Slide #1

THE STIMULATED EMISSION PUMPING AND DISPERSED FLUORESCENCE SPECTRA OF ACETYLENE ARE NOT INTRINSICALLY UNASSIGNABLE

IT’S WHAT YOU PLUCK!

A TUTORIAL ON

INTRAMOLECULAR DYNAMICS

FROM A QUANTUM MECHANICAL Heff

TO A CLASSICAL MECHANICAL Heff:

VIEWS OF INTRAMOLECULAR DYNAMICS

Slide #2

LOST IN A SEA OF FIT PARAMETERS

V Q 12

kiQ i2

i1

3N 6

16 i, j,k k ijkQiQ jQk Cubic

1

24 i,j,k,l kijkl QiQ jQkQl Quartic

4 ATOMS, 3N–6=6 MODES, IGNORE SYMMETRY

CUBIC: 6[iii] + 30[ijj] + 20[ijk] = 56 [total]

QUARTIC: 6[iiii] +30[ijjj] + 15[iijj] + 60[iijk] + 15[ijkl]

= 126 [total]

E V i1

3N 6 i vi 1

2 i, j1

3N 6 xij vi 1

2 v j 12

i, j,k yijk vi

12 v j

12 vk

12

i[6] xij[21] yijk[56]

V(Q) E(V) BY PERTURBATION THEORY

total [83]

Slide #3

PERTURBATION THEORY AND “RESONANCE”

H(0)n(0)En

(0)n(0)

n

n(0)

kn

Hkn(1)

En(0) Ek

(0)

k(0)

EnEn

(0) Hnn(1) Hkn

(1) 2

En(0) Ek

(0)

VALID IF 1Hkn

(1)

En Ek

(0)

DEFINE “ZERO-ORDER”BASIS SET

OTHERWISE“RESONANCE”

MUST DIVIDE BASIS STATES INTO QUASI-DEGENERATE GROUPS

H

H1eff H12 H13

H2eff H23

H3eff

VAN VLECK TRANSFORM AND DIAGONALIZE EACH

˜ H ieff

(0)

k n

Slide #4

IS RESONANCE BAD NEWS?

CAN’T USE NONDEGENERATE PERTURBATION THEORY

CAN’T DERIVE SIMPLE ENERGY LEVEL FORMULAS“x-k RELATIONSHIPS”

A FEW kijk AND kijkl ARE SINGLED OUT FOR SPECIAL TREATMENT

* NOT BECAUSE OF THEIR MAGNITUDE

* BECAUSE THEY ARE LARGE WRT AN ENERGY DENOMINATOR

“RESONANCE”

* RESONANCES ARE USUALLY SYSTEMATIC

* RESONANCES HAVE PROMINENT EFFECTS ON THE SPECTRUM AND THE EARLY TIME DYNAMICS

• MOST NON-RESONANT kijk and kijkl CAN BE IGNORED!

Slide #5

POLYADS

EXAMPLE:

1 22

k1,22Q1Q22

SELECTION RULE: v1 = ±1, v2 = ±2, 0

SCALING:

v1 Q1 v1 1 v1 1 1/2

v2 Q22 v2 2 v2 2 v2 1 1/2

NEAR DEGENERATE GROUPS OF BASIS STATES

P=4 POLYADv1 v2 P=2v1+v2 E0 0 0 0 1 1 1 0 2 0 2 2 1 1 3 0 3 3 2 0 4 1 2 4 0 4 4

4 K 2 • 2 • 1 1/20

4 K 1 • 4 • 3 1/2

4

ALL POLYADS EXPRESSED IN TERMS OF AND K

P = 10 POLYAD? HINT: 6 BASIS STATES

RESONANCE

COUPLING TERM

(20)

(12)

(04)

Slide #6

1 C201 20

(0) C121 12

(0) C041 04

(0)

2 C202 20

(0) C122 12

(0) C042 04

(0)

