1. More Spectra! A Lot More! Better Too! Now What?
Robert W. Field Department of Chemistry Massachusetts Institute of Technology 72nd International Symposium on Molecular Spectroscopy Talk TB01 June 20, 2017

2. More Spectra!
50 years ago, as a graduate student in the Klemperer Group, I worked more than 6 years in a failed attempt to observe a single line in a VUV-rf double resonance spectrum of a Λ-doublet transition in the CO A1Π state Pre-Laser days…

3. A Lot More!
As a postdoc in the Broida/Harris Group at UCSB, it took only two weeks for me to observe two Microwave Optical Double Resonance rotational transitions in BaO using an Ar+ laser. Then Don Jennings built a cw dye laser for me and the game was ON! Lots of lines. Maybe 20 per day. Spectral Velocity = [# resolution elements/second] = ~1000/10000=0.1 WOW! 100 cm-1 at 0.1 cm-1 resolution in 3 hours

4. Better Too!
In my talk MA02 “Welcome to Rydberg Land” at the 69th ISMS, I claimed a spectral velocity increase of 106 beyond what we had achieved 10 years ago in our two-laser double resonance sequential scans of Rydberg spectra of CaF. 3 years ago we achieved this factor of 106 increase by combining broad-bandwidth multiplexed Chirped Pulse mm-Wave Spectroscopy with a Buffer Gas Cooled Ablation Source. Better resolution (100 kHz), better relative intensities (±5%), upward vs. downward transitions distinguished by the phase of the Free Induction Decay signal, this ridiculous spectral velocity offers possibilities for multi- dimensional spectroscopies…

5. Now What? Previously unimaginable spectroscopic targets
Transition states
Permutation splittings [e.g. HaCbCcHd—HdCbCcHa]
Emergence of Large Amplitude Motion States
Local benders
Motion along an isomerization path
New classes of spectroscopic models and basis sets
Polyads
Broken polyads: trans-cis Isomerization on S1 HCCH
Partial pre-diagonalization: C1B2 State of SO2
Pull back the curtain on “ergodicity”

6. Introduction Cannot lead us far from the equilibrium region
50 years as a small-molecule spectroscopist
Astonishing improvements in technology, techniques, and theory
But we are still asking the many of the same old questions
It starts with assignments of eigenstates
“Assignment” has been based on energy level pattern or node- count
Cannot lead us far from the equilibrium region
Wrong questions, wrong models
Numerical description (“what?”) vs. Mechanism (“how, why, and when?”)
Experimentalists and theorists should BOTH use robust Heff fit models: e.g. the vibrational polyad model
Quantum number scaling of matrix elements and membership
Multi-Component Eigenvectors!
What is this molecule going to do when it grows up?

7. Three Ideas Polyads Transition States are where polyads break!
Robust
Predict and depict emergence of something special: Large Amplitude Motions
Transition States are where polyads break!
Trans-cis isomerization in the S1 state of acetylene
[Science 350, 1338 (2015)]
Vibronic Coupling: if it looks wrong, it is wrong
Chemical Intuition is based on the Diabatic Representation
Offenses to Chemical Intuition come from the Adiabatic Representation
Unequal SO bond lengths in C state of SO2
[JCP 144, , , and (2016)]
Talk MG09 Jun Jiang
Vinylidene to Acetylene Isomerization on S0: Talk TB08 Steve Gibson

8. Trans-Cis Isomerization in S1 HCCH
Josh Baraban, Bryan Changala, Georg Mellau, John Stanton, Anthony Merer, RWF

9. Polyad Heff: ROBUST Patterns
Accidental (on purpose?) near-degeneracies
Fermi (stretch-bend): 2ωa≈ωs [s=sym, a=antisym]
D-D (stretch-stretch and bend-bend): 2ωa≈2ωs
Usually between bond-sharing pairs of normal modes
Matrix element scaling and polyad membership
Eigenstate spectrum can reveal emergence of large amplitude motions along an isomerization path
Energy and structure of a transition state

10. Every Vibrational Level Up to the Energy of the HCCH S1
trans-cis Transition State is Observed, Assigned, and Fitted
ETS
State space
is dominated
by Bn polyads
Polyad Heff fit model generates a multi-component eigenvector for every eigenstate!

