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Spherical and symetric top
rotator energy levels
Hamiltonian and eigenvalues
Selection rules
Convention
Molecule is characterized by moments of inertia define in a molecule
based coordinates; the axes are labeled a, b, c and are made that IC
is always the largest moment of inertia and are made that inequality is hold:
IA<IB<IC
a, b, c
Practically, one evaluates moment of inertia in xyz coordinates
using the center of mass definitionand then assignment to an
appropriate a,b,c axes can be made.
Molecule and space-fixed angular momenta
(description of the symmetric top molecule)
Y
X
Z
XYZ –the space fixed laboratory system
x,y,z (or abc) –the molecular coordinate system
e.g.,
One can derive the transformation matrix S:
S
Using the matrix elements of S and the operator expressions for JX, JY, JZ
one can obtain J x,y,z in molecular frame and the corresponding expression
in the laboratory frame
Commutation relationships:
[Jx, Jy]=-ih/2πJz
Sign due to the direction cosine terms
[JX, JY]=ih/2πJZ
[J2, Jz]=0
[Jr, Js]=0, r=XYZ, s=xyz
The space fixed and molecular
Frame operators commute with
each other
The rigid rotor symmetric top Hamiltonian H is expressed in the molecular frame
and J2, Jz and JZ all commute with H. There is a set of simultaneous eigenfunctions that:
Kh/2π – is defined as the projection of J
Along the molecular z-axis
B>C
Spherical top
The energy does not depend on K or M and
each level has a degeneracy factor g = (2J +1)(2J +1).
The rotational constants A, B and C are defined as:
Schematic diagram representing the rotational level structure of different rotors.
The diagram also includes the degeneracy g of the rotational levels. Levels with
K> J do not exist.
Selection rules and rotational spectra
for symmetric tops
ΔJ=+/-1
ΔM=0, +/- 1
ΔK=0
The transitions are confined
to lie within a K-stack
Asymmetric top
The IJKM> functions are not eigenfunctions of ^HR. However, the IJKM> functions
form a complete set of basis functions and the wavefunctions ΨJτM of an asymmetric top
can be represented as a linear combination of these basis functions:
To obtain the energy levels of an asymmetric top, the following procedure can be
followed: one constructs ^HR in matrix form and obtains the energies by looking for the
eigenvalues of ^HR (matrix diagonalization).
Note that is not a good quantum number but simply an index running from -J to J with the
associated energy levels EJτM arranged in ascending order. In the literature, the double index
KaKc is used instead of , where Ka and Kc represent K in the prolate and oblate limit,
respectively, and τ= Ka - Kc.
Prolate-oblate correlation diagram
(labeling asymetric top levels)
Asymetric top energy levels