17_lecnotes_rwf

17_lecnotes_rwf - MIT Department of Chemistry 5.74 Spring...

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MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II Instructor: Prof. Robert Field 5.74 RWF Lecture #17 17 – 1 Normal Local Modes: 6-Parameter Models Reading : Chapter 9.4.12.5, The Spectra and Dynamics of Diatomic Molecules , H. Lefebvre-Brion and R. Field, 2 nd Ed., Academic Press, 2004. Last time: ω 1 , τ, ω 2 (measure populations) experiment ω 2 ω 1 ( abcd ) ← ( AB ) g two polyads. populations in (1234) depend on τ . E res, k could use f k = to devise optimal plucks for more complex situations E res (choice of plucks and probes) * multiple resonances * more than 2 levels in polyad Overtone Spectroscopy nRH single resonance nRH + 1RH double resonance dynamics in frequency domain Today: Classical Mechanics: 2 1 : 1 coupled local harmonic oscillators QM: Morse oscillator 2 Anharmonically Coupled Local Morse Oscillators eff . Antagonism. Local vs. Normal. H Local Whenever you have two identical subsystems, energy will flow rapidly between them unless something special makes them dynamically different: * anharmonicity * interaction with surroundings spontaneous symmetry - breaking eff Next time: H Normal .
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5.74 RWF Lecture #17 17 – 2 Two coupled identical harmonic oscillators: Classical Mechanics H T = ( ) PP R L , V + ( ) Q R L , ( ) R = Right, L = Left T = ( ) PP P P R L R L , 1 2 G geometry and masses 1 rr ( R 2 + P L 2 ) + 2 G P P = [ GP rr R L ] 2 1 RL ) F Q R V = ( QQ Q 2 L force constants 1 2 rr ( R + Q 2 ) + 2 F Q R Q = [ FQ L rr ] 2 rr R rr L + 2 2 2 2 () 0 rr R rr L Q 2 H = 1 GP 2 + 1 2 1 2 + 1 H R 0 H L rr R L + Q + rr R L kinetic potential (anharmonic) coupling coupling
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3 5.74 RWF Lecture #17 17 – 3 φ 2 1 1 1 1 1 + m 1 3 m G rr = + = + = = F rr k = µ m 1 m 2 mm m 3 m 3 13 F rr 1 k RL cos φ (projection of velocity of for stretch onto = G rr = m 3 direction) kinetic coupling gets small for large m or φ = π /2 Each harmonic oscillator has a natural frequency, ω 0 : 12 / 1 FG rr rr ] / = 2 1 π µ [ π c ω 0 = 2 and the coupling is via 1 : 1 kinetic energy and potential energy coupling terms.
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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17_lecnotes_rwf - MIT Department of Chemistry 5.74 Spring...

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