An approximate Hamiltonian for a nonrigid internal rotor has been derived. The potential energy has been expanded in a Taylor's series in the displacement coordinates and in a Fourier series in the angle of internal rotation Θ. The Hamiltonian was transformed by a contact transformation, and a second‐order Hamiltonian in which vibrations and rotations have been separated has been obtained. The Hamiltonian consists of terms which constitute the usual rigid internal rotational problem, of centrifugal distortion terms involving both over‐all and internal angular momentum, and of terms that arise because of the repulsive nature of the barrier. These repulsive terms enter as a single term, 2JFv(m|1 — cos3Θ|m), in the expression for the rotational transitions of symmetric rotors, where J is the total angular momentum quantum number and m is the pseudo‐quantum number for internal rotation. The repulsive constant, Fv, is given by the relation where Bxx(i)+Byy(i) is the derivative of the rigid rotor rotational constant with respect to the ith symmetry coordinate, and ai(1) is one‐half the displacement of the equilibrium position of the ith internal coordinate in going from Θ=0 to Θ=π/3.
The dependence of the barrier height upon the vibrational motion has also been studied.
REFERENCES
1.
2.
H. H.
Nielsen
, Phys. Rev.
40
, 445
(1932
). This potential is valid only for internal rotors in which one of the rotating groups is a symmetric rotor.3.
It is assumed that the torsional (hindered internal rotational) mode is a normal mode and thus independent of the other normal modes in a first approximation. If the torsional motion is not a normal mode, the treatment in this paper is not quite correct. It is almost correct for
4.
Vibrations, or normal modes, will refer to all normal vibrations other than torsional oscillations.
5.
6.
This Hamiltonian is a generalization of the one given by
E. B.
Wilson
, Jr., and J. B.
Howard
, J. Chem. Phys.
4
, 260
(1936
). Terms that include the effects of Coriolis force have been neglected, since, to the order discussed here, they will not affect the interactions between over‐all and internal rotations.7.
differs slightly from the instantaneous inverse inertia tensor even in the problem not involving internal rotation for it includes a contribution due to Coriolis interactions. See reference 6.
8.
D.
Kivelson
, J. Chem. Phys.
23
, 2236
(1955
). Hereinafter called III.9.
E. C. Kemble, The Fundamental Principles of Quantum Mechanics (McGraw‐Hill Book Company, Inc., New York, 1937), p. 394. See also reference 6.
10.
H. H.
Nielsen
, Revs. Modern Phys.
23
, 90
(1951
). also contains some effects of Coriolis interactions.11.
12.
13.
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© 1955 American Institute of Physics.
1955
American Institute of Physics
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