William F. Polik Department of Chemistry Hope College Holland, MI, USA Measuring, Modeling, and Computing Resonances in Excited Vibrational States of Polyatomic Molecules William F. Polik Department of Chemistry Hope College Holland, MI, USA

Outline Background Measurement Dispersed Fluorescence Spectroscopy H2CO, HFCO, D2CO, and HDCO Results Modeling Anharmonic Multi-Resonant Hamiltonian Polyad Quantum Numbers Computation Spectroscopically Accurate Calculations Applications

BACKGROUND

Potential Energy Surfaces Reactant Product Energy Reaction Coordinate Transition State The PES is a description of total molecular energy as a function of atomic arrangement Chemical structure, properties, and reactivity can be determined from the PES

Characterizing PES’s Reactant Product Energy Reaction Coordinate Transition State Vibrational States Measuring highly excited vibrational states characterizes the PES at geometries away from equilibrium

Characterizing PES’s Reactant Product Transition State Measuring highly excited vibrational states characterizes the PES at geometries away from equilibrium In general, a PES has 3N-6 dimensions

MEASUREMENT

Dispersed Fluorescence Spectroscopy Excite reactant molecules to higher electronic state Disperse fluorescence to vibrational levels Evibrational level = Elaser – Efluoresence

Experimental Setup Crystal Nd: YAG Laser Tunable Dye Laser Mirror Doubling Filter Sample + Ne Signal ICCD Computer Monochromator Frequency

Free Jet for Sample Preparation A free jet expansion cools the sample to 5K Molecules occupy the lowest quantum state and simplify the excitation spectrum

Lasers for Electronic Excitation A laser provide an intense monochromatic light source Promotes molecules to a single rovibrational level in an excited electronic state

Monochromator for Detection vib A monchromator disperses molecular fluorescence Evibrational level = Elaser – Efluoresence

H2CO DF Spectrum Dispersed Fluorescence Energy (cm-1)

Vibronic Selection Rules C2v E C2 xz yz A1 1 z (A-axis) A2 –1 B1 x (C-axis) B2 y (B-axis)

Rotational Selection Rules A-type rules: Ka=even, Kc=odd B-type rules: Ka=odd, Kc=odd C-type rules: Ka=odd, Kc=even One photon transition: J=0,1 From S1 000, each S0 vibrational state has at most one spectral transition; hence, PURE VIBRATIONAL SPECTROSCOPY

Rotational Congestion Rotational structure is superimposed on a vibrational transition

Rotational Congestion Only a single transition originates from J=0

Pure Vibrational Spectroscopy Only a single transition originates from J=0, eliminating all rotational congestion

H2CO Pure Vibrational Spectrum Dispersed Fluorescence Energy (cm-1)

H2CO Assignments Dispersed Fluorescence Energy (cm-1) 5000 6000 5500

H2CO Vibrational Modes Vibrational states are combinations of normal modes Example: 213162

H2CO Assignments Assignment Experiment Fit 4 Expt - Fit 00 -0.3 0.0 41 1167.4 1166.9 0.5 61 1249.6 1249.7 -0.1 31 1500.2 1499.7 21 1746.1 1745.8 0.3 … –12–52 5462.7 5464.2 -1.5 –314151–324161 5489.1 5489.4 -0.4 –113161–1151 5530.5 5529.5 1.0 213142+3262 5546.5 5544.0 2.5 213142 5551.4 5551.9 -0.5 21314161–214151 5625.5 5624.3 1.2 234361+224351 9865.8 9865.4 214661 9875.4 9875.0 2531 9987.8 9990.8 -3.0 233143+122241 10066.3 10067.5 -1.2

D2CO DF Spectrum 41 D2CO Dispersed Fluorescence Energy (cm-1)

HFCO DF Spectrum 31 HFCO Dispersed Fluorescence Energy (cm-1)

Summary of Assignments Molecule Previous # Current # Energy Range (cm-1) H2CO 81 279 0 - 12,500 D2CO 7 261 0 - 12,000 HFCO 44 382 0 - 22,500 HDCO 9 67 0 – 9,500 H2CO®H2+CO dissociation barrier » 28,000 cm-1 HFCO®HF+CO dissociation barrier » 17,000 cm-1

MODELING

Harmonic Oscillator Model Equally spaced energy levels

Anharmonic Correction Anharmonic Model “Real” molecules deviate from the harmonic model Energy levels are lowered and are no longer equally spaced X’s are negative Harmonic Energy Anharmonic Correction

Resonances The very strong interaction of two nearly degenerate states is called a resonance Example: k26,5 occurs in H2CO because modes 2 plus 6 are nearly degenerate with mode 5 w2 + w6 = 1756 + 1249 = 3005 cm-1 w5 = 2870 cm-1 (< 5% difference) Resonances cause energy level shifts, state mixing, and energy transfer

Classical Example Two oscillations of the pendulum 1 equal one oscillation of the pendulum 2 w1 » 2 w2 Resonant coupling by k1,22 results in energy transfer (1®2®1®...)

