Spin-free cumulants

The easiest and, in many respects, the most satisfactory way is to consider only the totally symmetric tensor components (i.e., the spin-free density matrices) and to define the spin-free cumulants in terms of these [17, 30]. This corresponds to replacing the considered state by an Ms-averaged ensemble. 

Keywords Unitary group adaptation Size-extensive SS and SU MRCC Spin-free RDM Spin-free cumulants Connectivity of cumulants Spin correlation of cumulants... 

Reduction of the expressions for the MRCC equations to explicitly connected ones via the decomposition of the n-RDMs into products of spin-free one-body RDMs and spin-free cumulants [80-82] and establish... 

For such Mi-averaged quantities, y " for both up and down spins are equal and hence each equals l/2y . With such simplifications, and introducing spin-free cumulants A in a similar manner ... 

Aspects of size extensivity in unitary group adapted multi-reference coupled cluster theories the role of cumulant decomposition of spin-free reduced density matrices... 

In the spin-free formulations of the UGA-MRCC theories, the use of CSFs entails that both the MRCC equations are in matrix form and the associated effective Hamiltonians will involve various w-body spin-free reduced density matrices (n-RDMs). n-RDMs are product separable and hence not size-extensive. From now on, we will refer to the spin-free RDMs as simply the ROMs. When spinorbital-based RDMs are discussed, we will explicitly indicate this. So, no confusions should arise. It is non-trivial to establish the extensivity of both the cluster operators and the effective Hamiltonian in spite of the occurrence of these n-RDMs. This paper will briefly review the formulation of the UGA-MRCC theories mentioned above and will present a comprehensive account of the aspects of connectivity which leads to extensivity. Although in some of our earlier papers [47] we sketched how size extensivity emerges after the cumulant decomposition of the n-RDMs, we will present here a detailed and thorough analysis of the underlying issues. 

The general structure of the cumulant decomposition of a spin-free RDM of given rank is discussed in some detail in the Appendix. In our discussions here, we will make use of the expressions explained therein and for the sake of continuity of reading refer to the appendix for further details. The cumulant decomposition of a 2-body RDM, is given by Eq. (35)... 

The spin-free form of the above cumulant decomposition requires careful handling. One may imagine that a straightforward spin-free form of the cumulant... 

In the general case of a fe-body spin-free RDM, the cumulant decomposition would require the concept of partitions i of k labels with Ni as the length of the upper and lower strings appearing in the products of yls in this partition. For the uniformity of notation, we would introduce the notation yli for y. The product of yds are all connected in each partition i due to connectivity of type (C). All the terms contributing to the partition are connected among... 

Here subscript cum in Nann means that this EUE index is produced by the so-called cumulant density matrix (6.45), as such RDM constmctions are termed in the current RDM theory [51]. For practical computations, within FCI or RAS-CI (restricted active space Cl), more suitable is a spin-free expression from [35]. 

Vinylidene complexes are electrophilic at the a-carbon. Therefore, they display reactivity more similar to that of Fischer-type carbene complexes than to that of Schrock-type carbene complexes. This trend fits with lire greater stability of the singlet spin state (and therefore higher HOMO-LUMO gap) of the free carbene in most Fischer carbene complexes and in vinylidenes. This trend also fits with the low oxidation state of the metal in most Fischer carbene complexes and vinylidene complexes. At the same time, some reactivity of vinylidine complexes is unique to those containing the cumulated unsaturation of a vinylidene unit. For example, the 3-carbon of a vinylidene ligand is nucleophilic and basic. 

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