13_562ln08

13_562ln08 - MIT OpenCourseWare http:/ocw.mit.edu 5.62...

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MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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5.62 Lecture #13: Nuclear Spin Statistics: Symmetry Number, σ . Low Temperature Limit for Rotational Partition Function Readings: Hill, pp. 153-156, 153-159 [466-472] Maczek, pp. 54-57 ORIGIN OF σ — NUCLEAR SPIN STATISTICS Quantum mechanics requires that the total wavefunction be either symmetric (does not change sign) or antisymmetric (changes sign) with respect to exchange of any two identical nuclei . ψ TOT must be symmetric if nuclei have integer spins (bosons) or antisymmetric if nuclei have 1/2 integer spins (fermions). ψ TOT (x) EXCHANGE if nuclei are BOSONS ⎯⎯⎯⎯ + ψ TOT ( x) number integer spin I homonuclear a symmetric function diatomic molecule an EVEN function nuclear spin quantum ψ TOT (x) EXCHANGE if nuclei are FERMIONS −ψ TOT ( x) 1/2 integer spin I an antisymmetric function an ODD function FOR EXAMPLE: O 2 The 16 O nuclei have spin I = 0 BOSONS Therefore ψ TOT for O 2 must be symmetric or EVEN ψ TOT = ψ TRANS ψ VIB ψ ELEC ψ ROT ψ NUCL.SPIN Need to investigate symmetry of each one of these wavefunctions for O 2 upon exchange of two 16 O nuclei. [ NOTE : we are permuting nuclei, not atoms. The Na atom is a Boson, but the Na nucleus in Na 2 is a Fermion.]
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5.62 Spring 2008 Lecture 13, Page 2 ψ TRANS depends only on CM coordinate of the molecule so this function is not affected by exchange — we will no longer consider it ψ VIB depends only on distance between the two nuclei so this coordinate is not affected by exchange — no longer consider it. [Be careful. In a polyatomic molecule, internal rotation of a methyl group does permute identical nuclei and it is a vibration.] ψ ELEC ground electronic state of O 2 is 3 Σ g . This term symbol tells us that the electronic wavefunction is antisymmetric or ODD with respect to exchange of the two nuclei. [The classification operator is σ v × i (both body-frame).] g and + u are both odd, + g and u are both even. , etc. consist of one manifold of even states (like + g ) and one nearly identical manifold of odd states (like g ). ψ ROT rigid rotor rotational wavefunctions are spherical harmonics ψ J=0,2,4,6 are symmetric with respect to exchange EVEN ψ J=1,3,5,7 are antisymmetric with respect to exchange ODD ψ NUCLEAR 16 O nucleus has a nuclear spin I = 0 a boson; there are (2I + 1)(I + 1) symmetric (EVEN) nuclear spin states: “ortho” (the states with larger nuclear spin degeneracy) (2I + 1) I
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This note was uploaded on 11/27/2011 for the course CHEM 5.43 taught by Professor Timothyf.jamison during the Spring '07 term at MIT.

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13_562ln08 - MIT OpenCourseWare http:/ocw.mit.edu 5.62...

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