158a: Lec' 26 Spectroscopy Selection Rules

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Monday Project Meeting? 11 pm?
158a Lec. 26 Spectroscopy Selection Rules
Rotational Spectrum
Vibrational
Spectrum Fred J. Grieman
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We want P2 c22 probability of being in
state 2
Solve for c2 c2 (1/i?)?-t? lt?2H??1gt
e-i(?12)t dt ?12 ?1 - ?2 Need to get H?
Now for light specifically __________
electric field interaction ? permanent dipole
(_________) change in dipole (___________)
electronic dipole (___________) E from
light Assume at first x polarized Ex Ex?
cos?t (Ex?/2) (ei?t e-i?t) just x
component Then H? -?x Ex ________________
f(x)f(t)
i? ?c2/?t e-i(?1-?2)t lt?2H??1gt
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c2 (-1/2i?) Ex? lt?2?x?1gt ?-t?
(ei?te-i?t) e-i(?12)t dt (-Ex?/2i?)
(?21)x sin(??t/2)/(??/2) ??
?-?12 c22 (Ex?/2?)2 (?21)x2
sin2(??t/2)/(??/2)2
_______ __________ _____________________
__
Look at (?21)x2 lt?2?x?1gt2
Rx212 (other texts)
__________________
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Spectrum ____________ of absorption/emission
vs. __________ Eint Ee Ev Er
_________________ spectroscopic
notation Rotational Spectroscopy of Diatomic
Molecules Energy of transition 1) F(J)
__________ B h2/(2Ihc) J 0, 1, 2,
gj 2J1
?rot YJ,m 2) Transition between upper ( '
) and lower ( " ) energy levels
?EJ,J ?(cm-1) F(J') F(J'')
____________________ Intensity on transition
(probability population) 1) Probability ?
Selection Rules ? J,J ______________
? ?xi ?yj ?zk Spherical Polar
µx µosin?cos? µy µosin?sin? µz
_______ µo ?
___________________

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Take µz (µz)Jm,Jm ?02? ?0? YJ,m
µocos? YJ,m sin? d? µo
?02? ?m ?m d? ?0? TJ,m cos?
TJ,m sin? d? permanent mm
?m 0 ________________
dipole recursion relation cos?
TJ,m c1 TJ1,m c2 TJ-1,m so, ?0?
TJ,m cos? TJ,m sin? d?
c1?0? TJ,m TJ1,m sin? d? c2?0? TJ,m
TJ-1,m sin? d? ___________ J J1
or J J-1
Non-zero only if __________ (Selection
Rule!!!) µx, µy give same result
except ?m 1 (only seen in magnetic field)
µJ,J ______________
(Kronecker delta)
General Selection Rule
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• 2) Population ? Boltzmann
• For J 0, gJ 2J1 1 E0
  0 so
• _________ (2J1)e-EJ/kBT
  ___________________
• Microwave Spectrum!!!!
• ?J 1 (absorption) J J1
• ? (cm-1) BJ(J1) BJ(J1) B(J1)
  (J2) BJ(J1)
• B2(J1) _____  J 1, 2,
  3,
• (_____________ between lines in
  spectrum)
• (Draw Spectrum on
  board.)
• Molecular constants from spectrum
• Determine _______________________ (bond
  distance!!!)
• ln(IJ/2J1) -hcB(J(J1)/kBT Plot ? T
  (temperature!!!)

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Correction to 1st order Model
________________________ As J? centrifugal
force? stretching bond which changes I Model
as bond force opposing stretching ½ kr2 vs.
BJ(J1) (Derivation in
handout.) F(J) _____________________
approximation D 4B3/?e2
(vibration constant) Spacing no longer 2B 
B 1 cm-1 D 10-6cm-1
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Vibrational Spectroscopy of Diatomic Molecules
Energy of Transition G(v)
____________ ?e h?/hc v 0, 1, 2,
gv1
?v Hv Gaussian ?Ev,v ?(cm-1)
G(v') G(v'') ?e(v'½) ?e(v''½)

________________ Intensity on
transition (probability population) 1)
Probability ? ___________________ For
vibration oscillating dipole (?µ/?r) ? 0
________________________ µe permanent
dipole at r 0 transition dipole ?
v,v lt?v??vgt µe lt?v?vgt (?µ/?r)r0
lt?v r ?vgt

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Go to reduced coordinate for selection rule r ?
y lt?v r ?vgt ? ___________________
remember y ______________
(lowering/raising operator) Then,
(?µ/?r)r0 lt?v y ?vgt (?µ/?r)r0 lt?v
½(a a-) ?vgt (?µ/?r)r0
c1lt?v?v-1gt c2 lt?v?v1gt
v v-1
v v1 _____________
_________________________
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2) Population ? Boltzmann For v 0,
E0 G(0) ½ ?e so
________________________________ Examples
T298 K kBT 300 cm-1 HCl
(?e 2886 cm-1) N1/N0 9 x 10-7
I2 (?e 215 cm-1)
N1/N0 0.355 Infrared Spectrum!!!! (Low
resolution no rot. structure)
Absorption
G(v') G(v'') ?e(v'- v'')
____________________ (one energy!)
Draw Spectrum on
board. Molecular constants from spectrum
_______________________________________________

? k (stiffness of bond) Next
ro-vibrational spectroscopy