Abstract:The paper accounts for the author's activity in developing bond order and valence indices since the early 80s.These indices represent an important conceptual link between the physical description of molecules as systems of electrons and nuclei and the chemical picture of molecules consisting of atoms kept together by bonds. They are also useful for a systematization and interpretation of the results obtained in the quantum chemical calculations, by permitting to extract from the wave function different pieces of information that may be assigned chemical significance. In some cases they can have some predictive power, too. Historically, the prototypes of such indices were introduced in the semiempirical quantum chemistry; the most important developments were Coulson's charge-bond order matrix in the simple Hückel theory and the Wiberg index in the CNDO framework. (Valence indices were also introduced in the semiempirical theory.) The definition of the ab initio bond order index emerged from the asymptotic term of the exchange energy component of the partitioning performed in the framework of the author's so-called "chemical Hamiltonian approach" using a "mixed" second quantization formalism for overlapping basis sets. They can also be introduced by studying the exchange part of the two-particle density (or of the second-order density matrix). Some properties of the bond order indices are discussed and the author's (until now unpublished) proof is also presented, showing the sufficient conditions under which the bond order index of a homonuclear diatomics is equal to the "chemist's bond order," i.e., the half of the difference between the number of electrons occupying bonding and antibonding orbitals. The ab initio valence indices are also introduced and discussed, and it is stressed that for correlated wave function the same "exchange only" definition of the bond order and valence indices should be used, which was introduced for the SCF case. The recent concept of the "atomic decomposition of identity" is also discussed and it is utilized for introducing bond orders and valences in the framework of the "3D analysis," when atoms are defined not by their basis orbitals but as regions of the three-dimensional (3D) physical space. Two versions of the 3D analysis are considered-the AIM (atoms in molecules)-type decomposing the space into disjunct atomic domains and the "fuzzy atoms" scheme in which there are no sharp boundaries between the atoms but they exhibit a continuous transition from one to another.
The definition and properties of the bond order and valence indices calculated from ab initio wave functions are summarized. Their physical interpretation, relationships to the exchange effects in bonding and generalization to correlated wave functions are also discussed. Some examples with typical bond order and valence values are shown.
I. The DefinitionsThe present communication deals with the concept of the bond order (multiplicity) and valence indices calculated from the ab initio wave functions of different molecules [l-81. These are parameters of direct chemical significance, because they can be put well into correspondence with the corresponding classical chemical notions. They may be useful both in studying the links between quantum mechanics and "ordinary" chemistry in general and in giving better interpretations and deeper understanding of the results of the actual quantum chemical calculations for particular systems.
Bond order between atoms
E ( B # A )
The concept of bond order and valence indices calculated at the ab initio level is discussed in connection with their close relationship to the nonclassical exchange effects in bonding. An improved definition of bond order and free valence indices is given for the open-shell SCF (UHF) case, and the generalization of the bond order and valence indices to correlated wave functions is also introduced.
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