Essentials of Computational Chemistry

15.3 TRANSITION-STATE THEORY 529
breaking (or making) bond, while secondary isotope effects cover all other possibilities. In
general, then, a KIE may be computed as
klight
kheavy
=
Q ‡
light
QR,light
Q ‡
heavy
QR,heavy
= Q‡
light
Q ‡
heavy
‡
−V
e light /kBT
‡
−V
e heavy /kBT
QR,heavy
QR,light
e −(ZPVE‡
light −ZPVE‡
heavy )/kBT
(15.33)
From a theoretical perspective, isotope effects are fairly trivially computed. The stationary
points on the PES and their electronic energies are independent of atomic mass, as are
the molecular force constants. Thus, one simply needs to compute the isotopically dependent
zero-point energies and translational, rotational, and vibrational partition functions, and
evaluate Eq. (15.33).
Primary isotope effects tend to be dominated by the difference in zero-point energies, as
illustrated in Figure 15.2. Because the reaction coordinate is the breaking bond, and because
there is little or no ZPVE associated with this mode in the TS structure, the full difference in
reactant ZPVEs enters into the difference in zero-point-including potential energy barriers.
E
1
2 hw heavy
∆V ‡
heavy
∆V ‡
light
1
2 hw light
Reaction coordinate
Figure 15.2 The dominant contribution to a primary KIE is the differential loss of zero-point vibrational
energy in the reaction coordinate when an isotopically substituted bond is broken. Because the
light isotopomer has a higher vibrational frequency, it has more ZPVE, and a lower potential energy
of activation (thus primary isotope effects expressed as klight/kheavy are essentially always greater than
1). Effects from other normal modes are ignored in this diagram (cf. Figure 15.3)