Page 201 - Academic Press Encyclopedia of Physical Science and Technology 3rd Organic Chemistry

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Encyclopedia of Physical Science and Technology EN011G-539 July 14, 2001 21:48

448 Organic Chemical Systems, Theory

also correspond to dissociation products, the sets of two or difference from the lowest vibrational level in the origi-
more smaller molecules formed from the same collection nal catchment basin to the lowest level available over the
‡
of nuclei and electrons. Some of the minima may have saddle point), and the activation entropy
S is a function
equal energies (e.g., a pair of enantiomers). In general, of the width of the saddle relative to that of the starting
one minimum is of lower energy than all the others and minimum.
corresponds to the most stable isomer of the molecule. Isotopic substitution does not have any effect on the
Each minimum is surrounded by its catchment basin potential energy surfaces as usually defined. However, it
(Fig. 6B). Within a given basin, the potential energy sur- does affect the dynamics of the vibrational process and the
face slopes toward the minimum in question. Each catch- dynamics of the motion over saddle points by changing
ment basin defines the range of geometries that correspond the magnitude of the energy of zero-point motion and the
to a given chemical species. spacing of the vibrational stationary states and thus the
density with which these states are packed. Isotopic sub-
stitution thus leads to changes in vibrational spectra and
3. Chemical Reactions
also to changes in reaction rate constants.
Under ordinary conditions molecules are frequently not Some vibrational stationary states have energies that lie
in stationary vibrational states. Vibrations and internal above some of the lower saddle points. In such states the
rotations are affected by collisions with neighboring molecule is free to travel from one catchment basin to
molecules, which add or subtract small amounts of vibra- another and in that sense has no fixed chemical structure.
tional energy more or less randomly. At thermal equilib- This sort of situation occurs, for instance, for rotations
1
rium the vibrational energy content is kT for each degree around single bonds at elevated temperatures.
2
of freedom. It is this supply of random thermal energy that The perfect quantum mechanical analogy to the rolling-
permits molecules to escape from one catchment basin to marble description given earlier is described in terms of
another in a thermal reaction. the motion of a wave packet, that is, of a nonstationary
In order to move from one minimum to another it is wave function initially more or less strongly localized in
necessary to overcome ridges that separate them (Fig. 6B). a particular region of nuclear geometries. Since the vibra-
This is done most easily by travel through saddle points, tional levels are spaced quite closely together, however,
which correspond to transition structures and are usually the classical description in terms of the rolling marble is
called transition states. Some of these lie only a little often quite adequate.
higher in energy than a starting minimum. Travel over In the Born–Oppenheimer approximation, the overall
these saddles is easy even at low temperatures, and a chem- rotation of a molecule can also be uncoupled from other
ical species corresponding to a shallow catchment basin kinds of motion. For an isolated molecule, the solution
may not be isolable at room temperatures or even below. of the Schr¨odinger equation for rotational motion leads to
Conformationalisomersareagoodexampleofsuchchem- a set of very closely spaced stationary levels. Transitions
ical species. Other catchment basins may be deeper and between these levels occur in the microwave region of the
the surrounding saddle points difficult to reach. They often electromagnetic spectrum. In solution, the quantization of
correspond to configurational or constitutional isomers. rotational motion is usually destroyed by intermolecular
An important attribute of a saddle point connecting two interactions, so that it has little importance in theoretical
catchment basins is the width of the saddle. Very nar- organic chemistry.
row saddles are difficult for the rolling marble to find So far we have concentrated on the lowest singlet hy-
and decrease the probability of travel from one catchment persurface E(S 0 ). However, motion can be studied simi-
basintoanother.Widesaddlesaccommodateamuchlarger larly on other hypersurfaces if they can be calculated to
flux under otherwise identical conditions and lead to fast start with. Such electronically excited states are usually
motion. produced either by absorption of light in the ultraviolet
For a reaction whose rate is not limited by the rate of and visible regions, as studied by electronic spectroscopy,
diffusion, the reaction rate is well described by Eyring’s or by energy transfer from another electronically excited
transition state theory. For the rate constant k,wehavethe molecule. The latter is particularly useful in the case of ex-
following: citation into a triplet state, since excitation from a ground
singlet to a triplet state by direct absorption of light has a

KT
S ‡
H ‡

k = κ exp − , very low probability.
h R RT
The study of the processes involving the higher poten-
where κ is the transmission coefficient, k the Boltzmann tial energy surfaces is the domain of photochemistry and
constant, T absolute temperature, h Planck’s constant, R photophysics. These are very difficult or impossible to
‡
the gas constant,
H the activation enthalpy (i.e., the understand in terms of the classical picture described in