The first question we may address is the relationship between the bond
modulus and the equilibrium bond length 
. If we take the
logarithm of Equation 
we obtain
A graph of 
against 
is
shown in Figure . If the regression coefficient is
calculated for this plot we obtain a value of -0.38 which
is statistically insignificant at a level of significance of
10%. Thus, there is no statistical evidence of a correlation between
and and hence we can
conclude that the bond modulus is independent of in this model.
The values of k are plotted against 
in Figure
. The regression coefficient for these data is 0.72 which
is statistically significant at a level of significance of 1%. Thus,
we can conclude that there is a correlation between the bond modulus
and overlap population. The best fit line for the data is
We can see that the best fit line does not pass through the origin,
but there is a small, negative value for the y-axis intercept. This
is not unreasonable, as the previous investigation in Section
showed that the minimum overlap population we can expect
between two bonded atoms is approximately 0.19e. Hence, we would expect
the bond modulus to reach zero before the overlap population.
Figure: Graph of bond modulus against Mulliken overlap
population. The best fit line 
is plotted for
comparison.
This analysis has shown that there is a linear relationship between overlap population and bond modulus within the harmonic approximation. The correlation between these values is not ideal which may be because population analysis takes no account of the detailed electronic structure of the bond and the model does not consider factors such as anharmonic effects. However, results show that the overlap population may be used as an approximate measure of the bond modulus.
