Spectroscopic signatures of bond-

breaking internal rotation in HCP.

Mark S Child and Matt P JacobsonOxford University UK

UK EPSRC

Outline

• Classical origin and validity of the polyad approximation for a Hamiltonian with the

angular periodicity of HCP.

• Vibrational/rotational level stucture in lowest (pure bending) polyad components.

Spherical pendulum Hamiltonian

Quantum states, semiclassical theory and

validity of the polyad approximation

Part 1

The model

HC

P

θ

ˆ ˆ ˆSep FermH H H Expressed in

trigonometric form

Hamiltonian components2 2

0

2 2s

2

ˆ ˆ sin2

ˆ ˆ +2

ˆ sin

Sep

s s

Ferm s

H BJ V

p q

H V q

Spherical pendulum (Bend)

CP stretch

Scaled parameter values0

0

100, 9, 0.25, 3

5 / 2

s

bende s

V B V

BV

Fermi resonance coupling

Energy unit ~ 147 cm-1

Spherical harmonic expansion , , , ,b bj

v k j k j k v k

Spherical pendulum states

Periodic orbits

Bifurcation diagramshowing onset and

frequencies of periodic orbits

Classical Fourier components closely related to ‘spherical pendulum’ matrix elements

Pendulum frequency variation

Semiclassical considerationsPoints are

Quantum level separations

Polyad attributes

Fraction of eigenstatenot attributable to a single

2:1 polyad

Reduced bending energy

/ 2s pE N

Vertical columns indicate ‘good polyads’

vs 2p s bN v v

Improved bending potential

and rotation-vibration coupling

parameters

Part 2

Extended RKR bending potential with bending frequency plot

Bending energy vs vibrational

angular momentum

HCP ‘monodromy’ plot

Coriolis splittings at nb =10 and nb

=40

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

Conclusions

• Trigonometric (spherical potential) Hamiltonian form imposes necessary periodicity.

• Eventually invalidates any harmonic oscillator based representation.

• 2:1 Fermi polyad model valid almost up to saddle point, provided matrix elements take account of angular periodicity.

• RKR based bending potential predicts large energy variation of vib-rotn parameters – in line with experimental observations.

RKR procedure2

2

2 2

0 02 2

0

( )

1/ 2

( ) ( ) ( )

( ) ( ) ( )2 2

( ) adiabatic stretching eigen function

RKR equation

( ) ( )

( ) bending level for

t

t

bend eff

r CP R GH

v U

stretch

dH f V

d

fr R

d dv

f U E v

E v v

0

Quantum monodromy plot

Note shape change as unit cell is transported around O

Accuracy of polyad approximation

Notes

• Barrier max at E=100 units

• 1 unit ~ 140 cm-1