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Observing Quantum Monodromy
An Energy-Momentum Map Built From ExperimentallyDetermined Level Energies Obtained from the ν7 Far-Infrared
Band System of NCNCS
Dennis W. Tokaryk, Stephen C. Ross
Department of Physics and Centre for Laser, Atomic, and Molecular Sciences
University of New Brunswick, Fredericton, NB Canada
Brenda P. Winnewisser, Manfred Winnewisser, Frank C. De Lucia
Department of Physics, The Ohio State University, Columbus, OH USA
Brant E. Billinghurst
Canadian Light Source, Inc., Saskatoon, SK Canada
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 1
Monodromy – you’ve seen it before!
A simple example from complex analysis: let z = x + iy = r eiθ.
Then the function,
ln(z) = ln(r) + iθ,
has a singularity at z = 0 + i0 = 0 eiθ!
(because ln(0) is undefined.)
Im(z)
z
y
r
θ
Re(z)
x
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 2
Monodromy – you’ve seen it before!
Make a circuit around the singularity, evaluating ln(z) en route:
At the start of the circuit θ = 0, so ln(z) = ln(r0) + i0 = ln(r0).
Im(z)
ln(z) = ln(r0)
z
θ
Re(z)
r0
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 3
Monodromy – you’ve seen it before!
At the end of the circuit, θ = 2πi, so ln(z) = ln(r0) + 2πi.
The value of ln(z) is not single-valued:
- the value depends on: how we got to z (i.e. upon its history).
Im(z)
θ
r0
Re(z)
z
ln(z)
= ln(r0) + 2πi
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 4
Chemistry
Molecules
(synthesis)
Physics/
Technology
Spectroscopic
data
Theoretical
spectroscopy
GSRB Hamiltonian
Mathematics
Topology of
the phase-space
surfaces of
constant E, J
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
Quantum
monodromy
as seen in
quasi-linear
molecules
June 20, 2016
Slide 5
Year
Timeline of LAM theoretical and experimental spectroscopy
1970
The Vibration-Rotation Problem in Triatomic Molecules Allowing for a Large-Amplitude Bending Vibration,
HBJ = Hougen, Bunker and Johns, J. Mol. Spectrosc. 34, 136 (1970)
The effective rotation-bending Hamiltonian of a triatomic molecule, and its application to extreme
centrifugal distortion in the water molecule, Hoy and Bunker, J. Mol. Spectrosc. 52, 439 (1974)
Semi-rigid bender (SRB) Hamiltonian by Bunker and Landsberg, J. Mol. Spectrosc. 67, 374 (1977)
1980
A reinterpretation of the CH2 photoelectron spectrum, Sears and Bunker, J. Chem. Phys. 79, 5265 (1983)
Analysis of the laser photoelectron spectrum of CH2, Bunker and Sears, J. Chem. Physics 83, 4866 (1985): We
NOW know that an energy-momentum map of their calculated rotation-bending energies in their Table V
shows quantum monodromy!
1990
Morse Oscillator Rigid Bender Internal Dynamics [MORBID], Jensen, J. Mol. Spectrosc. 128, 478 (1988).
OCCCS, NCNCS, NCNCO, and NCNNN as Semirigid Benders, [Hamiltonian now called: Generalized SemiRigid Bender (GSRB) ], Ross, J. Mol. Spectrosc. 132, 48 (1988)
Experimental confirmation of quantum monodromy: The millimeter wave spectrum of cyanogen
isothiocyanate NCNCS, B. and M. Winnewisser, Medvedev, Behnke, DeLucia, Ross and Koput, PRL, 243002
2000
(2005)
The hidden kernel of molecular quasi-linearity: Quantum monodromy, M. and B. Winnewisser, Medvedev,
DeLucia, Ross, Bates, J. Mol. Structure 798, 1-26 (2006)
2010
Analysis of the FASSST rotational spectrum of NCNCS in view of quantum monodromy, B. and M.
