DiatomicMoleculesLect8&9ME501F2011

DiatomicMoleculesLect8&9ME501F2011 - Purdue...

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Patterned Border Template 1 Purdue University School of Mechanical Engineering ME 501: Statistical Thermodynamics Lectures 8 and 9: Energy Level Structure of Diatomic Molecules Prof. Robert P. Lucht Room 2204, Mechanical Engineering Building School of Mechanical Engineering Purdue University West Lafayette, Indiana Lucht@purdue.edu , 765-494-5623 (Phone) September 12 and 14, 2011
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Patterned Border Template 2 Purdue University School of Mechanical Engineering Lecture Topics Classical analysis of the energy modes of a diatomic molecule. Two-particle SWE: some details that we skipped over in our hydrogen atom lecture. The Born-Oppenheimer approximation: electrons move much more rapidly than the two nuclei in the molecule. Rotational energy levels: solution of the angular part of the SWE. Vibrational energy levels: solution of the radial part of the SWE. Internal energy level structure of diatomic molecules.
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Patterned Border Template 3 Purdue University School of Mechanical Engineering Classical Analysis of Nuclear Motion - Energy of Translation, Rotation, and Vibration Consider a diatomic molecule as two heavy balls connected by a stiff (but massless) spring. The spring force is actually provided by the electron cloud that surrounds the nuclei. This electron “spring” provides a potential for the nuclei that is approximately a harmonic-oscillator potential: Vr k r r e ()  1 2 2 b g z x y (x 1 , y 1 , z 1 ) (x 2 , y 2 , z 2 ) r m 1 m 2 1 r 2 r
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Patterned Border Template 4 Purdue University School of Mechanical Engineering Classical Analysis of Nuclear Motion - Energy of Translation, Rotation, and Vibration Total (kinetic + potential) mechanical energy of system:    22 2 2 11 1 1 2 2 2 2 () mx y z m x y z V r    We will now introduce a new coordinate system, and manipulate the energy equation to obtain separate translational, rotational, and vibrational energy terms. The coordinates ( X, Y, Z ) define the center of mass of the molecule:   12 1 2 2 1 1 2 2 1 1 2 2 1 / / / mm X m xm x x X m Y m y m yy Y m y m Z m zm z z Z m    
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Patterned Border Template 5 Purdue University School of Mechanical Engineering Classical Analysis of Nuclear Motion - Energy of Translation, Rotation, and Vibration Introduce spherical coordinate system with origin at particle #1: z' x' y' (x 1 , y 1 , z 1 ) (x 2 , y 2 , z 2 ) r m 1 m 2 21 sin cos sin sin cos xxr yy r zz r 
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Patterned Border Template 6 Purdue University School of Mechanical Engineering Classical Analysis of Nuclear Motion - Energy of Translation, Rotation, and Vibration Combining the relations on the last two pages gives: xX m mm r yY m r zZ m r 1 2 12 1 2 1 2  sin cos sin sin cos  m r m r m r 2 1 2 1 2 2  sin cos sin sin cos Note that X , Y , Z , r , , and are all time dependent. Taking the derivatives of the above expressions and performing some algebra gives:     222 2 2 2 2 2 11 1 22 2
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DiatomicMoleculesLect8&9ME501F2011 - Purdue...

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