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Title:An effective-hamiltonian approach to CH5+, using ideas from atomic spectroscopy
Author(s):Hougen, Jon T.
Subject(s):Spectroscopy of Large Amplitude Motions
Abstract:In this talk we present the first steps in the design of an effective Hamiltonian for the vibration-rotation energy levels of CH$_{5}^{+}$. Such a Hamiltonian would allow calculation of energy level patterns anywhere along the path travelled by a hypothetical CH$_{5}^{+}$ (or CD$_{5}^{+}$) molecule as it passes through various coupling cases, and might thus provide some hints for assigning the observed high-resolution spectra. The steps discussed here, which have not yet addressed computational problems, focus on mapping the vibration-rotation problem in CH$_{5}^{+}$ onto the five-electron problem in the boron atom, using ideas and mathematical machinery from Condon and Shortley’s book on atomic spectroscopy. The mapping ideas are divided into: (i) a mapping of particles, (ii) a mapping of coordinates (i.e., mathematical degrees of freedom), and (iii) a mapping of quantum mechanical interaction terms. The various coupling cases along the path correspond conceptually to: (i) the analog of a free-rotor limit, where the H atoms see the central C atom but do not see each other, (ii) the low-barrier and high-barrier tunneling regimes, and (iii) the rigid-molecule limit, where the H atoms remain locked in some fixed molecular geometry. Since the mappings considered here often involve significant changes in mathematics, a number of interesting qualitative changes occur in the basic ideas when passing from B to CH$_{5}^{+}$, particularly in discussions of: (i) antisymmetrization and symmetrization ideas, (ii) $n,l,m_{l},m_{s}$ or $n,l,j,m_{j}$ quantum numbers, and (iii) Russell-Saunders computations and energy level patterns. Some of the mappings from B to CH$_{5}^{+}$ to be discussed are as follows. Particles: the atomic nucleus is replaced by the C atom, the electrons are replaced by protons, and the empty space between particles is replaced by an “electron soup.” Coordinates: the radial coordinates of the electrons map onto the five local C-H stretching modes, the angular coordinates of the electrons map onto three rotational degrees of freedom and seven bending vibrational degrees of freedom. The half-integral electron spins map onto half-integral proton spins or onto integral deuterium spins (for CD$_{5}^{+}$). Interactions: the Coulomb attraction between nucleus and electrons maps onto a Morse-oscillator C-H stretching potential, spin-orbit interaction maps onto proton-spin-overall-rotation interaction, and Coulomb repulsion between electrons maps onto some kind of proton repulsion that leads to the equilibrium geometry.
Issue Date:2016-06-20
Publisher:International Symposium on Molecular Spectroscopy
Genre:Conference Paper/Presentation
Rights Information:Copyright 2016 by the authors
Date Available in IDEALS:2017-01-26

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