A slightly more involved example is the strain of norbornane, bicyclo[2.2.1]heptane.
We can open this to heptane (two steps are hinted at here for clarity); molecules like
butane and heptane are used in the all-transoid, lowest-energy conformations:
H 3 CH 3 CH 3 CH 3 C–273.96832CH 3–79.83002–158.458042–276.39909
CH 3CH 3CH 3SE norbornaneðÞ¼#½ 273 : 96832 þ 4 ðÞ# 79 : 83002##½ 276 : 39909 þ 2 ðÞ# 158 : 45804¼# 593 : 28840 þ 593 : 31517 ¼ 0 : 02677 ¼ 70 :3 kJ mol#^1Khoury et al. report a value of 69.5 kJ mol#^1 (16.6 kcal mol#^1 ), fairly close to
the experimental value (60.2 kJ mol#^1 ), which they cite.
The two calculations shown here are simplified versions of the slightly more
involved methods of Khoury et al. [ 157 ], which attempt to make the bonds in the
reactants and the products more alike than in the very straightforward manner used
here; for example, the two C–C bonds in norbornane that we break are between a
secondary and a tertiary carbon, but the two C–C bonds we make to form two
butanes are between a secondary and a secondary carbon. Instead of using ethane
and ethane and making butane, we might have used ethane and propane and made a
bond between a secondary and a tertiary carbon in 2-methylbutane. This gives a
strain energy of 64.7 kJ mol#^1 , closer to the experimental one. In comparing the
strain in two hydrocarbon molecules, it is probably fairer to compare the strain per
C–C bond, because other things being equal, in a bigger molecule the strain is
more dispersed. Thus cubane, with six cyclobutane rings, has a strain energy of
622 kJ mol#^1 [ 162 ], while cyclobutane, with only one ring, has a strain energy of
110.0 kJ mol#^1 [ 157 ]. With these numbers, cubane is 5.7 times as strained as
cyclobutane. On a per-C–C-bond basis however, the strain energy of cubane and
cyclobutane are 622/12¼52 kJ mol#^1 and 110.0/4¼27.5 kJ mol#^1 ; using these
numbers, cubane is effectively only about twice as strained as cyclobutane. The role
of strain in connection with kinetic and thermodynamic stability has been discussed
for polyprismanes and superstrained C 5 molecules [ 163 ]. Calculations of the kind
we have done here are approximations to 0 K enthalpy changes (because ZPE and
thermal energy increases on going above 0 K are ignored).
Aromatic stabilization energy, ASE. We skirt the enormous literature on the
meaning and detection of aromaticity [ 164 ] and assert that a good measure of the
phenomenon is the aromatic stabilization energy, the energy change when an
306 5 Ab initio Calculations