Theory of Internal Over-All Rotational Interactions. II. Hamiltonian for the Nonrigid Internal Rotor

Abstract

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, 2JF_v(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, F_v, is given by the relation $$F_v\mathrm{=-\Sigma }\frac{1}{2}{\mathrm{[B}}_{\mathrm{xx}}^{\mathrm{(i)}}{\mathrm{+B}}_{\mathrm{yy}}^{\mathrm{(i)}}{\mathrm{]a}}_i^{\text{(1)}},$$ where B_xx⁽ⁱ⁾+B_yy⁽ⁱ⁾ is the derivative of the rigid rotor rotational constant with respect to the ith symmetry coordinate, and a_i⁽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.