Transcript Columbus_talk_FC09.ppt

The Dance of Molecules: New Dynamical Perspectives on
Highly Excited Molecular Vibrations
Michael Kellman
University of Oregon
Columbus, Ohio
June 22, 2007
Acknowledgment:
Vivian Tyng
Shuangbo Yang
Aniruddha Chakraborty
DOE Basic Energy Sciences
Combustion Program
Review article: M.E. Kellman and V. Tyng,
Accounts of Chemical Research April 2007 vol. 40
The “Dance of Molecules”: Goal is fundamental
understanding of molecular vibrations.
At low energy, near-harmonic normal modes.
Current research: What happens at high energy
with anharmonicity, mode coupling, chaos?
Long range goal: How do high energy vibrations
go over to chemical reactions?
Research goal: Theoretical tools to extract dynamics
of molecules, from analysis of experimental spectra.
Focus of talk: Acteylene spectra and dynamics, including
acetylene-vinylidene isomerization.
Guiding principles:
Frequency-domain spectra contain dynamical information.
High energy: new modes born in bifurcations of the lowenergy normal modes.
C2H2 Stretch-Bend System
w1 : w2 : w3 : w4 : w5
1. C-H symmetric
stretch
2. C-C stretch
K11/33
= 3372:1975:3289:608:729
~ 5:3:5:1:1
3. C-H antisymmetric stretch
3 polyad numbers:
K3/145 K1/244
K1/255 K14/35 …
4. Trans bend
DD-I, l,
5. Cis bend
Vibrational angular
momenta
l4, l5
DD-II
Ntot = 5n1+3n2+5n3+n4+n5
Ns= n1+n2+n3
l = l4+l5
Experimental link:
Effective fitting Hamiltonian with
diagonal terms and resonance couplings:
Hfit = H0 + S Vi
We use the fit to the spectrum developed by R.W. Field
and coworkers.
Spectra with high bend excitation are found empirically
to have nearly complete decoupling of stretches from bends.
A problem of pure bending dynamics.
Hfit = H0 + S Vi
C2H2 zero-order Hamiltonian:
H0 = S (ni + ½) w0i
+ S cii (ni + ½)2
+ S cij (ni + ½)(nj + ½)
Zero-order states:
(0, 0, 0,n4l 4, n5l 5)
Hfit = H0 + S Vi
Acetylene bend couplings:
generalized Darling-Dennison
l-resonance:
a4+a+4-a+5+a5-+ a+4+a4-a5+a+5Darling-Dennison DD-I:
a+4+a+4-a5+a5-+ a4+a4-a+5+a+5Darling-Dennison DD-II:
a+4+a+4+a5+a5++ a+4-a+4-a5-a5+a4+a4+a+5+a+5+ + a4-a4-a+5-a+5-
Semiclassical correspondence:
Hfit  Hclassical
Classical Hamiltonian from quantum fitting Hamiltonian:
Heisenberg correspondence between raising and lowering operators
and action-angle variables
a+
(n + ½)1/2 eif
a
(n + ½)1/2 e-if
Bifurcation analysis of classical Hamiltonian:
a normal mode becomes unstable with onset of chaos;
new modes born in branching or bifurcation of normal mode
Acetylene / vinylidene (HCCH / H2CC:) extremely important
in combustion
Acetylene an important intermediate in hydrocarbon combustion
Vinylidene isomer of acetylene is the precursor of the radical pool
formed by reaction with O2:
AV
V + O2    many radicals
I. Anharmonic molecular vibrations of acetylene C2H2
acetylene and its normal modes
new modes born in bifurcations:
l = 0 planar bends
II. Bifurcations and “moment of inertia backbending”
l > 0 bifurcation analysis
moment of inertia backbending: spectral signature
of bifurcation sequence "phase transition“
minimum energy surface “phase diagram”
III. Full rotation-vibration dynamics
CO2 normal modes, Fermi resonance, Coriolis resonance
IV. Future: Relation to reaction dynamics
I. Anharmonic molecular vibrations of acetylene C2H2
II.
Bifurcations and “moment of inertia backbending”
l > 0 bifurcation analysis
moment of inertia backbending: spectral signature
of bifurcation sequence "phase transition“
minimum energy surface “phase diagram”
III. Full rotation-vibration dynamics
C2H2 Orthogonal Mode, l > 0
Systems with intricate motions
including angular momentum,
have interesting possibilities!
e.g. the acetylene Orthogonal Mode?
“Moment of Inertia Backbending”
Realizations
Macroscopic objects (e.g. the human body)
Rotating nuclei phase transition
?
Pulsars (quark-nuclear matter phase transition)
?
Bifurcating Molecular Vibrations?
