Multi-reference coupled cluster methods

W. D. Laidig and R. J. Bartlett, A multi-reference coupled-cluster method for molecular applications. Chem. Phys. Lett. 104, 424-430 (1984). 

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. 

Although the past 20 years have witnessed a great progress in the Hilbert space multi-reference coupled cluster methods (see, for example, the work of Mukherjee and Pal [99],Paldus [101], Jeziorski and Paldus [102], Jankowski et al. [103],Paldus et al. [104], Paldus et al. [105], Meissner et al. [106], Kucharski and Bartlett [107], Balkovd et al. [108], Baikova and Bartlett [109], Balkovd et al. [110], Baikova et al. [Ill], Berkovic and Kaldor [112]) only a few applications of this approach have been reported, mostly oriented to the simple model systems exploiting a lowdimensional model space. Among the reasons for this paucity of applications are the choice of an appropriate model space, convergence difficulties arising from intruder state problems and from multiple solutions of non-linear coupled cluster equations. 

Specifically, if T] < 0.02, the CCSD(T) metliod is expected to give results close the full Cl limit for the given basis set. If is larger than 0.02, it indicates that the reference wave function has significant multi-determinant character, and multi-reference coupled cluster should preferentially be employed. Such methods are being developedbut have not yet seen any extensive use. 

A posteriori corrections can be developed for calculations performed by using the Brillouin-Wigner perturbation expansion. These a posteriori corrections can be obtained for the Brillouin-Wigner perturbation theory itself and, more importantly, for methods, such as limited configuration interaction or multi-reference coupled cluster theory, which can be formulated within the framework of a Brillouin-Wigner perturbation expansion. 

Abstract The purpose of this paper is to introduce a second-order perturbation theory derived from the mathematical framework of the quasiparticle-based multi-reference coupled-cluster approach (Rolik and Kallay in J Chem Phys 141 134112, 2014). The quasiparticles are introduced via a unitary transformation which allows us to represent a complete active space reference function and other elements of an orthonormal multi-reference basis in a determinant-like form. The quasiparticle creation and annihilation operators satisfy the fermion anti-commutation relations. As the consequence of the many-particle nature of the applied unitary transformation these quasiparticles are also many-particle objects, and the Hamilton operator in the quasiparticle basis contains higher than two-body terms. The definition of the new theory strictly follows the form of the single-reference many-body perturbation theory and retains several of its beneficial properties like the extensivity. The efficient implementation of the method is briefly discussed, and test results are also presented. 

The tautomerism of furoxan (l,2,5-oxadiazole-2-oxide) has been investigated by different computational methods comprising modern density functions as well as single-reference and multi-reference ab initio methods. The ring-opening process to 1,2-dinitrosoethylene is the most critical step of the reaction and cannot be treated reliably by low-level computations (Scheme 2). The existence of cis-cis-trans- 1,2-dinitrosoethylene as a stable intermediate is advocated by perturbational methods, but high-level coupled-cluster calculations identify this as an artifact <2001JA7326>. 

In order to get more detailed information about, e.g., bond strengths and equilibrium geometries in transition metal systems it is necessary to include electron correlation. This can be done either by traditional ab initio quantum chemistry models, e.g., Cl-methods and coupled cluster methods, or by density functional theory (DFT) based methods. Correlated ab initio methods are often computationally very demanding, especially in cases where multi-reference based treatments are needed. Also, the computational cost of these methods increases dramatically with the size of the system. This implies that they can only be applied to rather small systems. 

Szalay, P. G. Bartlett, R. J. Multi-reference averaged quadratic coupled-cluster method a size-extensive modification of multi-reference Cl, Chem. Phys. Lett. 1993, 214, 481-488. 

Multi-reference Brillouin-Wigner coupled-cluster method with a general model space Molecular Physics 103,2239 (2005)... 

Abstract The singlet-triplet splittings of the di-radicals methylene, trimethylene-methane, ortha-, meta- and para-benzynes, and cyclobutane-l,2,3,4-tetrone have become test systems for the applications of various multi-reference (MR) coupled-cluster methods. We report results close to the basis set limit computed with double ionization potential (DIP) and double electron attachment (DBA) equation-of-motion coupled-cluster methods. These diradicals share the characteristics of a 2-hole 2-particle MR problem and are commonly used to assess the performance of MR methods, and yet require more careful study unto themselves as benchmarks. Here, using our CCSD(T)/6-311G(2d,2p) optimized geometries, we report DIP/DEA-CC results and single-reference (SR) CCSD, CCSD(T), ACCSD(T) and CCSDT results for comparison. 

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