CC methods

In the coupled-cluster (CC) method [61J, one expresses the wavefunctlon In a somewhat different manner ... 

As Bartlett [ ] and Pople have both demonstrated [M], there is a close relationship between the MPPT/MBPT and CC methods when the CC equations are solved iteratively starting with such an MPPT/MBPT-like initial guess for these double-excitation amplitudes. 

These approaches provide alternatives to the conventional tools of quantum chemistry. The Cl, MCSCF, MPPT/MBPT, and CC methods move beyond the single-configuration picture by adding to the wavefimction more configurations whose amplitudes they each detennine in their own way. This can lead to a very large number of CSFs in the correlated wavefimction and, as a result, a need for extraordinary computer resources. 

D) MOST PERTURBATION AND CC METHODS ARE SIZE-EXTENSIVE, BUT DO NOT PROVIDE UPPER BOUNDS AND THEY ASSUME THAT ONE CSF DOMINATES... 

In contrast to variational metliods, perturbation tlieory and CC methods achieve their energies by projecting the Scln-ddinger equation against a reference fiinction (transition formula (expectation value ( j/ It can be shown that this difference allows non-variational teclmiques to yield size-extensive energies. 

Professor Bartlett brought the CC method, developed earlier by others, into the mainstream of electronic structure theory. For a nice overview of his work on the CC method see ... 

The implementation of the CC method begins mueh as in the MPPT/MBPT ease one seleets a referenee CSF that is used in the SCF proeess to generate a set of spin-orbitals to be used in the subsequent eorrelated ealeulation. The set of working equations of the CC teehnique given above in Chapter 19.1.4 ean be written explieitly by introdueing the form of the so-ealled eluster operator T,... 

The CC method, as presented here, suffers from the same drawbaeks as the MPPT/MBPT approaeh its energy is not an upper bound and it may not be able to aeeurately deseribe waveflinetions whieh have two or more CSFs with approximately equal amplitude. Moreover, solution of the non-linear CC equations may be diffieult and slowly (if at all) eonvergent. It has the same advantages as the MPPT/MBPT method its energy is... 

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... 

The exaet definition is slightly more eomplieated, sinee the wave funetion has to be properly antisymmetrized and projected onto the acmal basis, but for illustration the above form is sufficient. Such R12 wave funetions may then be used in eonnection with the Cl, MBPT or CC methods described above. Consider for example a Cl calculation with an R12 type wave funetion. The energy is given as... 

The only generally applicable methods are CISD, MP2, MP3, MP4, CCSD and CCSD(T). CISD is variational, but not size extensive, while MP and CC methods are non-variational but size extensive. CISD and MP are in principle non-iterative methods, although the matrix diagonalization involved in CISD usually is so large that it has to be done iteratively. Solution of the coupled cluster equations must be done by an iterative technique since the parameters enter in a non-linear fashion. In terms of the most expensive step in each of the methods they may be classified according to how they formally scale in the large system limit, as shown in Table 4.5. 

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. 

It can clearly be seen that the CISD curve is worse than either of the other two, which are essentially identical out to a AR of 1.3 A. The size inconsistency of the CISD method also has consequences for the energy curve when the bond is only half broken. Figure 11.11 illustrates why the use of Cl methods has declined over the years, it normally gives less accurate results compared with MP or CC methods, but at a similar or Irigher computational cost. Furthermore, it is difficult to include the important triply excited configurations in Cl methods (CISDT scales as M ), but it is relatively easy to include them in MP or CC methods (MP4 and CCSD(T) scales as M ). 

In the semiclassical centrifugal sudden (SCS) approximation some additional simplifications were made, which permit us to estimate the scattering phase by Eq. (5.50). Therefore the accuracy of SCS has to be checked separately. Fortunately, for the Ar-N2 system some cross-sections were calculated by the BFCP method [200] as well as by the CC method [206], which is considered to be the best. Using the same potential as in [209] the SCS cross-sections were found in [191] for fixed total energy of collisions E. The results are compared in Table 5.1. 

Quadratic Cl (QCI) and coupled cluster (CC) exemplify more complex methods that are not strictly variational in character, but include physical corrections similar to those of higher-order perturbation theory. Keywords for these methods also include a specification of substitutions from the reference FIF configuration, such as QCISD or CCSD, respectively, for QCI or CC methods with all single and double substitutions. More complete descriptions of these methods are beyond the scope of this appendix. 

An important advantage of MP2 and higher-order perturbation methods is their size-consistency at every order. This is in contrast to many variational Cl methods, for which the calculated energy of two identical non-interacting systems might not be equal to twice that of an individual system. Size-consistent scaling is also characteristic of QCI and CC methods, which are therefore preferable to standard Cl-type variational methods for many applications. 

Recently, quantum chemical computational techniques, such as density functional theory (DFT), have been used to study the electrode interface. Other methods ab initio methods based on Hartree-Fock (HF) theory,65 such as Mollcr-PIcsset perturbation theory,66,67 have also been used. However, DFT is much more computationally efficient than HF methods and sufficiently accurate for many applications. Use of highly accurate configuration interaction (Cl) and coupled cluster (CC) methods is prohibited by their immense computational requirements.68 Advances in computing capabilities and the availability of commercial software packages have resulted in widespread application of DFT to catalysis. 

The CC method was developed for the system of interacting particles. The basic equation for this theory is... 

CC) methods, which have largely superseded Cl methods, in the limit can also be used to give exact solutions but again with same prohibitive cost as full Cl. As with Cl, CC methods are often truncated, most commonly to CCSD (N cost), but as before these can still only be applied to systems of modest size. Finally, Moller-Plesset (MP) perturbation theory, which is usually used to second order (MP2 has a cost), is more computationally accessible but does not provide as robust results. 

Recent years have witnessed a considerable activity towards extending the standard single-reference coupled-cluster (CC) methods (1-9) to potential energy surfaces (PESs) involving bond breaking without invoking a multireference description (see, e.g., refs 9-31). Undoubtedly, it would be very useful if we could routinely calculate large portions of molecular PESs with the ease-... 

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