11. B2 Polyads in HCCH S1 [trans-bent]
Consist of (v4,v6) = (2,0), (1,1)*, and (0,2) vibrational levels
Mode 4 (torsion) and Mode 6 (cis-bend)
Add some quanta in trans-bend (Mode 3): Spectator?
3nB2
Polyad pattern should extrapolate from n to n+1
Surprise! Sometimes, it doesn’t!
As Physical Chemists we learn by breaking things
Polyads are a generalized form of pattern
Polyads break!
* Coriolis interaction

12. Excitation in v3 distorts (v4,v6) bending polyads
Steeves et. al., J. Mol. Spec., 256, 256, 2009.

13. Example of polyad breakdown The “reduced” matrix element K4466
v3 = 0 * *
v3 = 1
v3 = 2
* Expected behavior

14. Dip in frequency along the isomerization path, ωeff for mode 3 or 6, from Polyad Fits Spectator (ω4) vs. Isomerizing (ω3 & ω6) Modes
no dip
no dip
no dip
Excitation in BOTH modes 3 and 6 is
required to generate an isomerization dip.
Excitation in mode 4 is irrelevant.

15. Energy and Structure of Transition State
trans-bend cis-bend local bend

16. Summary: From Polyads to Transition State
Equilibrium structure: simple patterns
Dynamical secrets
Encoded in polyads
Polyads tell us what to look for and how to gain access to it
Polyads are more robust than normal modes
Fit parameters that define the first polyad predict all higher polyads, until a qualitative change in dynamics occurs!
Ah hah! What do Physical Chemists like to do?
Emergence of large amplitude local motions (LAM)
LAM states are unlike all other nearby vibrational states, thus they are surprisingly resistant to interaction
An isomerization path is a favorite habitat for a LAM
Isomerization dip: polyads break!

17. Why Does the C State of SO2 Have Unequal SO Bond Lengths?
Barratt Park, Jun Jiang, Catherine Saladrigas

18. The C 1B2 state of SO2 has a double-minimum potential in ν3
ν3 appears to be very low-frequency (~200 cm-1), as suggested by
Origin isotope shifts
Inertial defects
Coriolis Constants
Centrifugal Distortion Constants
(0,0,v3) progression appears to be staggered, as evidenced by Coriolis perturbations observed in bright (a1) states.
Many pattern-based assignments are incorrect, especially due to the lack of observation of any odd-v3 levels.
J. C. D. Brand, P. H. Chiu, A. R. Hoy, J. Mol. Spectr. 60, 43 (1976).
A. R. Hoy, J. C. D. Brand, Mol. Phys. 36, 1409 (1978).
K. Yamanouchi, M. Okunishi, Y. Endo, S. Tsuchiya, J. Mol. Struct. 352/353, 541 (1995).
K.-E. J. Hallin, Ph.D. Thesis, University of British Columbia, 1977.

19. Low-Lying Vibrational Levels of C-State SO2. Only a1 Levels: Regular?6
— a1 levels 2 2 4
(predicted)
— b2 levels 2 4
Tvib 2 2
v3 = 0
(v1, v2,v3) =
(0,0,v3)
(0,1,v3)
(0,2,v3)
(0,3,v3)
(1,0,v3)

20. Low-Lying Vibrational Levels of C̃ State SO2. Regular? Nope! Staggered6
— a1 levels 2 2 5 4
(predicted)
— b2 levels 2 1 4 3 1
Tvib 2 1 3 2 1 1
Odd v3 levels observed by
mmW-UV double resonance
v3 = 0
(v1, v2,v3) =
(0,0,v3)
(0,1,v3)
(0,2,v3)
(0,3,v3)
(1,0,v3)

21. Consequence of a double-well minimum along ν3
v3 = 0 1 2 3 4 5
ν3 staggering
Also
1:2 Fermi Resonance between ν1 and ν3
ν1≈2ν3
ν2≈ν3
Extensive c-axis Coriolis interaction between ν2 and ν3

22. Strong Coriolis interactions between ν2 and ν3
Horrible mess. Assignment would
have been impossible without an Heff
that included staggering.

23. Vibrational Assignments of the 3-D Wavefunctions Visual inspection fails!!! It must fail when there are strong anharmonic interactions! Nodes are view-dependent!
(1,0,4)r
(1,2,2)r
Projections of the wavefunction of the 2394 cm-1
eigenstate onto the q1-q3 plane at different values of q2.

24. Two-step Diagonalization: The eigenvector approach
Full 3D Hamiltonian
(Normal mode)
Diagonalize
Broad distribution of normal mode basis state characters in the single eigenstate at 2394 cm-1
This observed eigenstate is unassignable in the normal mode basis

25. Two-step Diagonalization: The eigenvector approach
Full 3-D Hamiltonian
(Normal mode)
Strong ν1~ν3 interactions (1:2 Fermi) + double-well
Partial 3-D Hamiltonian
(Normal mode)
A tailored basis which takes into account the most important interactions of the molecules
Diagonalize
The full Hamiltonian in the new tailored basis
Diagonalize
Energies and eigenvectors in the tailored basis
Weaker interactions between ν2 and ν3

26. Vibrational assignments in terms of eigenvectors in a prediagonalized basis
The vertical axis represents the squares of the basis state coefficients of one specific eigenstate.
*The vibrational wavefunctions of the C-state of SO2 are poorly described in the normal-mode basis (left panel).
*The partially-prediagonalized mode basis (right panel) takes into account the strong Fermi-resonance between the two stretching modes and the double-well along the antisymmetric-stretching coordinate.