Quantum Examples Molecular orbitals Molecular vibrations 2265 215164 All chemical bonds arise from mixing of atomic orbitals 2265 k26,5 « 215164 5263

Polyad Model Groups of vibrational states interacting through resonances are called polyads Polyad energy levels are calculated by solving the Schrödinger Equation 2265 k26,5 « 215164 5263  k44,66 224263 k26,5 « 21425162 425261 224461 214451

Matrix Form of Schrödinger Eqn Diagonal Elements: Off-Diagonal Elements: Harmonic Energy Anharmonic Correction Resonant Interactions

H2CO Anharmonic Polyad Model Fits Parameter Fit 1 Fit 2 Fit3 Fit 4 ω1° 2818.9 2812.3 2813.7 2817.4  ω6° 1260.6 1254.8 1251.5 1251.9 x11 -40.1 -29.8 -30.7 -34.4 x66 -5.2 -2.8 -2.1 -2.2 k26,5 148.6 146.7 138.6 k36,5 129.3 129.6 135.1 k11,55 140.5 137.4 k44,66 21.6 23.3 k25,35 18.5 Std Dev 23.4 4.34 3.34 2.80

Tacoma Narrows Bridge

Resonances Destroy Quantum Numbers Resonances destroy bridges … and quantum numbers 23 k2,66 « 2262 2164 66 What is v2? 3, 2, 1, 0 v6? 0, 2, 4, 6

Polyad Quantum Numbers Polyad quantum numbers are the conserved quantities after state mixing Example: k2,66 23 k2,66 « 2262 2164 66 Npolyad = 2v2 + v6 = 6

Determining Polyad Quantum Numbers Of the 3N-6 dim vibrational vector space, resonances couple a subspace, leaving the orthogonal subspace uncoupled More resonances = larger polyads = fewer polyad quantum numbers k2,66  Npolyad = 2 v2 + v6

H2CO and D2CO Polyad Quantum Numbers v1 v2 k36,5 v3 k26,5 Noop = v4 v4 k11,55 Nvib = v1+v4+v5+v6 v5 Nenergy = 2v1+v2+v3+v4+2v5+v6 v6 k44,66 k1,44 Nenergy still good! v1 v2 k1,44 v3 k44,66 NCO = v2 v4 k36,5 Nvib = v2+v3+v5 k’s NCO still good! v5 Nenergy = 2v1+2v2+v3+v4+2v5+v6 Nenergy still good! v6

H2CO and D2CO DF Spectra Dispersed Fluorescence Energy (cm-1)

H2CO and HDCO DF Spectra – Symmetry! Dispersed Fluorescence 41 HDCO Energy (cm-1)

COMPUTATION

Model Fits to Experimental Data

Ab Initio Calculations Compute force constants via numerical differentiation for Taylor expansion of PES with MOLPRO Calculate xij via perturbation theory and identify important kijk, kijkl with SPECTRO Compute excited vibrational states from w, x, k with POLYAD

Van Vleck Perturbation Theory ~0

Parallel Computing ernst (2003) mu3c (2006) mu3c-2 (2011) Force constants are computed as numerical derivatives, i.e., by calculating energies of displaced geometries PES calculation takes hours instead of weeks with parallel computing

Computation of PES and Vibrations

Ab Initio Computation of Molecular PES’s    Molecule Average Absolute Difference Energy Range Energy Level Standard Deviation ω° x k H2O 2.8 1.9 8.2 0 - 15,000 20.0 D2O 1.0 0.8 - 0 - 9,500 22.6 HDO 2.7 2.6 13.1 H2CO 5.1 3.5 15.2 0 - 10,000 23.0 D2CO 6.7 3.6 29.6 0 - 11,500 25.9 HDCO 4.0 4.9 22.9 12.0 HFCO 9.4 5.8 0 - 22,500 42.8 DFCO 1.8 7.7 6.2 SCCl2 3.9 0 - 20,000 18.6 Average 4.3 14.9 20.5

APPLICATIONS

Application: Quantum Numbers Quantum Numbers allow us to understand the microscopic world Atoms: n l ml s ms Molecules: rotation, vibration, electronic Normal mode vibrational quantum vi numbers apply near equilibrium Polyad vibrational quantum numbers apply for excited states Nspecial (oop bend, CO stretch, vib ang momentum) Nstretch (sum of high freq stretches) Nenergy (energy ratios)

Application: Kinetics Anharmonicity increases QA and QB Polyad quantum numbers decrease the accessibility of QA and QB

Application: Computational Chemistry Fastest growing chemistry subdiscipline Method and computer improvements imply high accuracy near equilibrium (±1 kcal/mol) Methods relatively untested away from equilibrium Validating methods on prototypical systems (H2CO, HFCO) will permit application to more complex systems

Conclusions Dispersed fluorescence spectroscopy is a powerful technique for measuring excited states (general, selective, sensitive) The multi-resonant anharmonic (“polyad”) model accounts for resonances and assigns highly mixed spectra (w, x, k) Polyad quantum numbers remain at high energy (Nenergy is always conserved) High level quartic PES calculations and the multi-resonant anharmonic model accurately predict excited vibrational states and potential energy surfaces

Acknowledgements H2CO D2CO HFCO HDCO Theory Funding Rychard Bouwens (UC Berkeley - Physics), Jon Hammerschmidt (U Minn - Chemistry), Martha Grzeskowiak (Mich St - Med School), Tineke Stegink (Netherlands - Industry), Patrick Yorba (Med School) D2CO Gregory Martin (Dow Chemical), Todd Chassee (U Mich - Med School), Tyson Friday (Industry) HFCO Katie Horsman (U Va - Chemistry), Karen Hahn (U Mich - Med School), Ron Heemstra (Pfizer - Industry) HDCO Kristin Ellsworth (Univ Mich – Dental School), Brian Lajiness (Indiana Univ– Med School), Jamie Lajiness (Scripps – Chemistry) Theory Ruud van Ommen (Netherlands – Physics), Ben Ellingson (U Minn – Chemistry), John Davisson (Indiana Univ – Med School), Andreana Rosnik (Hope College ‘13) Funding NSF, Beckman Foundation, ACS-PRF, Research Corporation, Dreyfus Foundation

Polik Group