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![Page 1: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/1.jpg)
Spectroscopic signatures of bond-
breaking internal rotation in HCP.
Mark S Child and Matt P JacobsonOxford University UK
UK EPSRC
![Page 2: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/2.jpg)
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.
![Page 3: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/3.jpg)
Spherical pendulum Hamiltonian
Quantum states, semiclassical theory and
validity of the polyad approximation
Part 1
![Page 4: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/4.jpg)
The model
HC
P
θ
ˆ ˆ ˆSep FermH H H Expressed in
trigonometric form
![Page 5: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/5.jpg)
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
![Page 6: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/6.jpg)
Spherical harmonic expansion , , , ,b bj
v k j k j k v k
Spherical pendulum states
![Page 7: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/7.jpg)
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
![Page 8: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/8.jpg)
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
![Page 9: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/9.jpg)
Improved bending potential
and rotation-vibration coupling
parameters
Part 2
![Page 10: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/10.jpg)
Extended RKR bending potential with bending frequency plot
![Page 11: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/11.jpg)
Bending energy vs vibrational
angular momentum
HCP ‘monodromy’ plot
![Page 12: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/12.jpg)
Coriolis splittings at nb =10 and nb
=40
![Page 13: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/13.jpg)
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
![Page 14: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/14.jpg)
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.
![Page 15: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/15.jpg)
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
![Page 16: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/16.jpg)
Quantum monodromy plot
Note shape change as unit cell is transported around O
![Page 17: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/17.jpg)
Accuracy of polyad approximation
Notes
• Barrier max at E=100 units
• 1 unit ~ 140 cm-1
![Page 18: Spectroscopic signatures of bond- breaking internal rotation in HCP. Mark S Child and Matt P Jacobson Oxford University UK UK EPSRC.](https://reader030.fdocuments.in/reader030/viewer/2022032722/56649f425503460f94c6198d/html5/thumbnails/18.jpg)