“Backbending and Upbending”
Backbending
(weak coupling)
Uncoupled
Upbending
(strong coupling)
Bifurcation sequence with moment of inertia upbending
l > 0 bifurcation analysis:
bifurcation sequence
Local Mode  Off-Great Circle  Orthogonal Mode
Like a “phase transition” in the “shape” of the vibrating
molecule.
Displays “moment of inertia upbending”
Minimum energy modes in C2H2
as a function of J
Nb = 32, variable J = l
22200
E (cm-1)
I = (J2)/2 E(J)
0.16
Bifurcation
22000
0.14
21800
Orthogonal
family
21600
Off-Great Circle
family
0.12
21400
Bifurcation
0.10
21200
Local family
0.08
21000
w2  E(J)/ J]2
J
0.06
20800
0
5
10
15
20
0
5e+3
1e+4
2e+4
2e+4
3e+4
3e+4
J = 1, Local Mode
I
J = 2, Local Mode
I
J = 3, Local Mode
I
J = 4, Local Mode
I
J = 5, Local Mode
I
J = 6, Local Mode
I
J = 7, Local Mode
I
J = 8, Local Mode
I
J = 9, Off-Great Circle Mode
I
J = 10, Off-Great Circle Mode
I
J = 11, Off-Great Circle Mode
I
J = 12, Off-Great Circle Mode
I
J = 13, Orthogonal Mode
I
J = 14, Orthogonal Mode
I
J = 15, Orthogonal Mode
I
J = 16, Orthogonal Mode
I
“Phase Diagram” of the Minimum Energy Surface
The L, OGC, and O modes are the lowest energy members
of their polyad of states [Nb, l].
Together they form the minimum energy surface (MES) in
the polyad space. (The l = 0 members form the
minimum energy “reaction path” to isomerization.)
We can make a “phase diagram” of the members of the
MES.
“Phase diagram of the minimum energy states”
of [Nb, l] has a tetracritical point!
Relation to Landau theory:
The normal modes and new modes born in bifurcations are
critical points of the reduced polyad Hamiltonian (polyad
number and conjugate angle projected out).
This is formally analogous to the Landau theory of phase
transitions, where phases and their transitions (bifurcations) are
described as critical points of a thermodynamic energy
function.
I. Anharmonic molecular vibrations of acetylene C2H2
II. Bifurcations and “moment of inertia backbending”
III. Full rotation-vibration dynamics, full stretch-bend
dynamics
Analytical Critical Points Analysis: Key to Higher Dimensions
Can we extend bifurcation analysis to more degrees of freedom? (rotation-
vibration spectra; full stretch-bend degrees of freedom). Difficulty is higherdimensional phase space. The polyad Hamiltonian is key to solution.
Normal modes and new modes born in bifurcations are critical points of
reduced polyad Hamiltonian (polyad number and conjugate angle projected
out) – solutions of analytical polynomial/trigonometric equations – no need
to integrate Hamilton’s equations to search for bifurcations – effort is
scalable with number of degrees of freedom.
I. Anharmonic molecular vibrations of acetylene C2H2
II. Bifurcations and “moment of inertia backbending”
III. Full rotation-vibration dynamics
IV. Future: Relation to reaction dynamics
IV. Future: Relation to reaction dynamics:
How do our “new modes” go over to chemical reactions?
e.g. H + O2  HO2  HO + O
“the most important reaction in combustion”
New work on fundamental transition state theory: Wiggins, Uzer, Wiesenfeld,
Jaffe (generalizes Pechukas theory of periodic orbit dividing surfaces
(PODS) for 2 degrees of freedom).
A rigorous phase space formulation of the transition state, reaction coordinate.
Future questions for us: What are the high energy HO2 modes?
How do they relate to reaction modes, reaction coordinates,
transition states?
Conclusions:
In high energy acetylene, new vibrational modes are born in bifurcations of the lowenergy normal modes.
Information about these is obtained from analysis of experimental spectra using
semiclassical correspondence and critical points analysis of the effective Hamiltonian.
With l > 0, moment of inertia backbending is spectral signature of a bifurcation
sequence. Analogy to Landau theory of phase transitions.
The analytical critical points method can be used for systems with higher
phase space dimension. We have carried this out for full rotation-vibration dynamics
of CO2.
Next challenge: connection to reaction theory.
Review article: M.E. Kellman and V. Tyng, Accounts of Chemical Research
April 2007 vol. 40
Conclusions:
Molecules at have interesting transitions in which new vibrational modes are born in
bifurcations of the low-energy near-harmonic normal modes.
Information about these transitions can be obtained from analysis of experimental
spectra using semiclassical correspondence and critical points analysis of the effective
Hamiltonian.
The critical points bifurcation analysis has a close analogy with the Landau theory of
phase transitions.
Rotating systems (nuclei, stars) with “shape” transitions show interesting moment of
inertia backbending effects.
