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CE 561 Lecture Notes
Fall 2009
p. 1 of 27
Days 9 and 10:
Potential Energy Surfaces (hypersurfaces)
Much of the microscopic theory of chemical kinetics depends on the idea of a
potential energy
surface
(often called a potential energy
hypersurface
, because it usually involves more than 3
dimensions).
These are based on the
BornOppenheimer
approximation of quantum mechanics.
This approximation states that, because the atomic nuclei are thousands of times heavier than the
electrons, we can separate the motion of the electrons from that of the nuclei.
The nuclear
motion is much, much slower then the motion of the electrons, so we can assume that the
electron cloud responds instantaneously to any changes in the nuclear position.
For now, we will
consider only the
ground electronic state
of a molecule – the one in which the electrons occupy
the orbitals of lowest energy.
An assembly of
N
atomic nuclei has 3
N
degrees of freedom – each of the
N
molecules can move
in 3 dimensions.
We could describe the system using 3 Cartesian coordinates for each atom.
However, if the state of the system only depends on the arrangement of the atoms relative to each
other, then 6 of these degrees of freedom can be eliminated (5 for a linear molecule).
That is
because three degrees of freedom correspond to overall translation of the whole system (in 3
dimensions) and three degrees of freedom (2 for a linear molecule) correspond to overall rotation
of the whole system.
Translating or rotating the whole molecule does not change the relative
positions of the atoms (the distance between any pair of atoms).
In the absence of any imposed
fields (magnetic, electrical, etc.) the position and orientation of a molecule will not affect its
energy.
Simple interatomic potentials:
The potential energy surface for a molecule or a reaction is a description of the total energy of
the atoms involved (all of the nuclei and all of the electrons) as a function of the positions of the
nuclei.
A chemical reaction can be described in detail as motion of the atoms on this potential
energy surface.
We will begin by considering the potential energy surface for a system of 2
atoms (a diatomic molecule if they form a chemical bond).
The energy of this system depends
on (3
N
5) = 1 degree of freedom – the distance between the atoms.
If we denote the potential
energy as
V
(or
V
(
r
), where
r
is the distance between the atoms) then the force acting on either
atom is
r
dV
F
dr
= −
.
For some types of interaction, particularly electrostatic interactions among
only two particles, we know what the interaction potential looks like:
Ionion interaction:
2
12
()
ZZe
Vr
r
=
Iondipole interaction:
1
2
cos( )
(, )
Ze
r
µθ
θ
−
=
(
µ
is the dipole moment,
is the angle between the dipole axis and a line joining the
centers of mass of the ion and dipole (along which r is measured)).
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This document was uploaded on 07/08/2011.
 Fall '09

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