notes17 - 5.73 Lecture #17 Perturbation Theory IV 17 - 1...

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17 - 1 5.73 Lecture #17 modified 9/30/02 10:12 AM Perturbation Theory IV Last time: Transition probabilities in the presence of cubic anharmonicity direct n = ±1 singly forbidden n = ±4, ±2, 0 doubly forbidden n = ±7, ±5, ±3, ±1 ψ n ( 0 ) n ±1 ( 0 ) n ( 0 ) n ( 0 ) + n ( 1 ) n ( 0 ) + n ( 1 ) n ( 0 ) + n ( 1 ) Today ** brief remarks about in the presence of anharmonic mixing: Ψ x, 0 () = n ( 0 ) * partial recurrences depend on * rate of dephasing of recurrences depends on , the fractional change in the average frequency. ω d dn ** Quasi degeneracy when H nk ( 1 ) E n ( 0 ) E k ( 0 ) ≈1 must diagonalize ** coupled oscillator example: POLYADS, IVR Possibility: Intramolecular Vibrational Redistribution in Acetylene Extra basis states mixed in by a x 3 anharmonicity denoted by underline. ** “x-k” relationships
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17 - 2 5.73 Lecture #17 modified 9/30/02 10:12 AM What about Quartic perturbing term bx 4 ? Note that E n ( 1 ) = nbx 4 n ≠0 and is directly sensitive to sign of b ! special hint What about wave packet calculations? ψ n expressed as superposition of k ( 0 ) basis state terms (perturbed eigenstates) Ψ x,0 () expanded as superposition of k ( 0 ) terms (state prepared at t =0 ) Ψ x,t oscillates at e iE n t h (evolving prepared state) E n = E n ( 0 ) + E n ( 1 ) + E n ( 2 ) A state which is initially in a pure n ( 0 ) will dephase, then exhibit partial recurrences at but recurrence is not perfect since EE nm −≠ h ω m m m n 2 2 π≈ = π ω ω t t where is an integer recurrence not quite integer multiples * time of 1st recurrence will depend on E ! * successive recurrences will occur with larger phase error for ω n,n–1 vs. ω n+1,n 1st recurrence phase discrepancy is δ 2nd recurrence phase discrepancy is 2 δ etc. because E n + E n +1 2 decreases as n increases
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17 - 3 5.73 Lecture #17 modified 9/30/02 10:12 AM On pages 16-5 through 16-11 I worked out how a block of H eff is corrected so that “out-of-block” off-diagonal matrix elements can be safely ignored. These corrections come in two forms: (i) Second-order perturbation theory corrections to diagonal matrix elements. One example is the “ x–k ” relationships by which the x ij vibrational anharmonicity constants are evaluated in terms of third and fourth derivatives of V( Q ). (ii) Van Vleck transformation of “quasi-degenerate” or “resonant” blocks of H eff . Something analogous to second-order perturbation theory is used to fold out-of-block off-diagonal matrix elements into polyad blocks along the diagonal of H . These corrections occur both on and off the diagonal within these quasi-degenerate polyad blocks. “Resonance” is not accidental. Once it appears it affects larger and larger groups of near-degenerate basis states.
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notes17 - 5.73 Lecture #17 Perturbation Theory IV 17 - 1...

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