Winnewisser, Medvedev, DeLucia, Ross and Koput, PCCP, 12, 8158 ( 2010)
Pursuit of quantum monodromy in the far-infrared and mid-infrared spectra of NCNCS using synchrotron
radiation, M. and B. Winnewisser, DeLucia, Tokaryk, Ross and Billinghurst, PCCP, 16, 17373 (2014)
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 6
Year
Timeline of Mathematics leading to Monodromy in Molecules
1970
Geometrical obstructions to global action-angle variables, Nekhoroshev, Trans. Moskow Math. Soc. 26, 180
(1972)
On global action-angle coordinates, Duistermaat, Comm. Pure and Appl. Math. 33, 687 (1980)
1980
The quantum mechanical spherical pendulum, Cushman and Duistermaat, Bull. Am. Math. Soc. 19, 475
(1988)
Monodromy in the champagne bottle, Bates, J. Appl. Math. and Physics 42, 837 (1991)
Classical energy-momentum map
1990
Quantum states in a champagne bottle, Child, J. Phys. A: Math. Gen. 31, 657 (1998)
Energy-momentum map of discrete quantum levels
2000
Quantum monodromy in the spectrum of HOH and other systems: new insight into the level structure
of quasi-linear molecules, Child, Weston and Tennyson, Mol. Phys. 96, 371 (1999)
Monodromy in the water molecule, Zobov, Shirin, Polyansky, Tennyson, Coheur, Bernath, Chemical
physics letters, 414, 193 (2005): Comparison of high temperature HOH data with predictions
2010
Hamiltonian monodromy as lattice defect, Zhilinskii, Topology in condensed matter, Springer
series: Vol. 150, 186 (2006)
Quantum monodromy and molecular spectroscopy, Child, Contemporary Physics 55, 212 (2014)
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 7
Classical motion in the champagne-bottle potential
The classical motion was studied in 1991 by Larry Bates.
(Bates, J. Ap. Math. Physics 42 (1991) 837)
He made an Energy-Momentum map and identified the ‘critical points’ at
which the radial momentum is zero. These form the parabolic curve + the
critical point (k = 0, e = the energy of the top of the potential energy hump)
shown here:
e: total
energy
Critical points
k: angular
momentum
Energy-Momentum map
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 8
Classical motion in the champagne-bottle potential
The classical action variable for radial motion is,
Ir ~ the frequency of a particle’s radial oscillation
(for the given energy e and angular momentum k).
• Evaluate Ir at a starting point
on the loop shown on the
e: total
energy
energy-momentum map.
• Move a small distance around
Ir
the loop, re-evaluate Ir.
Ir
• Continue all the way around
Ir
k: angular the loop,
Ir
momentum
- calculate Ir at each point.
• Result: Ir has a different value
when you return to the starting
point!
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 9
Classical motion in the champagne-bottle potential
Our classical action
experiences a branch cut as shown!
So, the vibrational periodicity Ir represents will not smoothly
change when crossing over this line.
Branch cut
e: total
energy
k: angular
momentum
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 10
Larry Bates
GSRB:
(University of Calgary)
Monodromy in the
champagne bottle
J. Appl. Math. and
Physics 42, 837 (1991)
Quantum Hamiltonian
1µ J 2
1µ J2
+
2 ρρ ρ
2 zz z
H=
+ V(ρ)
+ Many more terms
Classical Hamiltonian
=
Tvib
+ Tz-rot +
+ Many more terms
pR2
j2
H=
+
+ V(R)
2m 2mR2
= Tvib + Tz-rot +
 GSRB contains the essential
core which leads to…
• GSRB already accounted for Monodromy before we even knew it existed!
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 11
Quantum motion in the champagne-bottle potential
Mark Child quantized the classical problem, and made an energy-momentum
map of discrete quantum levels.
(Child, J. Phys. A: Math. Gen. 31 (1998) 657; Contemporary Physics 55 (2014) 212)
quantum level of energy E and
line of constant radial vibration
angular momentum k.
v=1
v=0
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 12
Spectra of the ν7 low-frequency high-amplitude bending mode and of the
of the ν3 hybrid band stretching mode, taken at the CLS.
