More Spectra A Lot More Better Too Now

More Spectra! A Lot More! Better Too! Now What? Robert W. Field Department of Chemistry Massachusetts Institute of Technology 72 nd International Symposium on Molecular Spectroscopy Talk TB 01 June 20, 2017

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 A 1Π state Pre-Laser days…

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 Ba. O 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

Better Too! In my talk MA 02 “Welcome to Rydberg Land” at the 69 th 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 Ca. F. 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 k. Hz), 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 multidimensional spectroscopies…

Now What? § Previously unimaginable spectroscopic targets § Transition states § Permutation splittings [e. g. Ha. Cb. Cc. Hd—Hd. Cb. Cc. Ha] § 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 S 1 HCCH § Partial pre-diagonalization: C 1 B 2 State of SO 2 § Pull back the curtain on “ergodicity”

Introduction • • • 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?

Three Ideas • Polyads • Robust • Predict and depict emergence of something special: Large Amplitude Motions • Transition States are where polyads break! • Trans-cis isomerization in the S 1 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 SO 2 [JCP 144, 144311, 144412, and 144313 (2016)] Talk MG 09 Jun Jiang • Vinylidene to Acetylene Isomerization on S 0: Talk TB 08 Steve Gibson

Trans-Cis Isomerization in S 1 HCCH Josh Baraban, Bryan Changala, Georg Mellau, John Stanton, Anthony Merer, RWF

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

Every Vibrational Level Up to the Energy of the HCCH S 1 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!

B 2 Polyads in HCCH S 1 [trans-bent] • Consist of (v 4, v 6) = (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? • 3 n. B 2 • 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

Excitation in v 3 distorts (v 4, v 6) bending polyads Steeves et. al. , J. Mol. Spec. , 256, 2009.

Example of polyad breakdown The “reduced” matrix element K 4466 B 2 B 3 v 3 = 0 -51. 678* -51. 019* v 3 = 1 -60. 101 -57. 865 v 3 = 2 -66. 502 * Expected behavior

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 Excitation in BOTH modes 3 and 6 is required to generate an isomerization dip. Excitation in mode 4 is irrelevant.

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

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!

Why Does the C State of SO 2 Have Unequal SO Bond Lengths? Barratt Park, Jun Jiang, Catherine Saladrigas

The C 1 B 2 state of SO 2 has a double-minimum potential in ν 3 • ν 3 appears to be very low-frequency (~200 cm-1), as suggested by - Origin isotope shifts - Inertial defects - Coriolis Constants - Centrifugal Distortion Constants • (0, 0, v 3) progression appears to be staggered, as evidenced by Coriolis perturbations observed in bright (a 1) states. • Many pattern-based assignments are incorrect, especially due to the lack of observation of any odd-v 3 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.

Low-Lying Vibrational Levels of CState SO 2. Only a 1 Levels: Regular? 6 2 4 Tvib (predicted) 2 2 4 0 2 0 v 3 = 0 (v 1, v 2, v 3) = (0, 0, v 3) (0, 1, v 3) (0, 2, v 3) (0, 3, v 3) (1, 0, v 3) — a 1 levels — b 2 levels

Low-Lying Vibrational Levels of C State SO 2. Regular? Nope! Staggered 6 Tvib 5 4 4 3 2 2 1 1 0 1 v 3 = 0 0 2 2 1 1 (predicted) 0 0 Odd v 3 levels observed by mm. W-UV double resonance (v 1, v 2, v 3) = (0, 0, v 3) (0, 1, v 3) (0, 2, v 3) (0, 3, v 3) (1, 0, v 3) — a 1 levels — b 2 levels

Consequence of a double-well minimum along ν 3 Also ν 3 staggering w 3, eff (cm-1) 400 1 350 2 3 4 5 ν 1≈2ν 3 1: 2 Fermi Resonance between ν 1 and ν 3 300 250 v 3 = 0 200 0 500 ν 2≈ν 3 1000 1500 2000 Extensive c-axis Coriolis interaction between ν 2 and ν 3

Strong Coriolis interactions between ν 2 and ν 3 Horrible mess. Assignment would have been impossible without an Heff that included staggering.

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 q 1 -q 3 plane at different values of q 2.

Two-step Diagonalization: The eigenvector approach Full 3 D 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

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

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 SO 2 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.

The origins of the unequal S-O bond-lengths The q 3 -mediated vibronic coupling model for interactions between the A 1 and B 2 symmetry diabatic electronic states A 1 B 2 Diabats Adiabats

The v 2 -dependence of the ν 3 staggering: the approach to a conical intersection The increase in the level staggering as v 2 (bend) increases implies that the barrier height of the doublewell potential increases as the bend angle increases. The q 3 -mediated vibronic coupling model explains the observed v 2 -dependent level staggering.

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

Summary: SO 2 • SO 2 is a simple molecule, but its C 1 B 2 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 ν 1~2ν 3 Fermi interactions and ν 2~ν 3 Coriolis interactions • Assignments based on dominant character or nodecount 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

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

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

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=2 vs+va Matrix element scales as ~vavs 1/2 • D-D P=2 vs+2 va 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)

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

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!

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

Vibrational wavefunctions of the SO 2 C -state obtained from a reduced-dimensional Hamiltonian (2 -D) 2 -D Potential Energy Surface of the SO 2 C -state The curved nodal patterns of the 2 -D wavefunctions are characteristic of Fermi-resonance: ν 1≈2ν 3

Coriolis mixing angle: ν 2 frequency at 377 cm-1 The staggering in mode 3 causes the effective ν 3 frequencies to go toward-and-away from resonance with the ν 2 frequency.

The zigzag pattern in the Coriolis perturbed C rotational constants C 000 The rotational signature (poor penmanship? ) of the presence of a doublewell structure on the PES