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# The vibrational energy is evib v where the vibrational

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Unformatted text preview: e internuclear axis (due to relativistic effects) is faster than the molecular rotation. (Example: OH.) Hund Case (b): The precession of the electron spin around the internuclear axis is slower than the molecular rotation (Example: H2+.) In the Case (a) molecules, Σ is a conserved quantum number, and hence so is Ω=Λ+Σ. For example, OH is a 2Π molecule and so the possible values of Ω are 1/2 and 3/2 (and their negatives – see later). We therefore know that J must be at least as large as |Ω| and have the same character (either integer or half ­integer). For a given value of J, the squared angular momentum perpendicular to the internuclear axis is J(J+1)−Ω2. We know that for each value of |Ω| the above analysis gives two sets of energy levels for each J. Since the parity transformation reverses the sign of Ω, there is (for Ω≠0) one positive parity and one negative parity level. The degeneracy between them is broken by a multitude of corrections associated with finite nuclear:electron mass ratio, etc. In the Case (b) molecules, N is a conserved quantum number. A similar argument to the above shows that N is an integer, and is at least |Λ|. For Λ≠0, there are two degenerate levels, one of positive and one of negative parity, a phenomenon known as Λ doubling. The total angular momentum J takes on values from |N−S| to N+S in steps of ΔJ=1. C. VIBRATION We next consider vibration. The molecule has a natural frequency of vibration, ω = (k/μ)1/2, where k is the force constant of the bond and μ the reduced mass. The vibrational energy is Evib = (v+½)ω, where the vibrational quantum number v=0,1,2,…. Typical vibrational frequencies are in the mid ­IR: ~1014 Hz, or somewhat les...
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