3 C203 20

(0) C123 12

(0) C043 04

(0)

BRIGHT AND DARK STATES

CHANGE IN GEOMETRY

Let MODE 1 BE F–C DARK, MODE 2 F-C BRIGHT

ELECTRONIC TRANSITION: FRANCK-CONDON FACTORS qv,v

EIGENSTATES:

I1 1 g2 C04

1 2 q04,g

I2 2 g2 C04

2 2 q04,g

I3 3 g2 C04

3 2 q04,g

ONLY 1 BRIGHT STATE IN EACH 1 22 POLYAD

P = 4 POLYAD

Slide #7

Acetylene Polyad Structure

Polyad Quantum NumbersNs = v1 + v2 + v3

“quanta of stretching excitation”

Nres = 5v1 + 3v2 + 5v3 +v4 + v5

“approximate energy”

= 4 + 5

“total vibrational angular momentum”

v1 = sym. CH stretchv2 = CC stretchv3 = anti-sym. CH stretchv4 = trans bendv5 = cis bend4/5 = vib. ang. momentum

[Fried and Ezra, JCP 86 (1987) 6270;Kellman and Chen, JCP 95 (1991) 8671]

1 2 3 4 5, , , ,v v v v v

H =

= bright state 0,v2,0,v4,000

Slide #8

NOW FOR THE REAL WORLDENERGY LEVEL PATTERNS FROM DIFFERENT POLYADS OVERLAP

INTER-POLYAD MIXING?CAN OVERLAPPING POLYADS BE

DISENTANGLED?

HIGH RESOLUTION: SEP

PUMP DUMP

LOW RESOLUTION: DF

FLUORESCEPUMP

Slide #9

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Slide #13

INTRINSICALLY UNASSIGNABLE?

E

J(J+1)

VIBRATIONAL LEVELS EXIST?

OR SCATTER PLOT?

STATISTICAL TESTSLEVEL SPACING DISTRIBUTIONINTENSITY DISTRIBUTION

QUANTUM CHAOS ! ?OR NOT ? !

Slide #14

origin3

3

3

3

42,000 cm–1

16,000 cm–1

pum

p

C CH

H

S1

S0

:C CH

H

2 = CC stretch3 = trans bend **

2 = CC stretch4 = trans bend **FRANCK-CONDON PLUCK

Dispersed Fluorescence Spectroscopy from S1 State of Acetylene

C CH H

• Dispersed fluorescence spectra recorded from J=1 levels of 5 S1-State vibrational levels.

• Dispersed emission recorded on an intensified Charge Coupled Device (ICCD) at 16 cm–1 and 7 cm–1 resolution.

• Frequency calibration (good to ~3 cm–1) accomplished using Hg, Ne, Kr, Xe, Th, Fe, and Ar frequency standards.

• Intensity calibration (good to ~20%) accomplished using Standard of Spectral Irradiance (quartz tungsten lamp).

Slide #15

How do we make sense of these spectra?

Internal Energy (cm–1)

JCP 107 8349 (1997) XCC

Our Approach: Numerical Pattern Recognition

Based on two (good) approximations:1. The acetylene effective Hamiltonian is block

diagonal (polyads)2. There is one bright state per polyad.

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Slide #20

Selected Eigenstates at Evib ≈ 14,500 cm–1

“local bend” “counter-rotation”???

Slide #21

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Slide #24

ASSIGNABILITY

WHAT DOES IT MEAN TO “ASSIGN” A SPECTRUM ?

RIGOROUSLY CONSERVED QUANTITIES

[H,A] = 0BORING

APPROXIMATELY CONSERVED QUANTITIES

[H(0),A] = 0 PATTERNS[H(1),A] 0 DYNAMICS

SEVERAL CHOICES OF PARTITIONINGS OF H INTO H(0) + H(1)

RISKY

EARLY TIME DYNAMICS

“THE PLUCK”

MECHANISM