Neutron Scattering

The Id rotor : solution based on 'free rotor' funetions
The canonical strategy diagonalizes the hamiltonian matrix. It is important to chose an appropriate
set of basis functions [4] .
The simplest molecule consists of two atoms and allows two orientations in a double minimum
potential (H2 , H20 . .) . The case was already considered in the section "Translational
tunnelling " . Here we want to consider a mcthyl group, the most intensively studied system . It
is characterized by its momentum of inertia () around its symmetry axis or - equivalently - by
its rotational constant B = 20 .
An obvious set of basis functions are the free rotor functions exp(ini~o) .
In the single
particle model the interaction with the surrounding is reflected in a static rotational potential
V. To get the Hamiltonian H the kinetic rotational energy bas to be added
H = _ OJ 2 + V
r2
d2
- 20 jcp~ + V
d 2
-B d ;2 + V (17 .39)
A dimensionless representation of the eigenvalue (Schrödinger) equation is
with the scaled rotational potential V' =
H
B w
E
Bw
(17 .40)
n,
V3 (1 - cos(3nço» (17 .41)
n,-1
The eigenfunetions are expanded into free rotor functions up to the order 2M+1
ni--M
with eigenvectors a. . Only a few matrix elements deviate from zero . They follow from orthognality
relation of the angular functions
exp(an~)h (~)exp(ana~)d
exp(an~)h (~)exp(ana~)d
a,exp(irreV) (17 .42)
1 ; r ,
Z 4'",
y;t -
4 d(Zts)
d 2
exp(incp) ~~2
exp(irrup)dcp = -m2 S,zrn, (17 .43)
17-15