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.