Wave H. Shaffer

Generalized orbital angular momentum and the n-fold degenerate quantum-mechanical oscillator: Part I. The twofold degenerate oscillator

Abstract An operational procedure is used to determine the eigenvalues and eigenfunctions for the quantum-mechanical twofold degenerate oscillator problem in polar coordinates. The procedure leads automatically to the determination of the nonvanishing matrix elements of the Cartesian coordinates and conjugate linear momenta of the oscillator.

The Calculation of Perturbation Energies in Vibrating Rotating Polyatomic Molecules

An account is presented of the calculation of the purely vibrational contribution to the eigenvalues of the vibration‐rotation Hamiltonian of a polyatomic molecule. Special attention is given to the expansion of the quantum‐mechanical Hamiltonian; and to the results of the contact transformation of the Hamiltonian which simplifies the calculation of the second‐order energy corrections. The results of the transformation theory have been presented in the form of tables. For each type of perturbation term which appears in the first‐order Hamiltonian it is possible to obtain the second‐order energy corrections by inspection. This procedure eliminates the need for applying the transformation to each specific problem. The use of these results is illustrated by working out in considerable detail the case of the X2YZ2 tetrahedrally symmetric molecular model.

Operational procedure for the determination of the matrix elements of the direction cosines for a rigid symmetrical rotator

Abstract An operational procedure is presented for obtaining the matrix elements (J′K′M′|λAα|JKM), where λAα is the direction cosine between the A-axis (A = X,Y,Z) and the α-axis (α = x,y,z) of a space-fixed XYZ reference frame and a body-fixed xyz reference frame used to describe the pure rotation of a rigid symmetrical rotator. Ψ(J,K,M) are the normalized symmetrical rotator wave functions.

Intensities of Rotation Lines in Absorption Bands

An alternate method of computing the influence of vibration‐rotation interaction on the intensities of lines in the band spectra of molecules from that discussed by Herman and Wallis and by Herman and Rubin is presented. The method consists of transforming the Hamiltonian of the molecule by a contact transformation so that the wave functions remain those for a harmonic oscillator‐rotator. The electric moment must then also be transformed by the same contact transformation before the matrix elements of the electric moment are computed. The method is applied to a diatomic molecule and gives the same results as those stated by Herman and Wallis.

Rotation‐Vibration Energies of the Pyramidal XY3 Molecular Model

Complete expressions have been derived for the rotation‐vibration energies of the pyramidal XY3 molecular model in such a way as to include, through second order of approximation, all contributions to the energies arising from Coriolis interactions, anharmonicities, etc.

Vibration‐Rotation Energies of the Planar XY3 Molecular Model

The vibration‐rotation energies of the planar XY3 molecular model are investigated to the second order of approximation, including the cubic and quartic anharmonic terms of the potential energy, the Coriolis interactions between the degenerate oscillations and the total angular momentum, the dependence of the moments of inertia upon the vibrational state, and the centrifugal expansion terms. The contact transformation developed by Shaffer, Nielsen, and Thomas is applied to the Wilson‐Howard Hamiltonian, and it is found that the energies are simply the diagonal elements of the transformed Hamiltonian. The energies are expressed in term value form, E/hc=Gvib+Frot—ζh/4π2cIzz0(l2—l4)K, Gvib, Frot, and ζ being given in terms of the constants appearing in the Hamiltonian. The dependence of the moments of inertia, Ixx and Izz, upon the vibrational states is determined, and it is found that the quantity Δ=Izz—2Ixx is independent of the anharmonic constants, depending only on the dimensions of the molecule, the no...

The Infra‐Red Spectra of Bent XYZ Molecules Part I. Vibration‐Rotation Energies

The rotation‐vibration Hamiltonian, complete to second order of approximation, is set up for the bent XYZ molecular model. The allowed energies are calculated and expressed in term‐value form, E=hc(G+F); the vibrational term G is given explicitly and the elements of the secular determinant are given for evaluation of the rotational term F. The valence‐force form of harmonic potential function is discussed for the bent XYY′ model and normal frequencies of HDO are calculated.

A normal coordinate analysis of cyclopentane

Abstract A normal coordinate analysis of cyclopentane is carried out using the rigid body translations, rotations, and internal motion of CH2 groups to describe internal motions of the molecule. The potential energy is written in terms of valence-like coordinates for the framework of the molecule and only interactions between similar coordinates are retained. Force constants for these valence-like coordinates are calculated from a tentative assignment of the observed vibrational frequencies.

Operational derivation of wave functions for a symmetrical rigid rotator

Abstract An operational procedure is presented for obtaining the normalized eigenfunctions Ψ(JKM) of a rigid symmetrical rotator and the matrix elements of the rectangular components of its angular momentum in both body-fixed and space-fixed axes.

Valence and Central Forces in Bent Symmetrical XY2 Molecules

Appropriate generalized coordinates and potential functions are set up for studying the zero‐ and first‐order vibration problem of the best symmetrical XY2 molecular model under the assumption of valence and central forces. The force constants in the quadratic and cubic parts of the potential function of the water vapor molecule are evaluated for these two types of forces by using the data of Darling and Dennison.