Single-reference many-body perturbation

Cluster expansion representation of a wave-function built from a single determinant reference function [1] has been eminently successful in treating electron correlation effects with high accuracy for closed shell atoms and molecules. The cluster expansion approach provides size-extensive energies and is thus the method of choice for large systems. The two principal modes of cluster expansion developments in Quantum Chemistry have been the use of single reference many-body perturbation theory (SR-MBPT) [2] and the non-perturbative single reference Coupled Cluster (SRCC) theory [3,4]. While the former is computationally economical for the first few orders of the 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. 

MR-MBPT multi-reference many-body perturbation theory MR-MPPT multi-reference Mpller-Plesset perturbation theory CCSD single-reference coupled cluster with single and double replacements... 

If we except the Density Functional Theory and Coupled Clusters treatments (see, for example, reference [1] and references therein), the Configuration Interaction (Cl) and the Many-Body-Perturbation-Theory (MBPT) [2] approaches are the most widely-used methods to deal with the correlation problem in computational chemistry. The MBPT approach based on an HF-SCF (Hartree-Fock Self-Consistent Field) single reference taking RHF (Restricted Hartree-Fock) [3] or UHF (Unrestricted Hartree-Fock ) orbitals [4-6] has been particularly developed, at various order of perturbation n, leading to the widespread MPw or UMPw treatments when a Moller-Plesset (MP) partition of the electronic Hamiltonian is considered [7]. The implementation of such methods in various codes and the large distribution of some of them as black boxes make the MPn theories a common way for the non-specialist to tentatively include, with more or less relevancy, correlation effects in the calculations. 

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... 

Note that all of the above expressions are written in terms of single electron functions and no reference is made to many-electron functions. This is a fundamental characteristic of the many-body perturbation theoretic approach to the correlation problem. 

Many-body perturbation theory (MBPT) has been utilized as a convenient way of taking account of electron correlation beyond the Hartree-Fock (HF) approximation. In particular, its single-reference version is now fully established. M0ller-Plesset perturbation method [11], up to the fourth order, is provided as a standard tool in most... 

It is now well established by numerous and extensive applications that the single reference (SR) based many-body methods, viz. many-body perturbation theory (PT) [1], coupled cluster (CC) theory [2], coupled electron-pair approximations (CEPA) [3], etc. provide rather accurate descriptions of the energy in and around the equilibrium geometry of the closed-shell states. In particular, the single reference coupled cluster (SRCC)... 

A further restriction on the use of many-body perturbation techniques arises from the (quasi-) degenerate energy structure, which occurs for most open-shell atoms and molecules. In these systems, a single reference state fails to provide a good approximation for the physical states of interest. A better choice, instead, is the use of a multi-configurational reference state or model space, respectively. Such a choice, when combined with configuration interactions calculations, enables one to incorporate important correlation effects (within the model space) to all orders. The extension and application of perturbation expansions towards open-shell systems is of interest for both, the traditional order-by-order MBPT [1] as well as in the case of the CCA [17]. 

Olsen et al. and others rests on the assumption that the utihty of lower-order Moller-Plesset perturbation theory can be inferred from the behaviour of the higher-order terms in the perturbation series. It is widely appreciated that MoUer-Plesset perturbation theory and the equivalent many-body perturbation theory for a single determinantal reference function are not as robust as, for example, configuration interaction. Some care must therefore be exercised in applications to ensure that an appropriate reference function is employed. 

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