Size extensive formulations general formulation

An important insight in the development of size-extensive formulations in a IMS was the realization that the intermediate normalization convention for the wave operator, viz. PflP = P, should be abandoned in favor of a more appropriate normalization [28,61]. For the IMS, in general, products of quasi-open operators may lead to internal excitations, or may even be closed, so that if we choose il = 2 exp(7 )l< X< l, with = Top-f Tq-op, then powers of Tq. op coming from the exponential might lead from 4>p to internal excitations to some other model function or it may contain closed operators. We would have to bear this in mind while developing our formalism, and would not force POP = P in our developments. 

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... 

The next set of open-shell cluster expansion theories to appear on the scene emphasized the size-extensivity feature (al), and all of them were designed to compute energy differences with a fixed number of valence electrons. Several related theories may be described here - (i) the level-shift function approach in a time-dependent CC framework by Monkhorst/56/ and later generalizations by Dalgaard and Monkhorst/57/, also by Takahasi and Paldus/105/, (ii) the CC-based linear response theory by Mukherjee and Mukherjee/58/, and generalized later by Ghosh et a 1/59.60.107/,(iii)the closely related formulations by Nakatsuji/50,52/ and Emrich/62/ and (iv) variational theories by Paldus e t a I / 54/ and Saute et. al /55/ and by Nakatsuji/50/. 

Because of its size-extensivity and faster convergence with respect to excitation level Coupled cluster theory has replaced Cl theory as the dominant approach in ab initio correlation calculations. Like MBPT the theory is still mainly applied in cases where the exact wave function is dominated by a single determinant, but multireference methods have been formulated and begin to enter mainstream quantum chemistry. Generalization of the algorithms to the relativistic no-pair level can again be achieved via the spinorbital formulation of the methods. I will first discuss the single reference method and then consider the Fock space method [40] that uses multi-reference wavefiinctions for ionized or excited states. 

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... 

In conclusion, in the unlinked formulation of coupled-cluster theory, the amplitude equations yield solutions that are size-extensive but not termwise so. In general, therefore, it is not trivial to make size-extensive approximations to the unlinked coupled-cluster equations and the linked equations are to be preferred for such purposes. 

The insertion of various isocyanates into chromium(lll) alkoxide M—O bonds has been reported.737 The complexes are prepared by refluxing the isocyanates with a suspension of the alkoxide in benzene. No structural data were given for the products. Unusual bimetallic alkoxides have recently been prepared738 by the reaction of Cr[Al(OPr )4]3 with alcohols and acetylacetone (166). A wide range of spectroscopic methods were used to study them. In general, the results were in accord with a monomeric formulation similar to (166) below Cr[Al(OMe)4]3 was grossly insoluble the small size of the methyl groups may permit extensive polymerization. 

Microemulsions are defined as isotropic, transparent, and thermodynamically stable (in contrast to conventional emulsions) mixtures of a hydrophobic phase (lipid), a hydrophilic phase (often water), a surfactant, and in many cases a co-surfactant. From a lipid formulation perspective, microemulsions are generally regarded as the ultimate extension of the decreased particle size/increased surface area mantra, because emulsion particle sizes are usually less than 50 nm. Microemulsions also have additional pharmaceutical advantages in terms of their solubilizing capacity [54, 55], thermodynamic stability, and capacity for stable, infinite dilution. 

Some of the most widely used indices are the molecular connectivity indices x, which were originally formulated by Randic [46] and subsequently generalized and extended by Kier and Hall [47]. The family of indices is extensive and incorporates information about size, branching, unsaturation, heteroatoms, and numbers of rings. Each atom is given two descriptors, <5 (the number of adjacent nonhydrogen atoms)... 

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