Computational Chemistry

of this region and its curvature. The shape of the hypersurface can be handled as an
analytic function of atomic coordinates

E¼fðq 1 ;q 2 ;...Þ

which has been fitted to a finite number of calculated (e.g. by ab initio methods)
points. The functionEcan in favorable cases be used to calculate reliable rate
constants (this kind of molecular dynamics calculation requires the ability to break
and make bonds using a quantum mechanical method, in contrast to the common
MD calculations which use only molecular mechanics).The situation can be com-
plicated by quantum mechanical tunnelling [ 210 ], which, particularly where light
atoms like hydrogen move, can speed up a reaction by orders of magnitude
compared to classical predictions. Furthermore, since 1992 [ 211 ] it has been
shown by molecular dynamics that the traditional concept of a potential energy
surface with a straightforward intrinsic reaction coordinate (minimum energy path)
may in some cases be inadequate and even incorrect. Briefly, reacting molecules
sometimes move on to a plateau region or a “bifurcated” region of the surface and
then head toward products in directions determined by their internal motions; the
details can be “quite complex” [ 212 ]. Such surfaces are probably exceptional and
the traditional picture ofChapter 2seems likely to be applicable in most cases. Here
we will merely attempt to apply some fundamentals of rate theory to unimolecular
reactions to illustrate how straightforward calculations can provide useful informa-
tion about the stability of molecules. For rigorous calculations of rate constants one
best utilizes a specialized program, for example Polyrate ([ 133 ], based on RRKM
theory [ 134 ]). There are many discussions of the theory of reaction rates, in various
degrees of detail [ 148 , 213 ]. In this section we will limit ourselves to gas-phase
unimolecular reactions [ 214 ] and examine the results of some calculations. We will
use the simplified Eyring (Section 2.2) equation

kr¼

kBT

h

e#DG

z=RT

$ð 5 : 197 Þ

wherekr¼unimolecular rate constant (units¼s#^1 )
kB¼Boltzmann constant, 1.381' 10 #^23 JK#^1
T¼temperature, K
h¼Planck’s constant, 6.626' 10 #^34 Js
DG{¼the transition-state-reactant free energy difference, kJ mol#^1 (similar
results are obtained from the ZPE-corrected 0 K energy difference,DEtotal0K, which is
the 0 K enthalpy difference)
R¼gas constant, 8.314 J K#^1 mol#^1
ForT¼298 K (“room temperature”), (kBT)/h¼6.22' 1012 s#^1 andRT¼2.478
kJ mol#^1
With these values Eq. (5.197) becomes

kr¼ 6 : 22 ' 1012 e#DG

z= 2 : 478

ð 5 : 198 Þ

5.5 Applications of the Ab initio Method 325