Dynamical correlations coupled cluster methods

First let us review static and dynamic electron correlation. Dynamic (dynamical) electron correlation is easy to grasp, if not so easy to treat exhaustively. It is simply the adjustment by each electron, at each moment, of its motion in accordance with its interaction with each other electron in the system. Dynamic correlation and its treatment with perturbation (Mpller-Plesset), configuration interaction, and coupled cluster methods was covered in Section 5.4. 

The CASSCF method itself is not very useful for anything else than systems with few electrons unless an effective method to treat dynamical correlation effects could be developed. The Multi-Reference Cl (MRCI) method was available but was limited due to the steep increase of the size of the Cl expansion as a function of the number of correlated electrons, the basis set, and the number of active orbitals in the reference function. The direct MRCI formulation by P. Siegbahn helped but the limits still prevented applications to larger systems with many valence electrons [20], The method is still used with some success due to recent technological developments [21], Another drawback with the MRCI approach is the lack of size-extensivity, even if methods are available that can approximately correct the energies. Multi-reference coupled-cluster methods are studied but have not yet reached a state where real applications are possible. 

The main challenge as to an improved theory of electron correlation as a basis of accurate numerical quantum chemistry have been mentioned in this review, namely (a) the explicit treatment of the correlation cusp, (b) the formulation of methods that scale with a low power of the number of particles, (c) the consistent combination of MC-SCF-theory for the nondynamic and coupled-cluster methods for the dynamic correlation. 

Ah initio quantum chemical calculations are primarily used to describe the electronic character of individual molecules in the gas phase. Quantum chemical methods can vary widely in their accuracy, depending on the specific approximations taken. Those most applicable to the study of ionic liquids are medium level methods such as DFT and MP2 [20]. Hartree-Fock(HF) level calculations may be carried out as a starting point or to obtain geometries but should be followed by calculations that include some level of electronic correlation. Higher level methods such as Coupled Cluster methods (ie CCSD(T)) are only just accessible, and will not be routine. They do however allow for an estimation of effects hard to recover with the lower level DFT and MP2 methods, such as dispersion, more dynamic correlation, and an estimation of other neglected effects (such as the stabilization afforded by mixing in excited electronic states). The method employed (HF< DFT < MP2 < CCSD(T)) and sophistication of the basis set used are typically used to indicate the quality of an ab initio quantum chemical calculation. 

The second step of the calculation involves the treatment of dynamic correlation effects, which can be approached by many-body perturbation theory (62) or configuration interaction (63). Multireference coupled-cluster techniques have been developed (64—66) but they are computationally far more demanding and still not established as standard methods. At this point, we will only focus on configuration interaction approaches. What is done in these approaches is to regard the entire zeroth-order wavefunc-tion Tj) or its constituent parts double excitations relative to these reference functions. This produces a set of excited CSFs ( Q) that are used as expansion space for the configuration interaction (Cl) procedure. The resulting wavefunction may be written as... 

It is appropriate at this point to compare some formal properties of the three general approaches to dynamical correlation that we have introduced configuration interaction, perturbation theory, and the coupled-cluster approach. First, we note that taken fax enough (all degrees of excitation in Cl and CC, infinite order of perturbation theory) all three approaches will give the same answer. Indeed, in a complete one-paxticle basis all three will then give the exact answer. We axe concerned in this section with the properties of truncated Cl and CC methods and finite-order perturbation theory. 

Is there no rigorous way to separate static and dynamic electron correlation Dynamic correlation is present in any system with two or more electrons, but static correlation requires degenerate or near-degenerate partially-filled orbitals, a feature absent in normal closed-shell molecules. So in this sense they are separate phenomena. In another sense they are intertwined methods that go beyond the Hartree-Fock in invoking more than one determinant, namely Cl and its coupled cluster variant, improve the handling of both phenomena. 

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