The Moles of Vibration of Butane and Pentane. ``Free Rotation'' about Carbon-Carbon Bonds and a New Type of Stereoisomerism

Abstract

The Hamiltonian function for a chain of four carbon atoms with the simple valence force system potential energy is transformed by successive changes of variables to such a form that the wave equation is nearly separable into five harmonic oscillator equations and a sixth equation governing the internal rotation. This rotation equation has been solved numerically for twelve different rotational levels of the nonvibrating molecule, some of which actually correspond to rotations, others to a torsional oscillation of restricted amplitude about either the cis or the trans position. The remaining terms in the wave equation are treated as a perturbation; the first-order perturbation energy is less than 1 cm-1 for all moderately low vibrational states. There are thus two isomeric forms of butane with plane carbon skeletons, and a third form in which the two ethyl groups are rotating about the central carbon-carbon bond. The vibration frequencies for the three possible plane forms of a five-carbon chain have been calculated. When the frequencies of C2, C3, C4 and C5 are calculated with the same two force constants throughout, and by taking account of the two plane forms of C4 and the three of C5, results are obtained in fair statistical agreement with observed Raman and infrared lines.