5.80 Small-Molecule Spectroscopy and Dynamics

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1 MIT OpenCourseWare Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit:

2 5.80 Lecture #11 Fall, 2008 Page 1 Lecture #11: Pictures of Spectra and Notation Last Time Selection, propensity and intensity rules M(R) if α Sb (θ,φ) Ω i Ω absolute rotational linestrengths S f Ji J f permanent Sum rule rotation P µ dipole 2 g i or S 2 dm typical form for vibration dm P dq vgi universal angular dq factor must electronic M sum to 1 or to if v i v f P M 2 q vi v f g i total degeneracy TODAY: Patterns in spectra Typical constants How to assign spectra - problems and tricks Notation PURE ROTATION µ-wave sources limit range of J sampled (more difficult to assign than expected) superposition of lines from vastly different energy regions, isotopomers population effects. How to tell up from down. Think about this! VIBRATION-ROTATION J range limited by T, not by radiation source. P and R branch structure - open, no heads, zero gap (easy to assign) PQR notation hot bands and isotopomers ELECTRONIC Band heads due to large B blue vs. red degraded sign of B (hard to assign because of overlapping lines) J Head vibrational Sequences vs. Progressions Qualitative Franck-Condon Principle Universal notation for all molecular spectroscopy. Upper level denoted by Lower level denoted by always stated first always stated second

3 5.80 Lecture #11 Fall, 2008 Page 2 Z Z Z (for any quantity, Z) J = J J J + 1 O 2 P 1 Q 0 i.e. R(J) means J R 1 S 2 Denote whether transition is absorption or emission by direction of arrow. Always state upper level first. PURE ROTATION SPECTRA µ-wave (e.g. heteronuclear diatomic) requires permanent dipole moment ν GHz [ ν = ν/c c = cm/s] λ 3cm 1mm cm-wave and mm-wave regions ΩΩ for 1 + J = ±1 only because Ω = 0. recall S Ji J f F J 2 (J + 1) 2 v (J) = B v J(J + 1) D v J J 1 F v centrifugal distortion (J) F v (J 1) = 2JB v 4J 3 D v J of higher level Typical B-value 1 cm 1 = 30 GHz B: Cs 0.01 cm 1 I cm 1 2 H 2 61 cm 1 B ( cm 1 ) 1 = 12 C = B 3 D ω 2 ~ 10 6 B 1 m ( ) µ = 1 m 2 R 2 (cm) µ amu m 1 + m 2 Microwave Spectrum: Klystron, Backward Wave Oscillator (BWO) typical tuning range < 1/2 octave ± ~25% 8-12 GHz not DC to 100 GHz. See only a small portion of pure rotation spectrum for any given Exptl. setup.

4 5.80 Lecture #11 Fall, 2008 Page 3 Level Diagram Spectrum 2 1 ω + 6B ω + 2B 1 typical Klystron tuning range J = 0, v = 1 ω + 0B 1 2B 0 4B 0 6B 0 2α e 4α e 6α e v = ν isotope heavy isotope larger µ smaller B 2 0 6B 0 B v = B e α e (v + 1/2) α e 0.01 B e 1 0 2B 0 distinguish hot band from isotope by size of expected isotope fractional shift. J = 0, v = 0 0B 0 hot bands interspersed among cold band, even though ω B Almost regular pattern of lines separated by 2B ( 4J 3 D makes lines draw closer together at high J) intensity NON-LECTURE e F( ΩΩ S ω ev J J kt ( J M ) ν ROT 2J +1 N v gj < (N J M N ) N v=0 e 2J +1 specific M kt J+1 J value 200 cm 1 vibration at RT population M = 0 (Z-polarized) Derive below (2J + 1) from M-degeneracy = J )/kt 1 F( J ) 1+ F( J ) F( J ) F( J ) = ν ROT e F( J )/kt kt kt = kt kt How to assign? Can t see entire pattern. ν = 2BJ Can usually guess R e to ~10% so B e to ~20% given a line at 15GHz, probably J = 2 1 next line would be at 3/2 15 = 22.5 GHz if it was 1 0, next line would be at 2/1 15 = 30 GHz if it was 3 2, next line would be at 4/3 15 = 20 GHz So assignment is based on a guess followed by at least one confirming measurement. Non- 1 + states and polyatomic molecules: other kinds of transitions possible

