Essentials of Computational Chemistry

energy is
10.5 TECHNICAL CAVEATS 377
Ufree rotor = 1
2RT (10.46)
i.e., only half that computed for a harmonic oscillator using Eq. (10.44). The contribution
of a free rotor to the entropy is given by
3 (8π IintkBT)
Sfree rotor = R ln
1/2
σinth
+ 1
2
(10.47)
where σint and Iint are the reduced symmetry numbers and moments of inertia associated
with the free rotor. The definitions for these quantities may be found in the definitive work
of Pitzer and Gwinn (1942) on this subject.
Pitzer and Gwinn have also provided tables to determine the thermodynamic contributions
of hindered rotors (those having torsional barriers on the order of kBT ) when such rotors
are well described by the torsional potential
Ehindered rotor = 1
2 V(1 − cos σintθ) (10.48)
where V is the torsional barrier height and θ is the torsion angle. For the most careful work,
this is the appropriate treatment to employ.
Note that if the torsional barrier is considerably greater than kBT , then the harmonic
oscillator approximation is as valid as for any vibration, and no special precautions need
be taken.
10.5.3 Equilibrium Populations over Multiple Minima
It is not uncommon for a single molecule to have multiple populations. At non-zero
temperatures, the population of different conformations will be dictated by Boltzmann
statistics. If we make the approximation that we may neglect the continuous character of
conformational space and simply work with discrete potential energy minima, we can replace
a statistical mechanical probability integral with a discrete sum, and the equilibrium fraction
F of any given conformer A at temperature T may be computed as
F(A) = e−GoA
/RT
i
e −Go
i /RT
(10.49)
where i runs over all possible conformers, each characterized by its own free energy G o .In
measurements on systems at equilibrium, it is rarely possible to determine the free energies
of individual components of the equilibrium. Rather, one refers to the free energy of the
whole equilibrium population, which may be written
G o {A}
=−RT ln
i∈{A}
e −Go
i /RT
(10.50)