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19s_secndordreff

19s_secndordreff - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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Lecture # 19 Supplement Second-Order E ff ects: Centrifugal Distortion and Λ –Doubling Centrifugal distortion originates from vibration-rotation interactions. In other words, it results from the fact that the rotational constant B isn’t a constant at all but rather a function of r and as a result can have matrix elements o ff -diagonal in v . Since di ff erences between vibrational energy levels are much larger than di ff erences between rotational energy levels, it is appropriate to introduce corrections to the rotational Hamiltonian matrix elements by second-order perturbation theory involving summations over vibrational levels of the form: D v | B ( r ) | v � � v | B ( r ) | v . (1) E v E v v v We must now examine our rotational Hamiltonian matrix to obtain the precise centrifugal distortion cor- rections appropriate to each of the matrix elements. The simple minded prescription: “Replace B ( r ) by B ( v ) D ( v ) J ( J + 1) wherever B ( r ) occurs” will be shown to be incorrect. We will use the 2 Π , 2 Σ Hamilto- nian again as an example. First consider corrections to the 2 Π 1 / 2 | H | 2 Π 1 / 2 matrix element. The relevant matrix elements o ff -diagonal in v (but diagonal in | Λ | and S) are v , 2 Π ± 1 / 2 | B ( r ) R 2 | v , 2 Π 1 ± / 2 v , 2 Π ± 1 / 2 | B ( r ) R 2 | v , 2 Π ± 3 / 2 . (2) Since our basis functions are actually product functions, and since B ( r ) only operates on v and R 2 only operates on Π Ω ± , we can factor these matrix elements. | v | B ( r ) | v 2 Π ± 1 / 2 | R 2 | 2 Π ± 1 / 2 v | B ( r ) | v 2 Π 1 ± / 2 | R 2 | 2 Π 3 ± / 2 . (3) 1
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5.76 Lecture #19 Supplement Page 2 2 The second order correction to 2 Π 1 / 2 , v | B ( r ) R 2 | 2 Π 1 / 2 , v is therefore 2 Π ± 1 / 2 2 v | B ( r ) | v 2 2 Π 1 ± / 2 | R 2 | 2 Π 1 ± / 2 + G ( v ) G ( v ) v | R 2 Π 3 / 2 3 / 2 | 2 Π ± 2 E (2) 1 / 2 , 1 / 2 = (4) .
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