1. 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

2. 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

3. BACKGROUND

4. 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

5. 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

6. 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

7. MEASUREMENT

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

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

10. 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

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

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

13. H2CO DF Spectrum
Dispersed Fluorescence
Energy (cm-1)

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

15. 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

16. Rotational Congestion
Rotational structure is superimposed on a vibrational transition

17. Rotational Congestion
Only a single transition originates from J=0

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

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

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

21. H2CO Vibrational Modes
Vibrational states are combinations of normal modes
Example:

22. 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
5546.5
5544.0
2.5
213142
5551.4
5551.9
-0.5
–214151
5625.5
5624.3
1.2
9865.8
9865.4
214661
9875.4
9875.0
2531
9987.8
9990.8
-3.0
-1.2

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

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

25. 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

26. MODELING

27. Harmonic Oscillator Model
Equally spaced energy levels

28. 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

29. 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 = = 3005 cm-1
w5 = 2870 cm-1 (< 5% difference)
Resonances cause energy level shifts, state mixing, and energy transfer

30. 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®...)

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

32. 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
425261
224461
214451

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

34. 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

35. Tacoma Narrows Bridge

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

37. 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

38. 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

39. H2CO and D2CO Polyad Quantum Numbersv1
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

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

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

42. COMPUTATION

43. Model Fits to Experimental Data

44. 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

45. Van Vleck Perturbation Theory~0

46. 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

47. Computation of PES and Vibrations

48. 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

49. APPLICATIONS

50. 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)

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

52. 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

53. 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

54. 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

55. Polik Group