27. The origins of the unequal S-O bond-lengths
The q3-mediated vibronic coupling model
for interactions between the A1 and B2 symmetry diabatic electronic states A1 B2
Diabats
Adiabats

28. The v2-dependence of the ν3 staggering: the approach to a conical intersection
The increase in the level staggering as v2 (bend) increases implies that the barrier height of the double-well potential increases as the bend angle increases.
The q3-mediated vibronic coupling model explains the observed v2-dependent level staggering.

29. Why?
There is a q3(b2)-mediated vibronic interaction of the C 1B2 state with the 2 1A1 (bound) and 3 1A1 (repulsive) states
The vertical energy separation along q3 between the 2 1A1 and C 1B2 diabatic states decreases rapidly as the OSO angle increases. This causes the effective barrier height to increase rapidly with excitation in ν2.
“Anomalies” in the C state provide information about the energy and shape of the remote-perturber potential energy surface!

30. Summary: SO2
SO2 is a simple molecule, but its C 1B2 state defies conventional vibration-rotation assignments
IR-UV double resonance is required to directly observe level staggerings in the antisymmetric stretch
In addition to the staggerings caused by the barrier, there are very strong ν1~2ν3 Fermi interactions and ν2~ν3 Coriolis interactions
Assignments based on dominant character or node- count in the ab initio wavefunctions fail
A partial diagonalization scheme defines a “good” basis set
Unequal SO bond lengths offend chemical intuition
Vibronic Coupling!
Pattern of broken patterns provides information about the potential surface of the remote perturbing state

31. Big Picture Summary
New technology provides more and better spectra Ask qualitatively new kinds of questions Both experimentalists and theorists need to rethink “assignment” Eigenvectors of a physical Heff model How transition states and dynamics are encoded in discrete eigenstate spectra A molecular structure or spectral pattern that offends chemical intuition usually implies vibronic coupling

32. Who did this?
Barratt Park Anthony Merer Josh Baraban John Stanton Bryan Changala Jun Jiang The acetylene mafia since 1979, all with DOE support! Office of Basic Energy Sciences, Division of Chemical Sciences, Geosciences, and Biosciences, US Department of Energy

34. Polyads Obey Matrix Element Scaling and Membership Selection Rules
Fermi Selection rule: Δvs=±1, Δva=-2Δvs
Darling-Dennison Selection rule: Δvs=±2, Δva=-Δvs
Polyad Number, P (a group of quasi-degenerate states):
Fermi P=2vs+va Matrix element scales as ~vavs1/2
D-D P=2vs+2va Matrix element Scales as ~vavs
Matrix Elements scale  as P 
Polyad Membership scales  as P 
Examples of membership scaling rules
Fermi P=6: (vs,va)=(3,0), (2,2), (1,4), (0,6)
Fermi P=8: (vs,va)=(4,0), (3,2), (2,4), (1,6), (0,8)
D-D P=6: (vs,va)=(3,0), (2,1), (1,2), (0,3)

35. Isomerization Transition State
Passage through a transition state occurs in femtoseconds.
Textbooks: “It is impossible to spectroscopically characterize a transition state.” WRONG!
When there is no vibrational continuum, the transition state spectrum consists exclusively of discrete eigenstates. So where is the dynamics?
The pattern of eigenstates in the spectrum encodes ultrafast dynamics!
But is this spectrum assignable? YES!
New kinds of patterns

36. Barrier Proximal States
States that encode isomerization must be both energetically and spatially close to the transition state (saddle point): “barrier proximal”
In classical mechanics, a particle at the exact energy and position of a saddle point will remain stationary forever at the saddle point
In quantum mechanics, quantized vibrational motion along the isomerization path will approach zero frequency at the transition state
Frequency dip! ωeff sharply approaches zero!

37. Full 3-D vibrational Hamiltonian used to fit to the experimental data
A Gaussian hump to mimic the double-well minimum along ν3

38. Vibrational wavefunctions of the SO2 C̃-state obtained from a reduced-dimensional Hamiltonian (2-D)
2-D Potential Energy Surface
of the SO2 C̃-state
The curved nodal patterns of the 2-D wavefunctions are characteristic of Fermi-resonance: ν1≈2ν3

39. Coriolis mixing angle:
ν2 frequency at 377 cm-1
The staggering in mode 3 causes the effective ν3 frequencies to go toward-and-away from resonance with the ν2 frequency.

40. The zigzag pattern in the Coriolis perturbed C rotational constants
The rotational signature (poor penmanship?) of the presence of a double-well structure on the PES