19_580ln_fa08

# 19_580ln_fa08 - 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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5.80 Lecture #19 Fall, 2008 Page 1 of 8 pages Lecture #19: Second-Order Effects Last time: perturbations = accidental degeneracy Today: effects of “Remote Perturbers”. What terms must we add to the effective H so that we can represent all usual behaviors with minimum number of parameters. Use the van Vleck transformation. Two effects to be discussed * centrifugal distortion of all zero- and first-order parameters. e.g. B D [explicit R-dependence of B(R)] A A D [implicit R-dependence of A(R)] [interaction with all v’s of same Λ -S state] * Λ -doubling and other 2nd-order parameters [interaction with all v’s of all other states] We will work with 2 , 2 s example Recipe * H eff in terms of E, B, A, ( λ , γ ), α , β * van Vleck transformation: diagrammatically in the form of “railroads” for each location in H eff * each term in van Vleck transformation is explicit function f v,J ) * H ev,e H e v ,ev v ( o o e , E ev E e v v new 2 nd order parameter e/f 2 3/2 2 1/2 2 s 2 + B v ( y 2 2 ) B v ( y 2 1 ) 1/2 −β s v v ( y 2 1 ) 1/2 2 3/2 E v + A 2 + B v ( y 2 ) α s + β s 1 ( 1) s y 2 1/2 E v A 2 s E v + B v y 2 ( 1) s y y J + 1/2 For simplicity we do not include γ terms ( λ terms are not possible for S < 1 states).
5.80 Lecture #19 Fall, 2008 Page 2 of 8 pages What do we do with these? interesting remote H mn m m H = n n follows rules for matrix multiplication VV E m o δ m + λ 1 H m + λ 2 H mn H n m + H mn H n m H m,m m m 2 o E n o E o E n o n E m m ~ λ 2 o H mn H nm n E m + E m o E n o 2 We are going to write H eff in terms of zero-order parameters E, B, A perturbation parameters α , β second-order parameters D, A D , o, p, q H = H ROT + H SO H = ( ROT ) 2 e/f dependent q ( Λ -doubling) ( ) 2 H e/f independent D (centrifugal distortion of B) o ( Λ -doubling) + ( H SO ) 2 λ (2nd-order spin-spin) + ( H ROT ) e/f dependent p ( Λ -doubling) e/f independent γ (2nd-order spin-rotation) A D (centrifugal distortion of A) Generate many 2nd-order parameters — not all are linearly independent.

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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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19_580ln_fa08 - MIT OpenCourseWare http:/ocw.mit.edu 5.80...

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