Coupled-cluster theory open-shell systems

For the treatment of electron correlation, Cizek uses classical techniques as well as techniques based on mathematical methods of quantum field theory, namely, a coupled-cluster approach. A rapid development and deployment of these methods during the past decade was stimulated by the realization of the importance of size consistency or size extensivity in the studies of reactive chemical processes. Although truly remarkable accuracy and development have been achieved for ground states of closed-shell systems, an extension to quasidegenerate and general open-shell systems is most challenging. Cizek also works on the exploitation of these approaches to study the electronic structure of extended systems (molecular crystals, polymers107). His many interests in-... 

G2(MP2)-RAD, which implements restricted-open-shell versions of both coupled cluster and perturbation theories. The latter method has been shown to generally yield reliable results when applied to open-shell systems [66]. 

P. Neogrady, M. Urban, and I. Hubac, /. Chem. Phys., 100, 3706 (1994). Spin Adapted Restricted Hartree-Fock Reference Coupled-Cluster Theory for Open-Shell Systems. 

T. D. Crawford, Ph.D. Thesis, University of Georgia, 1996. Many-Body Perturbation Theory and Perturbational Triple Excitation Corrections to the Coupled-Cluster Singles and Doubles Method for High-Spin Open-Shell Systems. 

Nonadditive effects in open-shell clusters have been investigated only recently and relatively little information is available on their importance and physical origin. From the theoretical point of view, open-shell systems are more difficult to study since the conventional, size-consistent computational tools of the theory of intermolecular forces, like the Mpller-Plesset perturbation theory, coupled cluster theory, or SAPT, are less suitable or less developed for applications to open-shell systems than to closed-shell ones. Moreover, there are many types of qualitatively different open-shell states, exhibiting different behavior and requiring different theoretical treatments. 

Coupled electron pair and cluster expansions. - The linked diagram theorem of many-body perturbation theory and the connected cluster structure of the exact wave function was first established by Hubbard211 in 1958 and exploited in the context of the nuclear correlation problem by Coester212 and by Coester and Kummel.213 Cizek214-216 described the first systematic application to molecular systems and Paldus et al.217 described the first ab initio application. The analysis of the coupled cluster equations in terms of the many-body perturbation theory for closed-shell molecular systems is well understood and has been described by a number of authors.9-11,67,69,218-221 In 1992, Paldus221 summarized the situtation for open-shell systems one must nonetheless admit... 

At first, aU these methods were developed for closed-shell systems only. Later research in this area was directed towards local methods for open-shell systems and excited states, local triples corrections beyond (T) (triples included in coupled cluster iterations), [138], local energy gradients for geometry optimizations of large molecules [139], combination of the local correlation method with explicitly correlated wavefunctions. It is evident from the discussion that these local 0 N) methods open the applications of coupled-cluster theory to entirely new classes of molecules, which were far ont-of-scope for such an accurate treatment before. Possible applications lie, for example, in the determination of the thermochemistry of reactions involving... 

Approaches which consider one state at a time are often referred to as one-state or state-selective or single-root . They were first proposed in the late 1970s. A paper published by Harris [113] in 1977, entitled Coupled cluster methods for excited states, first introduced the state-selective approach. Four papers which were published in 1978 and 1979 advancing the state-selective approach parts 6 and 7 of a series of papers entitled Correlation problems in atomic and molecular systems part 6 entitled Coupled cluster approach to open-shell systems by Paldus et al. [114] and part 7 with the title Application of the open-shell coupled cluster approach to simple TT-electron model systems by Saute, Paldus and Cfzek [115], and two papers by Nakatsuji and Hirao on the Cluster expansion of wavefunction, the first paper [116] having the subtitle Symmetry-adapted-cluster expansion, its variational determination, and extension of open-shell theory and the second paper [117] having the subtitle Pseudo-orbital theory based on sac expansion and its application to spin-density of open-shell systems. 

Except for the closed-shell CCSD theory of Section 13.7, the theory presented in this chapter has been that of spin-unrestricted coupled-cluster theory. Spin-unrestricted coupled-cluster theory has the advantage of conceptual simplicity and general applicability and is widely used for open-shell systems. Still, there are considerable disadvantages associated with the spin-unrestricted approach, making it worthwhile to look for an alternative approach for open-shell systems. First, spin-unrestricted coupled-cluster theory suffers from spin contamination, which may adversely affect the calculation of excitation processes and spin-dependent (magnetic) properties. Second, spin-unrestricted theory is expensive since, in the spin-orbital basis, we work with separate sets of orbitals for the alpha and beta spins. 

In the closed-shell theory of Section 13.7, we developed a spin-restricted theory in which both of these problems are solved. Thus, for closed-shell systems, we may calculate a singlet coupled-cluster wave function at a fraction of the cost of the corresponding spin-unrestricted wave function. Unfortunately, for open-shell systems, the problems are more complicated and it is no longer obvious how we should best satisfy the Schrbdinger and spin equations in coupled-cluster theory. 

Lindgren, A coupled-cluster approach to the many-body perturbation theory for open-shell systems, Int. J. Quantum Chem. Symp. 12 33 (1978). 

The effects of including the triple excitations in coupled cluster linear response theory for evaluating the dynamic polarizabilities have been assessed for a set of closed-shell (Ne, HF, N2, CO) and open-shell (CN, CO, O2) systems, in view of exploring a new accuracy regime for molecular properties. The main conclusions include that i) for systems with little or no static correlation, CC3 is nearly identical to CCSDT, ii) CC3 and PS(T) [pole shifted technique where the CCSD-LR poles are corrected by adding a noniterative correction due to the triples] methods perform better than CCSD but their relative accuracy is not determined yet, iii) differences between CCSD and CC3 results as well as the errors with respect to CCSDT drop when the basis set is increased, and iv) ROHF-based CC-LR approaches should be favored over their UHF counterparts while the dilfer-ences between the ROHF and UHF appear as an appropriate criterion for determining whether higher-order UHF-based CC calculations can be used. 

The coupled cluster (CC) approach is the most powerful and accurate of generally applicable electron correlation methods. This has been shown in many benchmark applications of 4-component relativistic CC methods to atoms [11-18] and molecules [19-31]. The CC method is an all-order, size-extensive, and systematic many-body approach. Multireference variants of relativistic 4-component CC methods capable of handling quasidegeneracies, which are important for open-shell heavy atomic and molecular systems, have been developed in recent years [15,17-19,21,31]. In particular, the multireference FSCC scheme [32,33] is applicable to systems with a variable number of particles, and is an ideal candidate for merging with QED theory to create an infinite-order size-extensive covariant many-body method applicable to systems with variable numbers of fermions and bosons [6,7]. 

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