The Rotational Energy Levels Diatomic Molecule

1
The hidden Kernel of Molecular Quasilinearity
Quantum Monodromy or Molecules in aChampagne
Bottle Potential withDonuts in Phase Space
Manfred Winnewisser and Brenda P. Winnewisser,
Ivan R. Medvedev, and Frank C. De
Lucia Department of Physics, The Ohio State
University, Columbus, OH Stephen C.
Ross Department of Physics, University of New
Brunswick, Fredericton, NB Larry M.
Bates Department of Mathematics and Statistics,
University of Calgary, Alberta
2

• Larry M. Bates Monodromy in the champagne
  bottle
• J. Appl. Math. and
  Phys. 42, 837-847 (1991)
• ________________________
  ___
• Classical mechanics and topology
• a) The classical Hamiltonian system of a
  two-dimensional
• harmonic oscillator is completely
  integrable because
• energy and angular momentum are
  conserved.
• b) For a two-dimensional anharmonic
  oscillator with a
• champagne bottle potential function,
  however, there can be
• no global action-angle variables because
  the energy surface
• below the top of the barrier belongs to
  the topological type
• S2S1 and above the top of the barrier
  to type S3.
• c) The top of the barrier is the critical or
  monodromy point.

3
Surface Topology
S3

S2 x S1
Champagne bottle potential is rotationally
invariant with r2 x2 y2
Phase space h, j map is projected onto h, j plane

• Region 1 is below the monodromy point while
  region 2 is above.
• denotes a loop enclosing the critical point.
  Since the monodromy
• matrix connecting the actions is non-unitary,
  with the form

Each pair of values of angular momentum j and
energy h defines a separate torus in phase space
(x, y, px, py)
we learn that the bundle of tori is twisted.
4
Champagne bottle potential surface
Surface topology
h E2
Annulus of torus
S3
Direction of precession
Critical or monodromy point
Trajectory of particle with angular momentum j
1
?
Outer boundary of annulus corresponds to outer
concave potential wall
Annulus of torus projected onto configuration spa
ce
S2 x S1
h E1
Inner boundary of annulus corresponds to inner
convex potential wall M. S. Child, J. Phys A
Math. Gen. (1998) 657 - 670
5
M Monodromy matrix

• Location of
• classical monodromy
• point

Partial image of the quantum lattice
energy-momentum map for the molecule NCNCS. The
two column vectors (?Ka, ?vb) (0, 1) and (?Ka,
?vb) (1, 0), indicated at unit cell a, are
transported in parallel fashion counterclockwise
(red circle) around the critical point.
6
Correlation between the bending-rotation energy
levels of linear and bent triatomic molecules,
drawn from model calculations.
7
(No Transcript)
8
(No Transcript)
9
(No Transcript)
10
(No Transcript)
11
(No Transcript)
12
(No Transcript)
13
(No Transcript)
14
(No Transcript)
15
Three dimensional image of the quantum lattice
for the end-over-end rotational energy
contribution
NCNCS
HCCNCO
16
Three dimensional image of the quantum lattice
for the end-over-end rotational energy
contribution
NCNCO
OCCCS
17
Three dimensional image of the quantum lattice
for the end-over-end energy contribution
NCNCO
18
Three dimensional image of the quantum lattice
for the end-over-end energy contribution
NCNCO