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
++
++
++ +
+
+
+
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