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Spectroscopic signatures of a saddle point. Modelled on HCP as a perturbed spherical pendulum. Spherical pendulum. P. C. θ. H. Outline. Model Hamiltonian Properties of spherical pendulum states Classical trajectories of the coupled model Anharmonic resonances Polyad structure - PowerPoint PPT Presentation
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Spectroscopic signatures of a saddle point
Modelled on HCP as a perturbed spherical pendulum
C
P
H
Spherical pendulum
θ
Outline• Model Hamiltonian
Properties of spherical pendulum statesClassical trajectories of the coupled modelAnharmonic resonances
Polyad structure • Rotation/vibrational dynamics of HCP bending
statesExtended RKR potential functionAnomalous magnitudes of vibn/rotn parameters
• Summary
Model Hamiltonian
2
int
2 2
2 20
2int
0 s
CP stretch2
( / ) sin HCP bend/rotation2
sin 1:n anharmonic resonanceParameters modelled on HCP 100 9 0.25
s b
s s s
b rot
s
rot
H H H H
H p q
H B J V
H V q
V B
bend 0
3Equilibrium bending frequency
5rot
V
B V
Quantum pendulum states
const | |j v k
Diagonalize in a spherical harmonic basis
E/V0
2.0
1.0
0.0
-1.0
const2 | |v v k
k
Semiclassical pendulum statesComplete analytical solution in terms of Elliptic integrals, which yields the following limiting formulae for k=0
0
02
ln 16 /(1 )
as 1
2(1 ) ln 16 /(1 )
rot
k
rot
v
B VEv E
VBEk
Surfaces of section and periodic orbits
Periodic orbit bifurcations
Periodic orbit frequencies
Polyad structure E<B
Mean polyad number np=2vs+vb
Inside Fermi res
Outside
Measuredfrom
lowestlevel ofpolyad
Polyad structure 0<E<2B
Rotatingstates
Vibratingstates
Importance of resonance terms
ΔE
E np
HCP extended RKR bending potential
HCP bendmonodromy
plot
l doubling
2 2 2
2 2
( , 0) ( )[ ( 1) ]
[ ( 1) ]bE E n k gk B k J J k
D J J k
Vibration rotation constants
Summary
• Classical and semiclassical methods used to illuminate dynamics of HCP-like model
• Classical bending frequency function and Heisenberg matrix elements used to model occurrence and strength of 1:n resonances
• RKR plus ab initio information used to determine realistic HCP bending potential
• Anomalously large vibn/rotn interaction parameters explained and predicted
Acknowledgements• M P Jacobson (UCSF)• C D Cooper (Oxford)• UK EPSRC
References1. M P Jacobson and M S Child JCP 114, 250 (2001)
2. M P Jacobson and M S Child JCP 114, 262 (2001)
3. M P Jacobson and M S Child JPC 105, 2834 (2001)
4. M S Child, M P Jacobson and C D Cooper JPC 105, 10791 (2001)