Single reference version

We have so far examined the performance of the canonical transformation theory when paired with a suitable multireference wavefunction, such as the CASSCF wavefunction. As we have argued, because the exponential operator describes dynamic correlation, this hybrid approach is the way in which the theory is intended to be used in general bonding situations. However, we can also examine the behavior of the single-reference version of the theory (i.e., using a Hartree-Fock reference). In this way, we can compare in detail with the related... 

The full-dimensional VCI calculations, mentioned above, were done with the single-reference version of Multimode. The calculations used a full-dimensional potential due to Ojamae, Shavitt, and Singer [57], version OSS3(p), and considered all four-mode couplings among the 15 normal modes. Until just recently, this was the only full dimensional potential surface for H5OJ. In many respects the OSS3(p) potential is quite realistic however, it is not spectroscopically accurate. 

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... 

By 1984, the palladium-catalyzed aUyhc alkylation reaction had been extensively studied as a method for carbon-carbon bond formation, whereas the synthetic utility of other metal catalysts was largely unexplored [1, 2]. Hence, prior to this period rhodium s abihty to catalyze this transformation was cited in only a single reference, which described it as being poor by comparison with the analogous palladium-catalyzed version [6]. Nonetheless, Yamamoto and Tsuji independently described the first rhodium-catalyzed decarboxylation of allylic phenyl carbonates and the intramolecular decarboxylative aUylation of aUyl y9-keto carboxylates respectively [7, 8]. These findings undoubtedly laid the groundwork for Tsuji s seminal work on the regiospecific rho-... 

There are two versions of MM. One, that we refer to as single-reference MM is based on the exact Watson Hamiltonian, which is the Hamiltonian in rectilinear... 

Several bacterial classification systems are known, but the one used throughout this book is that found in Bergey. This reference exists as a four volume series, and also a single volume version which is adequate for most laboratory purposes. 

The determinant 0) plays key role in the CC theory. It is called the reference determinant and in the standard version of the CC theory this is the determinant used to generate all necessary excitations in the wave function. For this reason the standard CC theory is called the single reference theory. Also, in the standard implementation of the theory, the T operator includes only single and double excitations from 0) (the CCSD theory),... 

Similar conclusions have been reached by Jankowski and Paldus in a careful comparative study involving CP-MET and various CEPA-type versions. These investigations were concerned with H4 as a model system and especially concentrated on the reliability of coupled pair approaches in the presence of quasi-degeneracy. It turned out that single-reference coupled pair methods except CEPA-0 and CEPA-2 performed unexpectedly well. 

Fig. 2. Dynamics of hard disks for periodic boundary conditions in the center of mass frame of reference. The broken circles show the initial positions of particles i and j, which collide at time ty. The broken lines show die particle paths from 1 = 0 to t=ty in the infinite-checkerboard version of periodic boundary conditions for which the particle coordinates are and The solid drcles show the position of the particles at ty including the image particles. In the single-cell version of periodic boundary conditions, the particles are at f (ty) and ij(ty) in the original cell.

The presented quasiparticle framework offers a simple way to develop the MR versions of the single-refer-ence-based correlation methods, such as the Cl or the CC approaches [37], To introduce the MR variant of the MBPT [6], first the Hamiltonian should be separated into a zeroth-oder and a perturbation part, H = + V. Following the... 

The second modern geminals-based approach was developed by Szalewicz and co-workers in the early 1980s. These authors are mostly interested in the closed-shell CCD method, its linearized version (LCCD), and the second-(MP2) and third-order (MP3) Mpller-Plesset perturbation theory approaches, which are contained in LCCD, In single-reference CCD theory, the wave function is written in the exponential form... 

To derive a multi-reference version of the bwcc theory, we shall follow a procedure which is similar to that used in the case of single-reference function. We begin again with eqs. (4.1)-(4.7) ... 

The formulation of a multi-reference bwcc theory can now proceed in two distinct ways. In the first option, we can formulate a multi-root version of the multi-reference BWCC theory which yields all roots of the d-dimensional 9 space simultaneously. This is the approach employed in most multi-reference coupled cluster formulations which are based on the Rayleigh-Schrodinger expansion. In the second option, we can use the state-specific wave operator (4.59) and formulate a state-specific (or single root) version of multi-reference bwcc theory [10]. 

To derive the SS-MRCEPA(2), we selectively consider the diagonal EPV terms. This implies that we retain terms of the type powers of Tifi acting on where excites to a l n- Thus, this coTold be termed as a quasi-linearized version of SS-MRCC theory. This approximation closely resembles Meyer s single-reference based CEPA(2) method [42]. The in this method is taken as... 

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