Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P...
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Transcript of Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P...
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