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Unformatted text preview: 1 Lecture 2 C566 Spring 2011 Rotational spectra of molecules See Chpt. 4 of Hollas We will start with diatomic molecules, then work our way up to polyatomics (Hollas dives right in to treat all molecules in the first section. Additional reference: G. Herzberg Molecular Spectra and Molecular Structure Vol. I. Diatomic Molecules Approximate treatment of molecules H = H el + H vib + H rot + H nuclear spin + = el vib rot nuclear spin = E el + E vib + E rot + E nuc spin + rot is approximated as a rigid rotor. Again, we shift from the 6 degrees of freedom to separate out translation of the diatomic (in centerofmass coordinates, X = M1 m i x i , etc. for Y,Z, M = m 1 + m 2 ) and rotational motion about the center of mass. R is fixed, reduced mass is = m 1 m 2 /M. m 1 r 1 = m 2 r 2 shows where the center of mass is along the internuclear bond. For heavy light, the COM is close to the heavy center, and the reduced mass is close to the light mass. For a homonuclear, the COM is in the center, and the reduced mass = m/2. The Hamiltonian operator now must include a full three dimensional treatment, but with no variation in R. V(r, , ) = 0, so the only operator is the T operator. m 1 m 2 R r 1 r 2 QC 451 .H53 z y x R 2 + + + = + + = sin sin 1 sin 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 R R R R z y x H h h R= constant, so all derivatives with respect to R are zero. R 2 = I, the moment of inertia. First, we assume that our wavefunction is separable into theta and phi variables. The next step is to divide through by and h 2 /2I. ) ( ) ( ) , ( ) , ( sin sin 1 sin 1 2 ) , ( 2 2 2 2 = = + = E E I H h 2 2 2 2 2 2 2 ) sin (cos sin 1 sin 1 h IE = + + Multiplying by sin 2 , then rearranging (bring over E term, set whole expression equal to 0 = M J 2 M J 2 ) you prove separability, and show that the phidependent portion of the...
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This note was uploaded on 01/18/2012 for the course C 566 taught by Professor Carolinechickjarroll during the Spring '11 term at Indiana.
 Spring '11
 CarolineChickJarroll

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