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PPT – 158a: Lec' 26 Spectroscopy Selection Rules PowerPoint presentation | free to view - id: 26e6c5-MmZiZ

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Monday Project Meeting? 11 pm?

158a Lec. 26 Spectroscopy Selection Rules

Rotational Spectrum

Vibrational

Spectrum Fred J. Grieman

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

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)

__________________

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 ?

___________________

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

- 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!!!)

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

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

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 _____________

_________________________

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

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