Generalized bond index

Another useful notion is the so-called generalized bond index Kab referred to the given atoms A and B. According to [36], in case of CIS excited states, Kab, more exactly, Kab[C1S], turns out to be connected with CT numbers Ia b and some additional quantities. Before giving expressions applicable also to RPA and related models, we briefiy clarify the meaning of the generalized bond index. Even within the elementary MO theory, such as the tt-electron model, the corresponding quantity, that is /sT,uy[MO], is not the same as the usual bond order Instead, the squared bond order makes its appearance, namely, by definition the orbital index... 

To define Kab and more general multicenter indices at a post Hartree-Fock level, one must invoke more sophisticated techniques [64-67]. We follow our work [36] in which the full Cl and some approximated models were considered in detail. The conventional definition of the generalized bond index is based on identification of Kab with a charge density fluctuation measured via the second-order joint statistical moment [64] (generally, the joint cumulant) ... 

General an Index The general a -index (GAI) applies to molecules that contain heteroatoms, multiple bonds, and cis/rrans-specified double-bond locations [30], This approach is based on the orbital interaction graph of linked atoms (OIGLA) and the orbital interaction matrix of linked atoms (OIMLA). The GAI is computed as the absolute value of the determinant of OIMLA ... 

Generally, only powders that form hard compacts under an applied pressure without exhibiting any tendency to cap or chip can be considered as readily compactible. The compactibility of pharmaceutical powders can be characterized by its tensile strength and indentation hardness, which can be used to determine three dimensionless parameters strain index, bonding index, and brittle fracture index. 

Lukovits, I. (1990a). The Generalized Wiener Index for Molecules Containing Double Bonds and the Partition Coefficients. Rep.Mol.Theory, 1,127-131. 

As explained in the preceding section, the bond index and the valency have definite physical meanings only when basis functions are orthonormalized. MOPAC satisfies this assumption(6), but general MO methods, including DV-Xa, don t satisfy it. Fortunately basis functions can be orthonormalized after MO calculation without any loss of accuracy, provided that all overlap integrals are known. 

Armstrong, Perkins and Stewart s bond index and valency, which have been used in MOP AC, were extended to general MO methods including DV-Xa, by the way of basis function orthonormalization. 

Variants and generalizations of the bond index were proposed by Mayer [Mayer, 1986b, 1986a Wang and Werstiuk, 2003 Ponec and Cooper, 2005]. For example, in the framework of the AIM theory, starting from the correlation function characterizing the extent of the electron sharing and defined as... 

Lukovits, I. (1990a) The generalized Wiener index for molecules containing double bonds and the partition coefficients. Rep. Mol. Theory, 1,... 

Growing interest has been focussed on transition-metal photochemistry in view of its potential relevance to the problem of solar energy fixation. Among the more general developments however, a new formalism of bond indexes proposed by Vanquickenborne and Ceulemans promises to rationalize the photochemistry of hexaco-ordinate complexes of transition metals. The ligand most readily expelled is deemed to be that having the smallest bond index. Vogler and Kunkely have... 

Reactivity/ bonding index General principle Special principle Principle of chemical reactivity/bonding... 

Construction of the generalized connectivity index x(HD) for 3,4-dimethylheptane is illustrated in Table 9.2 at left. After making a list of bonds (the first column of... 

Very recently, Shustorovich and coworkers have extended the conceptual framework of this initial BOC-MP model to prove that the quadratic exponential potential (QEP), expressed in terms of a unity bond index (UBI) after normalization, provides a general, accurate description of any two-center, quasi-spherical interaction. As a result, the formalism has been renamed the UBI-QEP method [19-22]. This recent work includes a newly developed formalism to calculate the heat of adsorption of polyatomic molecules, such as ethylene and acetylene, without bond energy partitioning [19,22]. The reader is invited to study these latter papers to learn the most recent and accurate applications of this approach, which will be referred to as the BOC-MP/UBI-QEP method in this chapter. In this section, only an overview of the initial BOC-MP method will be provided, and greater detail is provided in these publications of Shustorovich [16-18]. 

The first Hilbert space definition of bond order was put forward by Wiberg in 1968 and generalized by Armstrong, Perkins, and Stewart in 1973. Ignoring the nonorthogonalities among basis functions, the Wiberg bond index reads... 

Molecular Connectivity Indexes and Graph Theory. Perhaps the chief obstacle to developing a general theory for quantification of physical properties is not so much in the understanding of the underlying physical laws, but rather the inabiUty to solve the requisite equations. The plethora of assumptions and simplifications in the statistical mechanics and group contribution sections of this article provide examples of this. Computational procedures are simplified when the number of parameters used to describe the saUent features of a problem is reduced. Because many properties of molecules correlate well with stmctures, parameters have been developed which grossly quantify molecular stmctural characteristics. These parameters, or coimectivity indexes, are usually based on the numbers and orientations of atoms and bonds in the molecule. 

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