~68 a-type sub-bands
of the ν7 mode
(For experimental
details, see
Winnewisser et al,
PCCP Perspectives
16, 17373 (2014 ) )
a-type sub-bands
b-type sub-bands
b-type subbands
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 13
The energy-momentum map
How we connected Ka stacks and
different vibrational levels
with data from the CLS
Up to
Ka = 10
Ka = 10
Ka = 12
Ka = 12
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 14
The NCNCS energy-momentum map showing the quantum lattice for the
ν7 in-plane large amplitude bending mode
Small Dots: show J = Ka levels calculated from the analysis
of the pure rotational spectrum only
Centers of the Red Circles: show levels measured
directly from the assignment of the FIR spectrum
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 15
Analysis of the laser photoelectron spectrum of CH2,
Bunker and Sears,
J. Chem. Phys. 83 4866(1985)
We NOW know that an
energy- momentum map of
the calculated rotationbending energies in their
Table V shows quantum
monodromy
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 16
Monodromy plot for water
Child, Contemporary Physics, 55 212 (2014)
with data from Zobov et. al., 414 193 (2005).
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 17
The take-home part…
• We determined precise relative energies for the lowest seven
vibrational levels of the ν7 mode of NCNCS:
for Ka up to (a maximum of) 12.
• Our energy-momentum map contains all of the structure due
to quantum monodromy expected for a quasi-linear
molecule.
• Stephen Ross’ GSRB Hamiltonian proved excellent at
predicting the positions of these energy levels, even though
only pure rotational spectra were included in the fitting.
• If you are working on a new quasi-linear molecule, an
energy-momentum map will be a helpful aid to determining
the height of the barrier to linearity, since the general
structure of the map will be the same for every such
molecule.
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 18
Thanks
• Natural Sciences and Engineering Research Council of
Canada (NSERC)
– Discovery grants for Drs. Ross and Tokaryk
• Damien Forthomme and Colin Sonnichsen for help with the
experiments
• Staff at the CLS for technical support and accommodation
during the experiments.
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 19
Classical motion in the champagne-bottle potential
Mark Child explains why not.
(Child, J. Phys. A: Math. Gen. 31 (1998) 657; Contemporary Physics 55 (2014) 212)
.. but now, for the same initial
conditions, Δθ ~ +π, and direction of
rotation reverses!
For small values of k > 0, between
successive radial maxima, Δθ > 0,
but very small.
won’t depend on k (same radial
vibrational frequency for all small k,
positive or negative.)
Child shows that the derivative of
the action wrt k is negative if k>0,
and positive if k<0…
discontinuous at k = 0!
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 20
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 21
Back to the Microwave spectrum of the v7 mode of NCNCS
e
υb = 6 1 0
Bending hot bands
J = 12  11 υ = 4 0
e
υb = 5 1 0
1f 2
b
υb = 3
1f 2
1e
0
e
υb = 2 0 1
e
1
υ =1
2 1f
1e
1f 2
3
2 1f 3
0
5
6
6
7
= Ka
5 = Ka
5
5
5
3
4
4
4
4
4
3
3
3
1f
2
6
6
6
7
7
7
8
= Ka
9 = Ka
8
9
10
b
e
υb = 0 1
0
K=1 K=0
2
31f
4
5
8
K=1
MF09 – 71st International Symposium on Molecular Spectroscopy
9
= Ka
10
11
= Ka
= Ka
Why it took
25-30 years
to get
beyond v=3!
University of Illinois
June 20, 2016
Slide 22
Quantum motion in the champagne-bottle potential
What did our game mean??
That the quantum numbers v and k,
which we took as valid below the
monodromy point, were not
universal.
New quantum numbers of vibration
and angular momentum are
necessarily required above the
monodromy point.
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 23
Quantum motion in the champagne-bottle potential
Rules of the game:
1. Define a closed path γ through
a set of quantum levels circling
the monodromy point.
2.
From each point, increase k by
one unit, then increase v by
one unit. Make a closed
trapezium.
3.
The base of the previous
trapezium should not make a
drastic change of orientation
from the previous one.
Check the return point – once
again, your horse turns into a cow!
MF09 – 71st International Symposium on Molecular Spectroscopy
University of Illinois
June 20, 2016
Slide 24