expansion coefficients t, which are equivalent to the a, coefficients in eq. (4.1). 

Expanding out the exponential in eq. (4.46) and using the fact that the Hamilton operator contains only one- and two-electron operators (eq. (3.24)) we get 

The CC method [120] is one of the mathematically elegant techniques for estimating the electron correlation [102,121]. In this method, the FCI wavefunction is represented 

The cluster operator T is defined in terms of standard creation-annihilation operators as 

Coefficients t in the last expansion are called the CC amplitudes and they can be defined either variationally or by solving a system of linear equations. The total number of items in (5.15) equals the number of electrons N because no more than N excitations are possible. In most computer codes the nonvariational CC method is implemented since the variational one is technically very complicated. Formally operating on the flTfF with (1 -t-T) gives, in essence, the FCI wavefunction. However, the advantage of the CC representation (5.14) hes in the consequences associated with tnmcation of T [102]. When, in (5.15), only single or double excitations are involved, the method is called CCS or CCD, respectively, when single and double excitations -then CCSD, etc. Let us consider as an example the CCD approximation when T = T2 and the expansion (5.15) has the form 

Note that the first two terms in parentheses of (5.16) define the CID method. However, the remaining terms involve products of excitation operators. Each application of generates double excitations, so the product of two applications generates quadruple excitations. Similarly, the cube of J2 generates hextuple substitutions, etc. Such high-level excitations can not be practically included in Cl calculations (in this sense the RCI method is called nonsize consistent). 

The computational problem of the CC method is determination of the cluster ampUtudes t for all of the operators included in the particular approximation. In the standard implementation, this task follows the usual procedure of left-multiplying the Schrodinger equation by trial wavefunctions expressed as determinants of the HF 

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There is a variation on the coupled cluster method known as the symmetry adapted cluster (SAC) method. This is also a size consistent method. For excited states, a Cl out of this space, called a SAC-CI, is done. This improves the accuracy of electronic excited-state energies. 

Coupled Cluster methods, including doubles (energies and optimizations) or singles and doubles (energies only), and optional triples terms (CCD, CCSD, CCSD(T)). 

The structure of ozone is a well-known pathological case for electronic structure theory. Prior to the QCI and coupled cluster methods, it proved very difficult to model accurately. The following table summarized the results of geometry optimizations of ozone, performed at the MP2, QCISD and QCISD(T) levels using the 6-31G(d) basis set ... 

Since the singly excited determinants effectively relax the orbitals in a CCSD calculation, non-canonical HF orbitals can also be used in coupled cluster methods. This allows for example the use of open-shell singlet states (which require two Slater determinants) as reference for a coupled cluster calculation. 

Analogously to MP methods, coupled cluster theory may also be based on a UFIF reference wave function. The resulting UCC methods again suffer from spin contamination of the underlying UHF, but the infinite nature of coupled cluster methods is substantially better at reducing spin contamination relative to UMP. Projection methods analogous to those of the PUMP case have been considered but are not commonly used. ROHF based coupled cluster methods have also been proposed, but appear to give results very similar to UCC, especially at the CCSD(T) level. 

The use of Cl methods has been declining in recent years, to the profit of MP and especially CC methods. It is now recognized that size extensivity is important for obtaining accurate results. Excited states, however, are somewhat difficult to treat by perturbation or coupled cluster methods, and Cl or MCSCF based methods have been the prefen ed methods here. More recently propagator or equation of motion (Section 10.9) methods have been developed for coupled cluster wave functions, which allows calculation of exited state properties. 

Figures 11.9 and 11.10 compare the performance of the CCSD and CCSD(T) methods, based on either an RFIF or UHF reference wave function. Compared to the RMP plot (Figure 11.7), it is seen that the infinite nature of coupled cluster causes it to perform somewhat better as the reference wave function becomes increasingly poor. While the RMP4 energy curve follows the exact out to an elongation of 1.0A, the CCSD(T) has the same accuracy out to - 1.5 A. Eventually, however, the wrong dissociation limit of the RHF wave also makes the coupled cluster methods break down, and the energy starts to decrease.

The spin contamination makes the UCC energy curves somewhat too high in the intermediate region, but the infinite nature of coupled cluster methods makes them significantly better at removing unwanted spin states as compared to UMPu methods (Figure 11.8). 

Eliav, E., Kaldor, U. and Ishikawa, Y. (1994) Open-shell relativistic coupled-cluster method with Dirac-Breit wave functions Energies of the gold atom and its cation. Physical Review Letters, 49, 1724—1729 Including newer unpublished results from this group. 

Belpassi, L., Infante, I., Tarantelli, E. and Visscher, L. (2008) The Chemical Bond between Au(I) and the Noble Gases. Comparative Study of NgAuF and NgAu (Ng) Ar, Kr, Xe) by Density Functional and Coupled Cluster Methods. Journal of the... 

Stanton JF, Bartlett RJ (1993) The equation of motion coupled-cluster method - a systematic biorthogonal approach to molecular-excitation energies, transition-probabilities, and excited-state properties. J Chem Phys 98 7029... 

Notes CCSD(T) coupled cluster method. BLYP, B3LYP, mPWPW91, and TPSS Various density functional theory-based methods. 

The computational complexity of the coupled-cluster method truncated after a given excitation level m - for example, m = 2 for CCSD - may be discussed in terms of the number of amplitudes (Nam) in the coupled-cluster operator and the number of operations (Nop) required for optimization of the wavefunction. Considering K atoms, each with Nbas basis functions, we have the following scaling relations ... 

J. Noga, W. Klopper, and W. Kutzelnigg, in Recent Advances in Computational Chemistry, Vol. 3, Recent Advances in Coupled-Cluster Methods, R. J. Bartlett (ed.), World Scientific, Singapore (1997). 

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