Model space incomplete

SIZE-EXTENSIVE FORMULATIONS WITH INCOMPLETE MODEL SPACE (IMS)... 

Furthermore, since the same operators appear in all the n-valence sectors of the Fock-space, the projections onto the wave-functions can be performed at the very end, and it is possible to treat systems with varying n (i.e., different degrees of ionization) on the same footing, and in an explicitly size-extensive manner with respect to the valence electrons. As it has turned out, only the Fock—space approach has the potentiality to furnish explicitly connected size-extensive theories for a general incomplete model space/91-96/, which shows its flexibility and generality. 

There are mainly two inter-related reasons why on would like to formulate a size-extensive theory involving incomplete model spaces (i) to bypass intruders, (ii) for calculating the low-lying EE s of a closed-shell ground state, one feels that a set of hole-particle determinants would suffice as a choice for a reasonable model space, which involves valence holes as well as valence particles. This is an IMS. 

Mukherjee/91/ initially proved LCT for incomplete model spaces having n-hole n—particle determinants, showing also at the same time the validity of the core—valence separation. The corresponding open-shell perturbation theory of Brandow/20/ for such cases leads to unlinked terms and a breakdown of the core-valence separation, which used IN for O. Mukherjee emphasized that it is essential to have a valence-universal wave operator O within a Fock space formulation/91/ such that it also correlates the subduced valence sectors. Later on,... 

Kutzelnigg et. al795/ have discussed in detail the Fock—space classification of operators for quasi— complete model space. They also introduced a new type od IMS, called the isolated incomplete model space(IIMS). In IIMS, products of q—open operators are all q-open, never closed. As a result O. = 1., just as in a CMS. THe resulting CC equations for IIMS have thus exacly the same structure as in the CMS. 

U. Kaldor, Intruder states and incomplete model spaces in multireference coupled-cluster theory The 2p states of Be, Phys. Rev. A 38 (1988) 6013. 

Incomplete model spaces generate disconnected diagrams not only for the h used by Hose and Kaldor, but for other perturbative and CC formulations [20, 45, 46]. The effects of such diagrams on the size extensivity of the calculated energies depend both on the particular h and on the kind of model space used [21, 47-52]. Most of the recent work on... 

Recent developments include exact [12-14, 44, 90, 91] and approximate [14, 90, 92-94] iterative schemes to determine Hg, the intermediate Hamiltonian method [21, 24, 95], the use of incomplete model spaces [43, 44] and some multireference open-shell coupled-cluster (CC) formalisms [16-20, 96, 97]. Only some eigenvalues of the intermediate Hamiltonian H, are also eigenvalues of H. The corresponding model eigenvectors of H, are related to their true counterparts as in Bloch s theory. Provided effective operators a are restricted to act solely between these model eigenvectors, the possible a definitions from Bloch s formalism (see Section VI.A) can be used. 

It seems that intruders can be avoided almost entirely for states computed at fixed nuclear geometries by a judicious choice of the incomplete model space [27,28]. The situation is far less satisfactory, however, for generating potential surfaces. Here different intruder states plague the computations at different regions of the potential surface, and there is neither a unique nor a natural choice of an incomplete model space which avoids intruders at all geometries. It thus becomes necessary to switch over to different model spaces for describing the different regions of the surface [29]. 

Coupled Cluster based size-extensive intermediate hamiltonian formalisms were developed by our group [33-35] by way of transcribing a size-extensive CC formulation in an incomplete model space in the framework of intermediate hamiltonians. In this method, there are cluster operators correlating the main model space. There are no cluster operators for the intermediate space. This formulation thus is conceptually closer to the perturbative version of Kirtman... 

Molecular Applications of Multireference Coupled-Cluster Methods Using an Incomplete Model Space. Direa Calculation of Excitation Energies. (FS-MRCCSD.)... 

Recently, an important progress has been achieved by Li and Paldus [15,42,43], who generalized the Jeziorski-Monkhorst formulation to arbitrary incomplete model spaces. This brings two advantages—computational savings due to smaller number of... 

C-conditions for the incomplete model space [42,59] have to be employed. 

There have been several attempts to avoid the CAS restriction and achieve size extensivity for an incomplete model space (see, for example. Refs. [69,70]). The methods based on abandoning the intermediate normalization condition resulted in an excessively complex formalism and as far as we know were actually never implemented. Only recently this problem has been attacked successfully by Li and Paldus, who introduced so called C-conditions for the amplitudes of internal excitations [15,42,43,71]. When these conditions are incorporated into the MRCC amplitude equations of the Jeziorski-Monkhorst method, the solutions with a general incomplete model space become also exactly size extensive. The C-conditions are, however, not limited to the original Jeziorski-Monkhorst formulation [36], but rather they apply to any Hilbert-space MRCC... 

Debashis Mukherjee is a Professor of Physical Chemistry and the Director of the Indian Association for the Cultivation of Science, Calcutta, India. He has been one of the earliest developers of a class of multi-reference coupled cluster theories and also of the coupled cluster based linear response theory. Other contributions by him are in the resolution of the size-extensivity problem for multi-reference theories using an incomplete model space and in the size-extensive intermediate Hamiltonian formalism. His research interests focus on the development and applications of non-relativistic and relativistic theories of many-body molecular electronic structure and theoretical spectroscopy, quantum many-body dynamics and statistical held theory of many-body systems. He is a member of the International Academy of the Quantum Molecular Science, a Fellow of the Third World Academy of Science, the Indian National Science Academy and the Indian Academy of Sciences. He is the recipient of the Shantiswarup Bhatnagar Prize of the Council of Scientihc and Industrial Research of the Government of India. 

Excitation energies of atomic barium and radium were calculated in 1996 using the Fock-space coupled cluster method [57]. The model space in the 2-electron sector included all states with two electrons in the 5d, 6s and 6p orbitals, except the 6p states inclusion of the latter led to intruder states and divergence, so that incomplete model spaces had to be employed. In the intermediate Hamiltonian approach all these states (including 6p ) were in Pm, Pi was defined by adding states with occupied 7s-10s, 7p-10p, 6d-... 

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