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 –H 2 CO, HFCO, and D 2 CO Results Modeling –Anharmonic Multi-Resonant Hamiltonian –Polyad Quantum Numbers Computation –Spectroscopically Accurate Calculations Applications

BACKGROUND

Potential Energy Surfaces The PES is a description of total molecular energy as a function of atomic arrangement Chemical structure and reactivity can be determined from the PES Measuring highly excited vibrational states characterizes the PES at geometries away from equilibrium In general, a PES has 3N-6 dimensions

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

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

High Overtone Spectroscopy Weak absorbance ( cm 2 ) Rotational congestion (50 cm -1 ) Single resonance

Stimulated Emission Pumping Double resonance Laser gives high resolution (0.1 cm -1 ) Time consuming, noisy, saturation

Dispersed Fluorescence Double resonance Monochromator: medium resolution (2 cm-1) Fast, zero background, linear

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

MEASUREMENT

Dispersed Fluorescence Spectroscopy Excite reactant molecules to higher electronic state Disperse fluorescence to vibrational levels E vibrational level = E laser – E fluoresence

Dispersed Fluorescence Spectroscopy Excite reactant molecules to higher electronic state Disperse fluorescence to vibrational levels E vibrational level = E laser – E fluoresence Reactant Products Fluorescence Laser excitation Energy Reaction Coordinate E vib

Dispersed Fluorescence Spectroscopy Double resonance Monochromator: medium resolution (2 cm -1 ) Fast, zero background, linear

Experimental Setup

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

Molecular Beam

Sample is cooled through a reduction of collisions Simplifies spectral analysis

Lasers

Monochromator

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 A monchromator disperses molecular fluorescence E vibrational level = E laser – E fluoresence E vib

H 2 CO DF Spectrum H 2 CO DF Spectrum Dispersed Fluorescence Energy (cm -1 )

Vibronic Selection Rules C 2v EC2C2  xz  yz A1A1 1111z (A-axis) A2A2 11–1 B1B1 1 1 x (C-axis) B2B2 1–1 1y (B-axis)

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

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

H 2 CO Pure Vibrational Spectrum H 2 CO Pure Vibrational Spectrum Dispersed Fluorescence Energy (cm -1 )

H 2 CO Pure Vibrational Spectrum H 2 CO Pure Vibrational Spectrum Dispersed Fluorescence Energy (cm -1 )

H 2 CO Assignments Dispersed Fluorescence Energy (cm -1 )

H 2 CO Vibrational Modes Vibrational states are combinations of normal modes Example:

H 2 CO Assignments AssignmentExperiment Fit 4Expt - Fit … –1 2 – – – – – – …

D 2 CO DF Spectrum D 2 CO DF Spectrum 4 1 D 2 CO Dispersed Fluorescence Energy (cm -1 )

HFCO DF Spectrum 3 1 HFCO Dispersed Fluorescence Energy (cm -1 )

Summary of Assignments MoleculePrevious #Current # Energy Range (cm -1 ) Year H 2 CO , D 2 CO , HFCO , HDCO9670 – 7, H 2 CO  H 2 +CO dissociation barrier  28,000 cm -1 HFCO  HF+CO dissociation barrier  17,000 cm -1

Summary of Assignments MoleculePrevious #Current # Energy Range (cm -1 ) H 2 CO ,500 D 2 CO ,000 HFCO ,500 HDCO9670 – 9,500 H 2 CO  H 2 +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 Model “Real” molecules deviate from the harmonic model Energy levels are lowered and are no longer equally spaced Harmonic Energy Anharmonic Correction

Resonances The very strong interaction of two nearly degenerate states is called a resonance Example: k 26,5 occurs in H 2 CO because modes 2 plus 6 are nearly degenerate with mode 5  2 +  6 = = 3005 cm -1  5 = 2870 cm -1 (< 5% difference) Resonances cause energy level shifts, state mixing, and energy transfer

Classical Example  1  2  2 Resonant coupling by k 1,22 results in energy transfer (1  2  1  )

Classical Example  1  2  2 Resonant coupling by k 1,22 results in energy transfer (1  2  1  )

Quantum Examples Molecular orbitals Molecular vibrations k 26,5  k 26,5 

Polyad Model Groups of vibrational states interacting through resonances are called polyads Polyad energy levels are calculated by solving the Schrödinger Equation k 26,5  k 26,5   k 44, k 26,5  k 26,5   k 44, k 26,5 

Polyad Model Groups of vibrational states interacting through resonances are called polyads Polyad energy levels are calculated by solving the Schrödinger Equation k 26,5  k 26,5   k 44, k 26,5  k 26,5   k 44, k 26,5 

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

H 2 CO Anharmonic Polyad Model Fits ParameterFit 1Fit 2Fit3Fit 4 ω1°ω1°  ω6°ω6° x  x k 26, k 36, k 11, k 44, k 25, Std Dev

