Chemistry Reference
In-Depth Information
symmetry groups of nonrigid molecules proposed by Longuet-Higgins [
3
]. We start
by writing down in explicit form the Schrödinger Hamiltonian for a molecule:
H
=
T
e
+
T
N
+
V
NN
+
V
eN
+
V
ee
(5.14)
The
T
operators are the kinetic energy operators for electrons and nuclei:
∂
2
∂x
i
+
2
∂
2
∂y
i
+
∂
2
∂z
i
T
e
=−
2
m
e
i
(5.15)
∂
2
∂X
n
+
2
n
2
∂
2
∂Y
n
+
∂
2
∂Z
n
T
N
=−
1
m
n
The
V
operators are the Coulomb interactions between the particles:
Z
n
Z
m
e
2
4
π
0
|
V
NN
=
R
n
−
R
m
|
n<m
Z
n
e
2
4
π
0
|
V
eN
=−
(5.16)
R
n
−
r
i
|
i,n
e
2
V
ee
=
4
π
0
|
r
i
−
r
j
|
i<j
The Hamiltonian may contain additional terms describing coupling between orbital
and spin momenta. Longuet-Higgins stated that this full Hamiltonian must be in-
variant under the following types of transformations:
1. Any permutation of the positions and spins of the electrons.
2. Any rotation of the positions and spins of all particles (electrons and nuclei)
about any axis through the center of mass.
3. Any over-all translation in space.
4. The reversal of all particle momenta and spins.
5. The simultaneous inversion of the positions of all particles in the center of mass.
6. Any permutation of the positions and spins of any set of identical nuclei.
The complete group of the Hamiltonian is the combination of all these possible
symmetries. This derivation is directly evident from the mathematical form of the
Hamiltonian and expresses fundamental properties of molecular space and time. Yet
it took 40 years, from Schrödinger to Longuet-Higgins, to obtain a clear definition
of the molecular-symmetry group. Three kinds of symmetries may be identified:
•
Space symmetries. Space is uniform, isotropic, and has inversion symmetry. This
is clear from the fact that the kinetic energy Laplacian operators are trace oper-
ators of second derivatives; hence, they are invariant under a sign change of all
coordinates and isotropic under rotations. All potential-energy operators depend
only on relative distances between particles and thus do not change under trans-
lations, rotations, or inversion.
