Vibration-rotation Hamiltonians of linear molecules

Abstract

The construction of the molecular vibration-rotation Hamiltonian is considered, with particular reference to two alternative treatments of molecules with linear reference configurations. These can be considered to have either (i) 3N - 5 vibrational and 2 rotational degrees of freedom or (ii) 3N - 6 vibrational and 3 rotational degrees of freedom. In either case the classical kinetic energy consists of vibrational, rotational and translational parts given by The rotational part contains the angular velocity ω and the modified moment of inertia tensor I' of Wilson and Howard, which also occurs in the relation J α - πα = Σβ I'αβωβ involving the total (J) and vibrational (π) angular momenta. In case (i), I′ has a vanishing z row and column, where z is the axis of the molecule. This is associated with the Sayvetz condition that the total angular momentum about the axis is purely vibrational. These equations therefore contain only the two components ωx and ωy of ω, which can be eliminated to give the Hamiltonian form of the kinetic energy. In case (ii), the z row and column of the I' matrix are non-vanishing and I' is non-singular. The three components of ω can therefore be obtained in terms of the angular momenta, to give a treatment that is entirely analogous to that for a nonlinear reference configuration. The parameters of the case (ii) Hamiltonian are expressed in terms of those of case (i).