Analogous effects are predicted for molecules in which the “shape” transition is effected
by a bifurcation sequence.
Critical Points Bifurcation Analysis:
“Principal” periodic orbits (normal modes and new modes born in
bifurcations) are fixed points of Hamiltonian flow i.e. critical points
of reduced polyad Hamiltonian (polyad number and conjugate angle
projected out) – solutions of analytical polynomial/trigonometric equations –
no need to integrate Hamilton’s equations to search for bifurcations – effort
is scalable with number of degrees of freedom.
So far: acetylene bends is effectively only
2 degrees of freedom!
With energy conservation, n = 2 degrees of freedom has (2n – 1) =
3 phase space dimensions.
How many phase space dimensions in acetylene bends?
Acetylene bends: n = 4 bend modes (doubly-degenerate cis + trans)
So: (2n = 8 phase space dimensions) – (2 polyad numbers Nb, l) (2 trivial conjugate angles); - (energy)
Thus: ( (2 x 4) – 2 – 2 – 1) = 3 phase space dimensions – effectively
only 2 degrees of freedom!
CO2 full rotation-vibration dynamics:
effectively 3 degrees of freedom
2 x (6 vibrations + rotations) – (conserved polyad numbers,
angular momenta, trivial angles, energy)
(2 x 6 – 7) = 5 = (2 x 3 – 1) = 5 phase space dimensions
or effectively 3 degrees of freedom
This is the
BIG STEP!
Application of analytical critical points method
to higher degrees of freedom:
pure bends with l > 0.
l = 2 orthogonal
mode
So far, bifurcation analysis is for vibrations only.
We now can do bifurcation analysis for full rotation-vibration
effective Hamiltonian with rotation-vibration coupling.
Show results for J = 0, 1
Jacobson pure bend fit, J = 0
Ro-vibrational critical points for C2H2 pure bending, J=1
Method is analytic, due to polyad Hamiltonian –
efficient, scalable to more degrees of freedom.
(underappreciated point known to mathematicians,
used by Kellman et al., Ezra et al.)
Bifurcation and phase space analysis for full
stretch-bend degrees of freedom.
Acetylene-vinylidene isomerization dynamics
3
Energy (eV)
2
1
0
-1
Reaction Coordinate
Coulomb explosion: A / V cycles on reaction coordinate
(local bend) for ~ 1 m-sec. (Acetylene does not relax).
How to account for this? We need new tools.
Status of reaction mode:
l = 0 Local bend is natural
candidate for the reaction mode.
l = 0 Local bend is a stable mode.
At higher l, no longer 1-dim reaction mode! Also, Local
bend becomes unstable.
Stability may have profound effect on IVR
l = 2 orthogonal
mode
Components for Acetylene/Vinylidene dynamics (1):
Time-dependent dynamics: survival probability P(t) from
stretch-bend fitting Hamiltonian for: (n = 22 local bend);
(n = 22 local bend + 1 antisymmetric stretch)
22 local bend
22 local bend + 1 antisymmetric stretch
P(t)
1
0
0
1
2
3
Time (ps)
4
5
Conclusion: phase space structure of local bend/stretch
is a bottleneck to energy relaxation of A / V isomerization.
Survival probability of l overtone states
Components for Acetylene/Vinylidene dynamics (2):
Need to break polyad number in fitting Hamiltonian:
P = 5 n1 + 3 n2 + 5 n3 + n4 + n5
approximate constant of motion of spectroscopic
Hamiltonian. Another bottleneck to energy flow,
but not exact. Need to model “leakage” in fitting
Hamiltonian.
Components for Acetylene/Vinylidene dynamics (3):
Spectral patterns of multiple wells, above barrier
motion (Acetylene, Vinylidene, Above Barrier)
ns=1
900
-1
E(i+1)-E(i) [cm ]
750
600
450
300
150
0
3000
8500
14000
19500
25000
E(i) [cm-1]
S. Yang, V. Tyng, and M.E. Kellman, J. Phys. Chem. A 107, 8345 (2003)
Components for Acetylene/Vinylidene dynamics (3):
Spectral patterns of multiple wells, above barrier
motion (Acetylene, Vinylidene, Above Barrier)
ns=1
900
-1
E(i+1)-E(i) [cm ]
* Fermi resonance
* “cross-barrier
**
750
600
450
resonance”
300
150
0
3000
8500
14000
E(i) [cm-1]
19500
25000
Summary:
New modes born in bifurcations
Analytical bifurcation analysis scalable to
many degrees of freedom
Reaction mode mode in bifurcation – intuitively like
reaction coordinate
Future directions: spectroscopic Hamiltonian for full
isomerization dynamics
Polyad-breaking terms
Fitting Hamiltonian for multiple wells, above barrier
The End