The degree or order of bonding between two atoms cannot only be determined through analysis of the molecular wavefunction, but also through using the bond distance/bond order correlation established by Linus Pauling. This empirical scheme is particularly useful for comparing calculated and experimentally determined data, but also has some value as a "post factum" analysis method. The approach is based on the assumption that bond orders vary exponentially with bond distances. This can be expressed in a straightforward manner with equation (1):
nx = no EXP((ro - rx)/c) (1)
In equation (1) the bond order nx of a bond of length rx is a function of a reference bond of length ro, whose bond order is defined as no. The constant c determines, how steeply the bond orders change with bond distances. Very frequently the reference bond distance is that of a single bond with no=1.0.
The original equation suggested by Pauling for single and double bonds uses a value of c = 0.3 (black line). For the analysis of bond orders in transition states this value leads to bond orders which are too small. For these structures a value of c = 0.6 appears to be more appropriate (blue line).
The calculation of bond orders n for ground state molecules will be demonstrated using ethylene, butadiene, and benzene as example. The following structural data have been calculated for these systems at the Becke3LYP/6-31G(d) level of theory (distances in pm):
The double bond in ethylene is used here as the reference systems for bonds with bond order 2 at a bond length of 133.09 pm. Using Paulings original expression (c = 0.3) the bond orders in butadiene amount to 1.937 (C1-C2) and 1.312 (C2 - C3), and to 1.607 for benzene. It is clear from this example, that the calculated bond orders always depend on the chosen reference system!
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