Tacoma Narrows Bridge

Resonances Destroy Quantum Numbers Resonances destroy bridges … and quantum numbers What is v 2 ? 3, 2, 1, 0 v 6 ? 0, 2, 4, k 2,66  k 2,66  k 2,66 6

Resonances Destroy Quantum Numbers Resonances destroy bridges … and quantum numbers What is v 2 ? 3, 2, 1, 0 v 6 ? 0, 2, 4, k 2,66  k 2,66  k 2,66 6

Polyad Quantum Numbers Polyad quantum numbers are the conserved quantities after state mixing Example: k 2,66 N polyad = 2v 2 + v 6 = k 2,66  k 2,66  k 2,66 6

Of the 3N-6 dim vibrational vector space, resonances couple a subspace, leaving the orthogonal subspace uncoupled Determining Polyad Quantum Numbers k 2,66  N polyad = 2 v 2 + v 6

H 2 CO and D 2 CO Polyad Quantum Numbers H 2 CO D 2 CO k 1,44 N vib = v 2 +v 3 +v 5 ( ultimately destroyed ) k 44,66 N CO = v 2 ( remains good! ) k 36,5 N res = 2v 1 +2v 2 +v 3 +v 4 +2v 5 +v 6 ( remains good! ) k 36,5 N oop = v 4 ( destroyed by k 44,66 ) k 26,5 N vib = v 1 +v 4 +v 5 +v 6 ( destroyed by k 1,44 and k 1,66 ) k 11,55 N res = 2v 1 +v 2 +v 3 +v 4 +2v 5 +v 6 ( remains good! )

H 2 CO and D 2 CO Polyad Quantum Numbers H 2 CO D 2 CO v 1 v 2 k 36,5 v 3 k 26,5 N oop = v 4 k 44,66 v 4 k 11,55 N vib = v 1 +v 4 +v 5 +v 6 k 1,44 v 5 N energy = 2v 1 +v 2 +v 3 +v 4 +2v 5 +v 6 N energy still good! v 6 v 1 v 2 k 1,44 v 3 k 44,66 N CO = v 2 v 4 k 36,5 N vib = v 2 +v 3 +v 5 k ’s N CO still good! v 5 N energy = 2v 1 +2v 2 +v 3 +v 4 +2v 5 +v 6 N energy still good! v 6

v 1 v 2 k 36,5 v 3 k 26,5 N oop = v 4 v 4 k 11,55 N vib = v 1 +v 4 +v 5 +v 6 v 5 N energy = 2v 1 +v 2 +v 3 +v 4 +2v 5 +v 6 v 6 H 2 CO and D 2 CO Polyad Quantum Numbers H 2 CO D 2 CO k 44,66 k 1,44 N energy still good! v 1 v 2 k 1,44 v 3 k 44,66 N CO = v 2 v 4 k 36,5 N vib = v 2 +v 3 +v 5 k ’s N CO still good! v 5 N energy = 2v 1 +2v 2 +v 3 +v 4 +2v 5 +v 6 N energy still good! v 6

H 2 CO and D 2 CO DF Spectra Dispersed Fluorescence Energy (cm -1 )

H 2 CO and HDCO DF Spectra – Symmetry! Energy (cm -1 ) Dispersed Fluorescence 4 1 HDCO 4 1 H 2 CO

COMPUTATION

Model Fits to Experimental Data

Ab Initio Calculations 1.Compute force constants via numerical differentiation for Taylor expansion of PES with MOLPRO 2.Calculate x ij via perturbation theory and identify important k ijk, k ijkl with SPECTRO 3.Compute excited vibrational states from , x, k with POLYAD

Van Vleck Perturbation Theory ~0

Parallel Computing 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 ernst (2003)mu3c (2006) mu3c-2 (2011)

Computation of PES and Vibrations

Ab Initio Computation of Molecular PES’s Molecule Average Absolute Difference Energy Range Energy Level Standard Deviation νxk H2OH2O , D2OD2O , HDO , H 2 CO , D 2 CO , HDCO , HFCO , DFCO , SCCl , Average

Ab Initio Computation of Molecular PES’s Molecule Average Absolute Difference Energy Range Energy Level Standard Deviation ω°ω°xk H2OH2O , D2OD2O , HDO , H 2 CO , D 2 CO , HDCO , HFCO , DFCO , SCCl , Average

APPLICATIONS

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

Application: Kinetics Anharmonicity increases Q A and Q B Polyad quantum numbers decrease the accessibility of Q A and Q B

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 (H 2 CO, 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 ( , x, k) Polyad quantum numbers remain at high energy (N energy is always conserved) High level quartic PES calculations and the multi-resonant anharmonic model accurately predict excited vibrational states and potential energy surfaces

Acknowledgements H 2 CO Rychard Bouwens (UC Berkeley - Physics), Jon Hammerschmidt (U Minn - Chemistry), Martha Grzeskowiak (Mich St - Med School), Tineke Stegink (Netherlands - Industry), Patrick Yorba (Med School) D 2 CO 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