Hilbert space approach

Single-root MR BWCC theory Hilbert space approach 9... 

SINGLE-ROOT MR BWCC THEORY HILBERT SPACE APPROACH. 

For the sake of completeness, we recall that the idea of the single-root formalism exploiting the Hilbert space approach was also proposed by Banerjee and Simons [31] and Laidig and Bartlett [34,35]. In both approaches they start from the complete active space MC SCF wave function, however, in order to eliminate redundant cluster amplitudes they approximate the wave operator by... 

Superficially it might appear that Lindgren s formulation is a Hilbert space approach involving redundant S operators, and the redundancy is eliminated by a set of augmented equations which at the same time ensures the connectedness of H. However, as noted by Mukherjee, and Haque an ... 

We define an operator as closed , if its action on any model function G P produces only internal excitations within the IMS. An operator is quasi-open , if there exists at least one model function which gets excited to the complementary model space R by its action. Obviously, both closed and quasi-open operators are all labeled by only active orbitals. An operator is open , if it involves at least one hole or particle excitation, leading to excitations to the g-space by acting on any P-space function. It was shown by Mukheijee [28] that a size-extensive formulation within the effective Hamiltonians is possible for an IMS, if the cluster operators are chosen as all possible quasi-open and open excitations, and demand that the effective Hamiltonian is a closed operator. Mukhopadhyay et al. [61] developed an analogous Hilbert-space approach using the same idea. We note that the definition of the quasi-open and closed operators depends only on the IMS chosen by us, and not on any individual model function. 

The surface relaxation effects do not affect the second-order magnetic properties of AIMs such as magnetic susceptibility and chemical shielding. The transferability of these properties provides a theoretical basis for the empirical Pascal rules. A Hilbert space approach to the partitioning of magnetic susceptibility into atomic contributions has also been proposed. ... 

These two possible formulations of multi-reference Brillouin-Wigner coupled cluster theory are discussed further below. In Section 4.2.2.1, we present a multiroot formulation of Brillouin-Wigner theory. This formalism is employed in Section 4.2.2.2 to develop a multi-root, multi-reference Brillouin-Wigner coupled cluster theory, using a Hilbert space approach. In Section 4.2.2.3, we discuss the basic approximations employed in the multi-reference Brillouin-Wigner coupled cluster method. 

Multi-root multi-reference Brillouin-Wigner coupled cluster Hilbert space approach... 

L. Meissner, S. A. Kucharski, and R. J. Bartlett, A multireference coupled-cluster method for special classes of incomplete model spaces, J. Chem. Phys. 91, 6187 (1989) L. Meissner and R. J. Bartlett, A general model-space coupled-cluster method using a Hilbert-space approach, ... 

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