5 5.80 Lecture #11 Fall, 2008 Page 4 (always M z,ii Ω = 0) for linear polyatomics and symmetric tops For Ω 0, etc. (parity doubling) J = Ω can have J = 0 Q branches J = Ω + Asymmetric tops very complicated No electric dipole? e.g. O 2 X 3 g = Ω 0 fine structure transitions due to magnetic dipole. VIBRATION-ROTATION spectra IR cm 1 requires dm z,ii /dq 0 for 1 + J = ±1 only v = ±1 strongest G(v) = ω e (v + 1/2) ω e x e (v + 1/2) 2 + F v (J) = B v J(J + 1) D v J 2 (J + 1) 2 B v = B e α e (v + 1/2) NOTE: Signs. Because almost always see ω e x e, D v, α e > 0 as defined above! typical ω e 1000 cm 1 Cs 2 42cm 1, I cm 1, H cm 1

6 5.80 Lecture #11 Fall, 2008 Page 5 J 3 G(v ) + F v (J ) v R branch P branch v 1st line in R branch J = J + 1 J = 1 J = 0 R(0) 2B J G(v ) + F v (J ) 1st line in P branch J = J 1 J = 0 J = 1 P(1) R(J ) P(J ) blue on left zero gap red on right Lines spaced by 2B with zero-gap of 4B where Q(0) would be. always negative R branch J = +1 R(J ) = G + F = G + BJ 2 + (2B + B)J + 2B P branch J = 1 P(J ) = G + F = G + BJ 2 (2B B)J notation G = G(v > ) G(v < ) G v > + v < 2 G(1/2) = ω e 2ω e x e = ν vα e small negative quadratic term lines in R branch gradually pull closer together, branch goes to blue lines in P branch gradually pull further apart, branch goes to red 2B large linear term

7 5.80 Lecture #11 Fall, 2008 Page 6 Double Humped Appearance of Vibration-Rotation band gradually pulling together gradually spreading seldom see band heads blue BJ(J + 1)/kT I ~ (2J + 1)e gap R ν P λ red Finer details: Hot Bands: 1 0 looks like 2 1 except is shifted 2ω e x e to red 2. constant difference between rotational lines is smaller by 2α e (< ~1%) 3. identical quadratic term B = α e 4. kt intensity down by e ω e Isotopomer Bands B v µ 1 ω e µ 1/2 intensity depends on isotopic abundance, NOT on T Overtone Bands near 2ω e 2. linear term (2B ± B) in rotational branches almost same as for v = +1 band. 3. quadratic term 2 as large can sometimes see bandheads for high overtone bands 4. intensity weaker than v = 1 band, usually by 10 to 100 ROTATIONAL ASSIGNMENT trivial because of * presence of zero gap * no overlap of band with itself Electronic Spectra for now deal with non- 1 + soon. No restriction on types of molecules. [Symmetry restrictions, g u, + +, S = 0, Λ = Ω = 0,±1] ν 10,000 cm 1 100,000 cm 1 VIS & UV & VUV (not X-ray) because not sharp lines

8 5.80 Lecture #11 Fall, 2008 Page 7 Rotational Structure of one vibrational band of electronic transition: just like VIBRATION- ROTATION band but more can happen. J = ±1 only Band origin T e + G (v ) G (v ) R(J) = ν 0 + BJ 2 + (2B + B)J + 2B P(J) = ν 0 + BJ 2 (2B B)J Fortrat parabola ambiguity about assignment Q(J) = ν 0 + BJ 2 + BJ (not for ) starts toward blue starts toward red usually compact B can be large and have either sign because B B α e Heads form! B > 0 R e < R e B < 0 R e > R e Head in P Head in R degrades to blue degrades to red blue red blue red By inspection, can tell sign of B bonding nature of excited state. At what J does head form? How far is head from band origin? Extrenum ν = dp(j) = 2 BJ (B + B ) dj P J HEAD R J HEAD 2 = B + B P if B > 0, then J HEAD ~ B 2 B B >0 = (3B + B ) R B if B < 0, then J HEAD ~ 2 B B <0 ( P ) ν 0 = B B + B (B + B ) B + B P J HEAD 2 B 2 B (B + B ) 2 = < 0 seldom negligible ~ B2 4 B B ( R (3B B ) 2 B 2 R J HEAD ) ν 0 = + 2B > 0 ~ + 4 B B

9 5.80 Lecture #11 Fall, 2008 Page 8 Can't use easily picked-out head as approximation for band origin! Hard to find origin because no zero gap. Covered by returning branch.

10 5.80 Lecture #11 Fall, 2008 Page 9 type transition ν 0 Q type transition ν 0 Vibrational structure next time.