## Pauling bond order

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):

n_{x} = n_{o} EXP((r_{o} - r_{x})/c)
(1)

In equation (1) the bond order n_{x} of a bond of length r_{x}
is a function of a reference bond of length r_{o}, whose bond order is
defined as n_{o}. 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 n_{o}=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!

**references**

L. Pauling,

"Atomic Radii and Interatomic Distances in Metals"

*J. Am. Chem. Soc.* **1947**, *69*, 542 - 553.

J. Wilkie, I. H. Williams,

"Transition-State Structural Variation in a Model for Carbonyl
Reduction by Lactate Dehydrogenase: Computational
Validation of Empirical Predictions Based upon
Albery-More O'Ferrall-Jencks Diagrams"

*J. Am. Chem. Soc.* **1992**, *114*, 5423 - 5425.

K. N. Houk, S. M. Gustavson, K. A. Black,

"Theoretical Secondary Kinetic Isotope Effects and the
Interpretation of Transition State Geometries. 1. The Cope
Rearrangement"

*J. Am. Chem. Soc.* **1992**, *114*, 8565 - 8572.

S. S. Glad, F. Jensen,

"Kinetic Isotope Effects and Transition State Geometries.
A Theoretical Investigation of E2 Model Systems"

*J. Org. Chem.* **1997**, *62*, 253 - 260.

`last changes: 01.04.2008, AS
questions & comments to: axel.schulz@uni-rostock.de`