Antibonding orbital defined

Problem 9-1. The bond order of a diatomic molecule is defined as [number of electrons in binding orbitals - number of electrons in antibonding orbitals]. Show that the following values are correct ... 

The fuzzy frontier between the molecular and the nanometric level can be elucidated from an electronic point of view. Molecules and small clusters can be described as systems in which the metal atoms form well-defined bonding and antibonding orbitals. Large clusters or small nanoparticles (quantum dots) with dimensions of a few nanometers are intermediate between the size of molecules and bulk material, presenting discrete energy levels with a small band gap owing to quantum-mechanical rules. Finally, larger particles tend to lose this trend and display a typical band structure similar to that of the bulk material. 

The material of the previous subsections may now be assembled into a useful result. If the energy zero for equation (18) is defined to be Uh then a simplification takes place we may write, remembering ah was defined to be the antibonding orbital, simply... 

The admixture of singly-excited configurations O, to Og to represent the HF wave function indicates electron transfer from bonding to antibonding orbitals and, accordingly, can be interpreted in terms of electron delocalization, which can be measured by the ratio CjlCG. Alternatively, one can define interbond populations by equation 1647 80 ... 

The bond order in a diatomic molecule is defined as one-half the difference between the number of electrons in bonding orbitals and the number of antibonding orbitals. The factor one-half preserves the concept of the electron pair and makes the bond order correspond to the multiplicity in the valence-bond formulation one for a single bond, two for a double bond, and three for a triple bond. Fractional bond orders are allowed, but are not within the scope of this discussion. 

We considered that nucleophilic addition is promoted when electron transfer from the reagent to the substrate is facilitated.41 Consequently, we proposed that X, Y and Z be classified according to their electrophilicity, as measured by the energy of their antibonding orbitals o cx, o CY and o cz. Thus, the best electron acceptor is considered to be the bulkiest group. This approach uses only one parameter and eliminates the tricky problem of defining effective bulk. See, however, the discussion on pp. 168-169. 

We define extended bond orbitals to be bond orbitals corrected in perturbation theory for those matrix elements that couple bonding and antibonding orbitals. This is done directly by using Eq. (1-16). Then corrections are added to every bond orbital 6> these arise from the interaction, 14, with neighboring antibond orbitals la),... 

For each bond orbital jb) there remains a sum over antibonding orbitals antibonding orbital in the same site. Let us consider that term first. The matrix elements can be written in terms of matrix elements of the hybrids on the two atoms by using the coefficients defined in Eqs. (3-13) and (3-16) for the bonding and antibonding states ... 

The topological or graph-theoretical approaches attempt to define the bonding and antibonding orbitals available to a cluster in terms of a sort of valence-bond description, where hybridized atomic orbitals are directed in space to form either localized or multicenter functions. (See also Hybridization) The analysis of a complex structure involving delocalized bonding may, however, require some seemingly subjective decisions about how the atomic orbitals overlap, where the multicenter bonds should be formed, and so forth. 

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