Model space general

G. Hose and U. Kaldor, /. Phys., B Atom. Mol. Phys., 12, 3827 (1979). Diagrammatic Many-Body Perturbation Theory for General Model Spaces. 

Finally, let me mention our recent contribution toward the MR CC methodology that is based on a completely general model space (GMS) [202,223]. In contrast to an IMS, which results by a truncation of the CMS, we define GMS as a model space that is spanned, at least in principle, by a set of arbitrarily chosen configurations. 

We have also presented a recently developed size-extensive and size-consistent SS-MRCC approach based on a general model space. For this, the intermediate normalization convention of the wave operator has to be abandoned in favor of some appropriate size-extensive normalization. Suitable operators, defined in Fock space—described as closed, open and quasi-open— have to be introduced to ensure that the effective operator furnishing the target energy on diagonalization is a closed operator. 

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

G. Hose and U. Kaldor, Diagrammatic many-body perturbation theory for general model spaces, J. Phys. B 12,3827 (1979) A general model space diagrammaticpeturbation theory, ... 

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, ... 

G. Hose and V. Kaldor, A general-model-space diagrammatic perturbation theory, Physica Scripta 21 357 (1980). 

All predictions must be taken for what they are, namely, generalizations based on current knowledge and understanding. There is a temptation for a user to assume that a computer-generated answer must be correct. To determine whether this is in fact the case, a number of factors concerning the model must be addressed. The statistical evaluation of a model was addressed above. Another very important criterion is to ensure that a prediction is an interpolation within the model space, and not an extrapolation outside of it. To determine this, the concept of the applicability domain of a model has been introduced [106]. 

In actual practice a number of tests must be passed at various nodes before final classification takes place. Also, a prohibitive time would be required to search a large database of models for ones which most closely approximated the actual data set. For this reascxi the concept of similarity nets is introduced. In this case, a more general model is first chosen, one which is clearly not conpletely absurd. A subset of other models which are variations of this first general model then provides the index for the final choice of model. Such a reduction in the model lists greatly reduces the search space for the closest fit. 

The main reason why existing MR CC methods as well as related MR MBPT cannot be considered as standard or routine methods is the fact that both theories suffer from the Intruder state problem or generally from the convergence problems. As is well known, both MR MBPT/CC theories are built on the concept of the effective Hamiltonian that acts in a relatively small model or reference space and provides us with energies of several states at the same time by diagonalization of the effective Hamiltonian. In order to warrant size-extensivity, both theories employ the complete model space formulations. Although conceptually simpler, the use of the complete model space makes the calculations rather... 

In the development of a general state-space representation of the reactor, all possible control and expected disturbance variables need to be identified. In the following analysis, we will treat the control and disturbance variables identically to develop a model of the form... 

Equations (3.23) and (3.24) are valid also for a model space containing several unperturbed energies, e.g. several atomic configurations. These equations will form the basis for our many-body treatment. The generalized Bloch equation is exact and completely equivalent to the Schrodinger equation for the states considered. 

In general, the corresponding energies are obtained by diagonalizing this effective Hamiltonian in the model space. If the model functions are already found, then the energies can be calculated directly starting with Eq. (3.29) and evaluating the matrix element, i.e. 

We will formulate the problem simply but generally. Consider a system of structureless, classical particles, characterized macroscopically by a set of thermodynamic coordinates (such as the temperature T) and microscopically by a set of model parameters that prescribe their interactions. The two sets of parameters play a strategically similar role it is therefore convenient to denote them, collectively, by a single label, c (for conditions or constraints or control parameters in thermodynamic-and-model space). 

The general idea of restriction to a model space underlies the development... 

Sometimes the model space has a natural basis which is not orthogonal, and a transformation to achieve orthogonalization may be desired. There are different orthogonalization procedures, e.g., as reviewed by Lowdin [117], but generally orthogonalization results in a transformation of the initial Hamiltonian H with overlap operator S to a new Hamiltonian... 

The matrix defined in Eq.(47) represents a general, non-relativistic spin-independent TV-electron Hamiltonian given by Eq.(2) in a specifically defined model space. The one-electron orbital space is spanned by N orthonormal localized orbitals 4>j, j = 1,2,..., TV and the TV-electron orbital space is one-dimensional with the basis function... 

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