An effective-hamiltonian approach to CH5+, using ideas from atomic spectroscopy

Hougen, Jon T.

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https://hdl.handle.net/2142/91179 Copy

Description

Title

An effective-hamiltonian approach to CH5+, using ideas from atomic spectroscopy

Author(s)

Hougen, Jon T.

Issue Date

2016-06-20

Keyword(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.

Publisher

International Symposium on Molecular Spectroscopy

Type of Resource

text

Language

En

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http://hdl.handle.net/2142/91179

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Copyright 2016 by the authors

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