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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 center-of-mass coordinates, X = M-1 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 phi-dependent 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