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

ACETYLENE IVR IN STATE˜ X 1g

IMPORTANT RESONANCES

Q,P ˆ Q , ˆ P a,a†, AND N

Heff IN NORMAL MODE BASIS SETS

CORRELATION DIAGRAM

VISUALIZATIONS OF QUANTUM DYNAMICS

BEST BASIS?

HEISENBERG’S CORRESPONDENCE PRINCIPLE

SURFACES OF SECTION

A CAUSE OF SOME BIFURCATIONS

ACETYLENEVINYLIDENE ISOMERIZATION

CHAOS AND BIFURCATIONS

QM Heff TO CM eff VIA

Slide #3

IMPORTANT RESONANCES

NORMAL MODES

1 2 3 4 5

+g +

g +u  

g  g

NRES = 5v1 +3v2 +5v3 +v4 +v5 NSTRETCH = v1 +v2 +v3

EACH TERM IN V(Q) MUST BE CUBIC TERMS QUARTIC TERMS

QiQjQk ΔNRES QiQjQkQ ΔNRES

133233122112144155244255345

571733113

1,2441,2553,24511,3344,55

00000

MOST IMPORTANTINTRAPOLYAD RESONANCES

NONE ARE RESONANT!

Slide #4

REPLACE Q, P, H BY DIMENSIONLESS

ˆ Q 2c

h

1/2

Q

ˆ P h2c 1/2 P

1

2ck 1/2 cm 1

ˆ H h2c 1H

n1a† n n1 1/2

n 1a n n1/2

n a†a n n N a†a

CREATION (a†) ANNIHILATION (a),AND NUMBER (a†a) OPERATORS

EXAMPLE

k122Q1Q22k k122

2c11

2h

1/2 2c 2 2

2h

a1 a1† a2 a 2

† 2

constants quantumnumbers

Slide #5

Acetylene Bending Effective Hamiltonian

Slide #6

Acetylene Bending Effective Hamiltonianin local mode coordinates

Slide #7

5,800

5,600

5,400

5,200

5,000

5,800

5,600

5,400

5,200

5,000

Inte

rnal

Ene

rgy

(cm

–1)

Inte

rnal

Ene

rgy

(cm

–1)

15,600

15,200

14,800

14,400

14,000

15,600

15,200

14,800

14,400

14,000

normal modebasis set eigenstates local mode

basis set

normal modebasis set eigenstates local mode

basis set

Correlation Diagrams

Slide #8

VISUALIZATIONS OF QUANTUM DYNAMICS

(t), |(t)|2 ARE TOO COMPLICATED. WHY?

NEED 1-D VISUALIZATIONS

SURVIVAL PROBABILITY

TRANSFER PROBABILITY

EXCITATION OF ONE MODE

RESONANCE OPERATOR

TRANSFER RATE OPERATOR

Pi(t) i(t) i(0) 2

Pi f i(t) f (0) 2

N i t (t)ai†ai (t)

†k k

(1) †Hk k k

Slide #9

y = 0.56

An Unusual Trend in IVR

Bright States: (0,0,0,v40,00)

Frequency Domain Time Domain

y = 0.16

y = 0.24

v4

Slide #10

Slide #11

Slide #12

Slide #13

REDUCED DIMENSION

TRAJECTORIESSURFACES OF SECTION

QM Heff

CM Heff Hexact

ai, ai, aiai† †

Slide #14

CLASSICAL MECHANICS

Q, P

ACTION, ANGLE I,

CONJUGATE VARIABLES

IT IS MOST CONVENIENT TO GO FROM QM TO CM VIA THE ACTION, ANGLE REPRESENTATION

Slide #15

HEISENBERG’S CORRESPONDENCE PRINCIPLE

QUANTUM CLASSICAL

CONSERVED NOT CONSERVEDCollaboration with C. Jung, UNAM, and H. S. Taylor, USC.

Slide #16

SURFACE OF SECTION

WAY OF DISPLAYING STRUCTURE IN EXPLORATION OF PHASE SPACE

REDUCED DIMENSION VIEW OF CLASSICAL TRAJECTORIES

* REGULAR (QUASIPERIODIC) VS. CHAOS

* CLASSES OF REGULAR MOTION

* BIFURCATIONS

APPEARANCE OF NEW CLASSESDISAPPEARANCE OF OLD CLASSES

(Q1, P1;Q2,P2;…Qn,Pn)[Q,P] [Ji,i]

Slide #17

* 2 2-D BENDS 4-D CONFIGURATION SPACE

* 2 GOOD QUANTUM NUMBERS: 4-2 = 2-D ACCESSIBLE CONFIGURATION SPACE* 2-D CONFIGURATION SPACE 4-D PHASE SPACE

4–1(ENERGY) = 3-D* TRAJECTORIES IN 3-D PHASE SPACE

(Qi,Pi) (Ji, i) ACTION, ANGLE

* SURFACE OF SECTION

* ONE TRAJECTORY FAMILY OF POINTS ON s. of s.

* SAMPLE MANY TRAJECTORIES

HCCH PURE BEND PHASE SPACE

NBEND & TOTAL

Ja n4 – n5 Jb 4 – 5

CHARACTERISTIC PATTERNUNCORRELATED DOTS

QUASIPERIODICCHAOS

COLOR CODED INITIAL CONDITIONS

“STRUCTURE OF PHASE SPACE”

(Jb, b, Ja, a)

COLOR

Jb VS. b PLANE (2–D) AT a = 0, 0,da

dt

Slide #18

Onset of Classical Chaos

Slide #19

Classical Dynamics Near 15,000 cm–1

Slide #20

0 5 10 15 20

600

650

700

750

Quanta of Bend Excitation (vb)

Eff

ecti

ve F

requ

enci

es (

eff)

eff = E/vb

Effective Frequencies as Means of Identifying Normal/Local Transition

cisbend

transbend local

bend

counter-rotation

++

++

++ +

+

+

+

Slide #21

v4 (quanta of trans bend)

v5 (quanta of cis bend)

Inte

rnal

En

ergy

(cm

–1)

Bright State

(0,0,0,00,100)l4 = –l5

Zero-Order Energies Eigenenergies(and spectrum)

Analysis of IVR: (0,0,0,100,00) Bright State

Slide #22

Slide #23

Dynamics

Å

Å

Halonen, Child, and Carter surface, Mol. Phys. 47, (1982), p. 1097.

Energetics

Isomerization Coordinate

25,000

20,000

15,000

10,000

5,000

0

Inte

rnal

Ene

rgy

(cm

–1)

:C CH

HC C H Isomerization CoordinateH

Slide #24

0 5 10 15 20

1200

1300

1400

1500

Nb

E(N

b+1)

– E

(Nb–

1)[c

m–1

]Effective Anharmonicities as Means of Identifying Normal/Local Transition

cisbend

transbend local

bend

counter-rotation

++

++

++ +

+

+

+