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)