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Nils Walter: Chem 260z (IC)y (IB)x (IA)linear: IA=0; IB=ICsymmetric: IA=IB≠≠≠≠IC≠≠≠≠0Rotor in 3-Dspherical: IA=IB=IC ≠≠≠≠0Rotational spectroscopy:The rotation of moleculesasymmetric: IA≠≠≠≠IB≠≠≠≠IC≠≠≠≠0Nils Walter: Chem 260Balancing equation: m1r1 = m2r2⇒⇒⇒⇒r1r2C(enter of mass)rm1m2The moment of inertia for a diatomic (linear) rigid rotorMoment of inertia: I = m1r12 + m2r22definition: r = r1 + r2222121rrmmmmIµ=+=µµµµ = reduced massNils Walter: Chem 260(No potential, only kinetic energy)⇒⇒⇒⇒EJ= hBJ(J+1); rot. quantum number J = 0,1,2,...Solution of the Schrödinger equation for the rigid diatomic rotor[Hz]Ψ=Ψ−EVh222µIhBπ4=⇒⇒⇒⇒EJ+1-EJ= 2hB(J+1)Nils Walter: Chem 260⇒⇒⇒⇒a molecule must be polar to be able to interact with lightLight is a transversal electromagnetic wavea polar rotor appears to have an oscillating electric dipole Selection rules for the diatomic rotor:1. Gross selection ruleNils Walter: Chem 260⇒⇒⇒⇒∆∆∆∆J = ±±±±1light behaves as a particle: photons have a spin of 1, i.e., an angular momentum of one unit andSelection rules for the diatomic rotor:2. Specific selection rulethe total angular momentum upon absorption or emission of a photon has to be preservedandrotational quantum number J = 0,1,2,…describes the angular momentum of a molecule(just like electronic orbital quantum number l=0,1,2,...)Nils Walter: Chem 260Transition dipole moment Okay, I know you are dying for it: Schrödinger also can explain the∆∆∆∆J = ±±±±1 selection ruleτµµdiffi∫ΨΨ=final stateinitial state⇒⇒⇒⇒only if this integral is nonzero, the transition is allowed; if it is zero, the transition is forbiddenNils Walter: Chem 260⇒⇒⇒⇒classical rotational energy The angular momentum221ωIE=ωIP =classical angular momentum EIP2=)1(+=JhBJEJand ⇒⇒⇒⇒IhBπ4=with hJJP)1(+=But P is a vector, i.e., has magnitude and directionmJJ=1Nils Walter: Chem 260DegeneracyEnergy only determined by J⇓⇓⇓⇓all mJ= -J,…,+Jshare the same energy⇓⇓⇓⇓2J+1 degeneracy Selection rule: ∆∆∆∆mJ= 0, ±±±±1J=3 J=2 mJ= mJ=Nils Walter: Chem 260∆∆∆∆J = ±±±±1kTEloweruppereNN∆−=The allowed rotational transitions of a rigid linear rotor and their intensityincreasingly degenerate2J + 1statesNils Walter: Chem 260The non-rigid linear rotor rotor:Centrifugal forces elongate the bond and increase the moment of inertia∆∆∆∆J = ±±±±1 still holds!Nils Walter: Chem 260J = 0,1,2,…K = J,J-1,…,-JThe symmetric


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U-M CHEM 260 - Lecture Note

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