A saddle point (SP) on a
potential energy surface (PES)
• TS structures can be
determined by searching
for first-order SP on the
PES.
• Minimum in all
dimensions but one.
• Maximum along the
lowest energy path
between R and P on the
PES
PES: z = x2 - y2
TST approximation to the rate constant
Z: integrals of the Boltzmann factors over the specified region of
configurational space
This is the basic equation of TST when a hyperplanar dividing
surface is used
The harmonic approximation
- the harmonic oscillator Hooke´s law: F = -kx
for the restoring force for an extension, x
k
m
m
Diatomic molecule
If the spring is stretched a distance x > 0,
the restoring force will be negative,
pull the mass back to its equilibrium
position
The potential energy, V, is obtained by integrating:
F = -dV/dx = -kx
to give:
V = 1/2 kx2
V
The harmonic approximation
- the harmonic oscillator k
Another derivation
Diatomic molecule
m
m
Taylor expansion V about the minumum energy:
V(x) has a minimum at x = 0, V’(0) = 0 :
Generalization
• For a system with more than 1 vibrational degree we calculate
normal modes coordinates
• Normal modes can be calculated everywhere on the PES
• At the minimum R:
• At the saddle point :
Harmonic TST (HTST)
TST:
Harmonic
approximation:
HTST:
Harmonic TST (HTST)
For each of the normal modes:
So,
Harmonic TST (HTST)
The factors of kBT cancel out and all factors of 2! except for one:
Here, the square root of the effective mass for the normal modes has
been multiplied both in the numerator and the denominator.
Harmonic TST (HTST)
The vibrational frequency is:
So,
HTST and Arrhenius relation
Arrhenius plot
Slope: activation energy
Intercept: vibrational entropy
Beyond classical mechanics
• At low T the classical treatment of vibrations is often not good
enough, where quantum effects start to play an increasing role.
• Zero point energy (ZPE) of the vibrational ground state.
• In most extreme cases: the system might tunnel from R to P.
• The simplest inclusion of ZPE can be made if the system is most
likely in the lowest energy level both initially and at the TS.
• Then one can replace Ea = Vsp - Vmin with: