6.6 Effect of Hindered Rotational Motions
341
6.6 Effect of Hindered Rotational Motions
Complex molecules often contain groups of atoms connected to one another or
to a rigid molecular framework by single-electron chemical bonds. These groups
can exercise rotations relative to one another or to the rigid framework in addition
to participating in the overall rotational motion of the molecule. Such relative
rotational motions are referred to as ‘internal’ rotations: if two rotating groups
are sufficiently close to one another for their electronic charge clouds to interact,
then the rotational motion of one group relative to another may generate a potential
energy that opposes, or hinders, free rotational motion about the common bond. It
seems reasonable to expect that the greater the separation between the two groups
of atoms, the freer will the internal rotation be.
For molecules in thermal equilibrium, the magnitude of the potential energy
barrier to free rotation relative to thermal energy, k B T , may be employed to delineate
three categories of the resultant internal motion of a group of atoms. On the one
hand, if the potential energy barrier is large relative to k B T , we say that there is a
high barrier to rotation, so that the internal motion will essentially be torsional, and
may be treated in the same fashion as any other vibrational motion in the molecule.
Typical of this category would be twisting motions about a common multiple bond:
perhaps the simplest example of this category of internal motion is the twisting
motion of the CH 2 groups in ethene, C 2 H 4 , about the carbon–carbon double bond.
On the other hand, if the potential energy barrier is very small relative to k B T ,
it may simply be ignored to a good approximation, and the internal motion may
be treated as free rotational motion. Typical of this category would be the relative
rotational motion of two chemical groups separated by more than one bond length:
the simplest example of this category of internal motion is dimethyl acetylene,
H 3 C–C≡C–CH 3 , whose barrier to rotation is essentially zero. The most common
category of internal motion is that for which the potential energy barrier to rotation
is neither much larger nor much smaller than k B T , in which case all three regimes
of motion may be accessed as the temperature is varied, thereby leading to a motion
that is quite complex, often requiring numerical evaluation to be carried out. Such
calculations often necessitate the use of an appropriate molecular model.
Because we have rotational motion about a bond, the potential energy associated
with it must be periodic with period 2π/σ , where σ is the symmetry number
associated with the group of rotating atoms. The hindering potential V (φ) must
therefore satisfy the condition
V (φ +
2π
σ ) = V (φ) .
(6.6.1)
We shall only have occasion to consider periodic potential energy functions of the
form 5
5 This form for V (φ) represents the leading term of a Fourier series expansion of the periodic
interaction. Although the inclusion of additional terms of the expansion would give a more
accurate representation of the interaction, Eq. (6.6.2) normally suffices for comparison with
existing experimental data associated with this phenomenon. For C 2 H 6 and C 2 Cl 6 , for example,
341
6.6 Effect of Hindered Rotational Motions
Complex molecules often contain groups of atoms connected to one another or
to a rigid molecular framework by single-electron chemical bonds. These groups
can exercise rotations relative to one another or to the rigid framework in addition
to participating in the overall rotational motion of the molecule. Such relative
rotational motions are referred to as ‘internal’ rotations: if two rotating groups
are sufficiently close to one another for their electronic charge clouds to interact,
then the rotational motion of one group relative to another may generate a potential
energy that opposes, or hinders, free rotational motion about the common bond. It
seems reasonable to expect that the greater the separation between the two groups
of atoms, the freer will the internal rotation be.
For molecules in thermal equilibrium, the magnitude of the potential energy
barrier to free rotation relative to thermal energy, k B T , may be employed to delineate
three categories of the resultant internal motion of a group of atoms. On the one
hand, if the potential energy barrier is large relative to k B T , we say that there is a
high barrier to rotation, so that the internal motion will essentially be torsional, and
may be treated in the same fashion as any other vibrational motion in the molecule.
Typical of this category would be twisting motions about a common multiple bond:
perhaps the simplest example of this category of internal motion is the twisting
motion of the CH 2 groups in ethene, C 2 H 4 , about the carbon–carbon double bond.
On the other hand, if the potential energy barrier is very small relative to k B T ,
it may simply be ignored to a good approximation, and the internal motion may
be treated as free rotational motion. Typical of this category would be the relative
rotational motion of two chemical groups separated by more than one bond length:
the simplest example of this category of internal motion is dimethyl acetylene,
H 3 C–C≡C–CH 3 , whose barrier to rotation is essentially zero. The most common
category of internal motion is that for which the potential energy barrier to rotation
is neither much larger nor much smaller than k B T , in which case all three regimes
of motion may be accessed as the temperature is varied, thereby leading to a motion
that is quite complex, often requiring numerical evaluation to be carried out. Such
calculations often necessitate the use of an appropriate molecular model.
Because we have rotational motion about a bond, the potential energy associated
with it must be periodic with period 2π/σ , where σ is the symmetry number
associated with the group of rotating atoms. The hindering potential V (φ) must
therefore satisfy the condition
V (φ +
2π
σ ) = V (φ) .
(6.6.1)
We shall only have occasion to consider periodic potential energy functions of the
form 5
5 This form for V (φ) represents the leading term of a Fourier series expansion of the periodic
interaction. Although the inclusion of additional terms of the expansion would give a more
accurate representation of the interaction, Eq. (6.6.2) normally suffices for comparison with
existing experimental data associated with this phenomenon. For C 2 H 6 and C 2 Cl 6 , for example,