To illustrate how choices of coordinate systems for calculations of chemical bonds can immensely affect the results and consequently engender ill-defined descriptors of the bonds, sample calculations for n-butane and cyclobutane are shown in this section. Note that it is known that the all the four equivalent C-C bonds in cyclobutane are weaker than any of the two distinct C-C bonds in n-butane; therefore, juxtaposition and evaluation of the strength of the C-C bonds in this C4 system can exemplify how force constants fail and how compliance constants do not. The tables immediately below are results that are calculated at MP2/aug-cc-pvtz level of theory based on typical force constants calculation.
Tables 1 and 2 display a force constant in N/cm between each pair of carbon atoms (diagonal) as well as the coupling (off-diagonal). Considering natural internal coordinates on the left, the results make chemical sense. Firstly, the C-C bonds are n-butane are generally stronger than those in cyclobutane, which is in line with what is expected. Secondly, the C-C bonds in cyclobutane are equivalent with the force constant values of 4.173 N/cm. Lastly, there is little coupling between the force constants as seen as the small compliance coupling constants in the off-diagonal terms.
However, when z-matrix coordinates are used, the results are different from those obtained from natural internal coordinates and become erroneous. The four C-C bonds all have distinct values in cyclobutane, and the coupling becomes much more pronounced. Significantly, the force constants of the C-C bonds in cyclobutane here are also larger than those of n-butane, which is in conflict with chemical intuition. Clearly for cyclobutane—and numerous other molecules, using force constants therefore gives rise to inaccurate bond descriptors due to its dependence on coordinate systems.