E. Bright Wilson

Some Mathematical Methods for the Study of Molecular Vibrations

Developments which reduce the labor of calculating the vibration frequencies of complex molecules are described. In particular a vectorial scheme is given for obtaining the reciprocal of the matrix of the kinetic energy in terms of valence‐type coordinates. A general rule for writing down the coefficients of the transformation to symmetry coordinates is derived together with a method of obtaining the kinetic energy reciprocal matrix (G) in terms of symmetry coordinates with a minimum of algebra. A treatment of redundant coordinates is developed. In addition, reduction of the secular equation by the splitting out of high frequencies, a new type of isotope product rule, and the determination of normal coordinates are discussed. The molecule CH3Cl is worked out as an illustration.

A Method of Obtaining the Expanded Secular Equation for the Vibration Frequencies of a Molecule

A method is described for obtaining the secular equation for the vibration frequencies of a molecule directly in expanded form, i.e. as an algebraic rather than a determinantal equation. The force constants occur literally and the masses may occur either literally or numerically, as desired. The symmetry may be employed as usual to factor the secular equation. Several methods of obtaining approximate roots are described for which the expanded form is particularly suitable. Finally, an example, the nonlinear general triatomic molecule, is worked out algebraically.

Approximate Treatment of the Effect of Centrifugal Distortion on the Rotational Energy Levels of Asymmetric‐Rotor Molecules

A first‐order treatment yields the relation W=W0+A1W02+A2W0J(J+1)+A3J2(J+1)2+A4J(J+1)〈Pz2〉+A5〈Pz4〉+A6W0〈Pz2〉 for the rotational energy W of a nonrigid asymmetric rotor. The As are constants independent of the rotational quantum numbers (J, K−1, K+1) while W0 is the rigid‐rotor energy. Pz is the operator for the component of angular momentum along the axis of quantization. Formulas are given for 〈Pz2〉 and 〈Pz4〉, based on continued fractions, as well as expansions useful for nearly symmetric cases. As a special case, the corrections are derived for transitions between the components of asymmetry doublets.

The Stark Effect for a Rigid Asymmetric Rotor

The Stark effect, arising from the interaction of a uniform electric field with a permanent electric dipole that is arbitrarily oriented within a rigid asymmetric rotor, and with a dipole induced in the rotor by the field, has been evaluated by perturbation methods. Tables are given for the perturbation of the energy levels so that, for J≤2, the rotational energies of an asymmetric molecule in an electric field may be readily approximated to terms quadratic in the electric field.The effect of accidental degeneracy upon the Stark effect and line intensities has been considered. A qualitative discussion of certain features of Stark patterns is given that may be useful in the identification of rotational spectral lines.

The Experimental Determination of the Intensities of Infra‐Red Absorption Bands I. Theory of the Method

With spectrographs of available resolving power, the apparent integrated absorption coefficients of infra‐red bands usually differ greatly from the true coefficients because the spectrograph yields at each setting a weighted average of the fractional light transmitted in a band of frequencies, whereas what is desired is the unweighted average of the logarithm of the fractional light transmitted. It is shown that true absorption coefficients can be obtained by (a) eliminating the violent fluctuations in intensity with frequency by broadening the rotational lines with a non‐absorbing foreign gas and (b) eliminating the error due to the intensity variation of the envelope by extrapolating the apparent integrated absorption coefficient divided by the partial pressure to zero partial pressure of the absorbing gas. These two steps permit vibrational intensities to be measured to a reasonable accuracy even with a spectrograph of low resolving power.

The Statistical Weights of the Rotational Levels of Polyatomic Molecules, Including Methane, Ammonia, Benzene, Cyclopropane and Ethylene

A general method is described for calculating the statistical weights (degeneracies) of the energy levels of polyatomic molecules. Wave functions for a molecule are assumed to be expressible as linear combinations of products of the electronic, vibrational, rotational and nuclear spin functions. By using standard methods of group theory, the number of linear combinations of these products are found having the correct symmetry with respect to those permutations of identical atoms which are equivalent to rotations of the molecule. It is not necessary to find the combinations themselves. The molecules CH4, CD4, CH3D, CHD3, CH2D2, CH3X, CD3X, NH3, ND3, C6H6, C3H6 and C2H2 are treated. In addition noncombining species in polyatomic molecules and the splitting of energy levels due to the multiplicity of equilibrium configurations are discussed. Tables of statistical weights are given for the above molecules which could be used to interpret alternating intensities in rotation‐vibration spectra or for more exact ...

Theory of Centrifugal Distortion Constants of Polyatomic Rotor Molecules

The distortion constants ταβγδ are expressed in terms of inverse force constants and of the derivatives of the inertia tensor with respect to interatomic parameters. Harmonic potential functions are assumed. Relations that simplify the procedure for obtaining the required derivatives are obtained. Furthermore, rules are given that enable one to write down directly the derivatives with respect to certain common internal displacement coordinates. Group theoretical considerations are used to obtain some knowledge of the distortion constants.

Microwave Spectrum, Structure, Dipole Moment, and Quadrupole Coupling Constants of Formamide

The microwave spectra of formamide, and the isotopic species 1, 2 N di‐deutero formamide, 1 N mono‐deutero formamide (cis) and (trans), have been investigated. The rotational constants obtained from the frequencies of assigned rotational transitions lead to the following effective structural parameters: for the bond distances, there results C–O = 1.243±0.007 A, C–N = 1.343±0.007 A, N–H = 0.995±0.007 A, and for the bond angles ∠HNH = 118.98±0.50°, ∠ NCO = 123.58±0.35°, ∠NCH = 103.9±1.2°. The molecule is planar. Stark effect measurements yield values for the dipole moment components which, when combined with assumed approximate bond moments and the structural parameters, give the magnitude and the direction of the dipole moment. The dipole moment is thus equal to 3.714±0.06 Debyes and makes an angle of 39.6° with the C–N bond. The quadrupole coupling constants, Xa = 1.9 mc, Xb = 1.7 mc, Xc = ‐3.6 mc, were calculated from the frequency shifts of the hyperfine splittings observed for several rotational transi...

Relative Intensity Measurements in Microwave Spectroscopy

A method is described for the measurement of peak intensity ratios of gas phase microwave absorption lines. Large errors associated with multiple reflections in the waveguide have been eliminated by careful design and the use of ferrite isolators. Detector nonlinearities have been neutralized by using a precisely calibrated step attenuator allowing both lines involved in a ratio to be displayed with approximately the same amplitude. Means of minimizing additional sources of error are discussed. Application of the method to a number of problems indicates that ratios accurate to a few percent can be obtained.

Microwave Spectrum of Propionaldehyde

The microwave spectrum of propionaldehyde, CH3CH2CHO, has been investigated in the frequency region 8–38 Gc/sec. The existence of two stable rotamers has been demonstrated. The cis rotamer, in which the four heavy atoms are coplanar, is more stable by 900±100 cal/mole. An approximate treatment of the coupling of the —CH3 and C–C torsional motions in the cis rotamer is presented. Using this treatment the barrier to internal rotation in the cis rotamer is found to be 2280±110 cal/mole. The dihedral angle in the higher energy, gauche rotamer is about 131°. Evidence is given for coupling of the C–C torsional momentum and over‐all rotational angular momentum in the gauche rotamer. Other aspects of the spectrum are also discussed.