Concise Physical Chemistry

c19 JWBS043-Rogers September 13, 2010 11:29 Printer Name: Yet to Come

TRANSITION STATES 313

determined by X-ray scattering and that determined by neutron scattering (which

yields the nuclear locations) is more than 0.1A in some cases (Allinger, 2010). ̊

Geometric parameters from different sources should not be used indiscriminately.

Motion of the entire molecule in the crystalline lattice is calledrigid-body motion.

The amplitude of rigid-body motion is not spherically symmetrical, so the position

of the molecule is not represented as a sphere but as an ellipsoid.

As molecular spectroscopy has contributed force constants to molecular mechan-

ics, so MM contributes vibrational frequencies to spectroscopy. Because force param-

eters are not the same as force constants, MM values of bond stretching frequencies

are not the same as known experimental values but they generally agree to within

±25 cm−^1. Knowing the geometry and atomic masses, angular momenta can also

be calculated along with spectral intensities, heat capacities, entropies, Gibbs free

energy functions, and temperature variations of the thermodynamic functions.

19.9 TRANSITION STATES

Existence of several potential energy minima, one for each of the atoms in a more or

less complicated molecule, implies a potential surface analogous to a mountainous

terrain with peaks, valleys, and mountain passes. Each pass represents a potential

path from one minimum to another. In chemistry, these paths are said to be along

a possiblereaction coordinate. A transition from one minimum potential energy to

another goes over a relative maximum called the transition state. A transition state

is a maximum relative to the two minima it connects but, analogous to a mountain

pass in high country, it is a least energypathbetween the minima, Mathematically,

all second derivatives of the energy are positive except one, leading to vibrational

frequencies all of which are real except for one which is an imaginary number.^2

An example is conversion of the “chair” form of cyclohexane to the “boat” form:

For this change to take place, cyclohexane must go through a planar conformation

that is higher in energy than either the chair or the boat. The planar conformation is

the transition state. One way (not necessarily the best way) of finding the transition

state energy is by starting with a planar input geometry along with the chair and boat.

The MM4 difference between the chair and boat forms is 5.7 kcal mol−^1 =

23 .8kJmol−^1 at 226 K, and the experimental value is 5.5 kcal mol−^1 = 23 .0kJmol−^1.

The transition enthalpy from chair to boat is a little less than twice this amount and

involves somewhat more complicated geometries like the twist boat conformation

(Allinger et al., 1996, p. 650 ff).

(^2) Please do not think that there is something wrong with it: The term “imaginary” is just mathematical
jargon meaning that if you square it, you get a negative number.