18_lecnotes_rwf

18_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 #18 18 – 1 eff eff Transformation between H Local and H Normal Reading : Chapter 9.4.13, The Spectra and Dynamics of Diatomic Molecules , H. Lefebvre-Brion and R. Field, 2 nd Ed., Academic Press, 2004. Last time: 2 identical coupled subsystems 1. Classical mechanical treatment of two 1 : 1 coupled local Harmonic Oscillators Simple transformation decouples the sub-systems. 2. quantum Mechanical treatment of Morse oscillator± V(r) = D e (1 – exp – ar) E(v)/hc = ω ( v + 1/2) + x ( v + 1/2) 2 ( x is usually negative)± 3 Expand V(r) , use 1st-order p.t. for r 4 and 2nd-order for r . Get exact result for energies! Justifies use of harmonic oscillator basis set even when diagonal anharmonicity appears in E (0) + E (1) 3. 2 anharmonically coupled local Morse oscillators± eff± 3 parameter H Local (relationships or constraints among traditional fit parameters) Today: Transformation H Local H eff eff Normal Why? Good description of the pluck (e.g. overtone vs. SEP). antagonism between term that lifts degeneracy in polyad vs. term that has off-diagonal intrapolyad matrix elements± 2-level illustration± 3 parameter model – some inconsistencies± eff 6 parameter model H eff Local H Normal
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5.74 RWF Lecture #18 18 – 2 Incompatible Terms in H eff Often we have a choice between two zero-order basis sets. These correspond to two limiting cases for the dynamics. In one case, one term causes an on-diagonal energy splitting that preserves the limiting case and another term causes an off-diagonal matrix element that destroys the limiting case. In the other limiting case, the roles of the two terms are reversed. Consider the following illustration for a 2 level system: HH + A + B E 0 A 0 0 B = 0 E + 0 A + B 0 Limit #1: Diagonalize H ° + B (i.e. set A = 0) 2 12 ±= / [ 1 ± 2 ] ( diagonalize by inspection - often possible in a limiting case ) ± ( H ° + B ) ± 2 ) = ± 1 ( 2 E B E ( as required ) 2 ± A ± = 1 ( AA ) = 0 2 ± A m + ) = = 1 ( AA A 2 + A A T is the transformation H 1 T H = + EB E that diagonalizes H ° + B B lifts the degeneracy and preserves the #1 limit A
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18_lecnotes_rwf - MIT Department of Chemistry 5.74, Spring...

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