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Measuring, Modeling, and Computing Measuring, Modeling, and Computing
Resonances in Excited Vibrational States Resonances in Excited Vibrational States
of Polyatomic Moleculesof Polyatomic Molecules
William F. PolikDepartment of Chemistry
Hope College
Holland, MI, USA
OutlineOutline
• Background
• Measurement
– Dispersed Fluorescence Spectroscopy
– H2CO, HFCO, and D2CO Results
• Modeling
– Anharmonic Multi-Resonant Hamiltonian
– Polyad Quantum Numbers
• Computation
– Spectroscopically Accurate Calculations
• Applications
BACKGROUNDBACKGROUND
Potential Energy SurfacesPotential 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
Reactant
Product
Ene
rgy
Reaction Coordinate
Transition State
Characterizing PESCharacterizing PES’’ss
• 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
En
erg
y
Reaction Coordinate
MEASUREMENTMEASUREMENT
Dispersed Fluorescence SpectroscopyDispersed Fluorescence Spectroscopy
• Excite reactant molecules to higher electronic state• Disperse fluorescence to vibrational levels
• Evibrational level = Elaser – Efluoresence
Reactant
Products
Fluorescence Laser
excitation
En
erg
y
Reaction Coordinate
Evib
Experimental Setup Experimental Setup
Crystal
Nd: YAG Laser
Tunable Dye Laser
Mirror
Mirror
Doubling
Filter
Sample + Ne
Sig
nal
ICCD Computer Monochromator
Frequency
Free Jet for Sample PreparationFree Jet for Sample Preparation
• A free jet expansion cools the sample to 5K• Molecules occupy the lowest quantum state and simplify the
excitation spectrum
Reactant
Products
En
erg
y
Reaction Coordinate
Lasers for Electronic ExcitationLasers for Electronic Excitation
• A laser provide an intense monochromatic light source• Promotes molecules to a single rovibrational level in an excited electronic
state
Laser excitation
Reactant
Products
En
erg
y
Reaction Coordinate
Monochromator for DetectionMonochromator for Detection
• A monchromator disperses molecular fluorescence
• Evibrational level = Elaser – Efluoresence
Fluorescence Laser
excitation
Reactant
Products
En
erg
y
Reaction Coordinate
Evib
0 4000 8000 12000
41 H2CO
DF
In
ten
sity
S0 Energy (cm-1)
HH22CO DF SpectrumCO DF Spectrum
Dis
pers
ed F
luor
esce
nce
Energy (cm-1)
Vibronic Selection RulesVibronic Selection Rules
C2v E C2 xz yz
A1 1 1 1 1 z (A-axis)
A2 1 1 –1 –1
B1 1 –1 1 –1 x (C-axis)
B2 1 –1 –1 1 y (B-axis)
C
O
HH
z(A)
y(B)
A1 A2 B1 B2 S0
S1 41
C-type B-type A-type
1
2
1
2
1
2
1
12vibelecvibelec210 A
B
A
A
A
B
B
BA :COH 4 For
1A
Rotational Selection RulesRotational Selection Rules
• From S1 000, each S0 vibrational state has at most one spectral transition; hence,
PURE VIBRATIONAL SPECTROSCOPY
A1 A2 B1 B2 S0
S1 41
C-type B-type A-type
000
110
111
101
000
One photon transition: J=0,1
A-type rules: Ka=even, Kc=odd
B-type rules: Ka=odd, Kc=odd
C-type rules: Ka=odd, Kc=even
Rotational CongestionRotational Congestion
• Rotational structure is superimposed on a vibrational transition
1 0
1
2
3
0
v
1 0
2
3
J HCl
2600 2700 2800 2900 3000 3100
Energy (cm-1)
Abs
orba
nce
Rotational CongestionRotational Congestion
• Only a single transition originates from J=0
1 0
1
2
3
0
v
1 0
2
3
J HCl
2600 2700 2800 2900 3000 3100
Energy (cm-1)
Abs
orba
nce
Pure Vibrational SpectroscopyPure Vibrational Spectroscopy
• Only a single transition originates from J=0, eliminating all rotational congestion
1 0
1
2
3
0
v
1 0
2
3
J HCl
2600 2700 2800 2900 3000 3100
Energy (cm-1)
Abs
orba
nce
0 4000 8000 12000
41 H2CO
DF
In
ten
sity
S0 Energy (cm-1)
HH22CO Pure Vibrational SpectrumCO Pure Vibrational Spectrum
Dis
pers
ed F
luor
esce
nce
Energy (cm-1)
HH22CO AssignmentsCO AssignmentsD
ispe
rsed
Flu
ores
cenc
e
Energy (cm-1)
5000 60005500
HH22CO Vibrational ModesCO Vibrational Modes
• Vibrational states are combinations of normal modes
• Example: 213162
Symmetric C-H Stretch
C
O
H H
C=O Stretch
C
O
H H
C-H2 Bend
C
O
H H
C
O
H H
C
O
H H
Antisymmetric C-H Stretch
C
O
H H
C-H2 Rock Out-of-plane Bend
+
+ +
-
HH22CO AssignmentsCO AssignmentsAssignment Experiment Fit 4 Expt - Fit
00 -0.3 0.0 -0.341 1167.4 1166.9 0.561 1249.6 1249.7 -0.131 1500.2 1499.7 0.521 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 1.0213142+3262 5546.5 5544.0 2.5
213142 5551.4 5551.9 -0.521314161–214151 5625.5 5624.3 1.2
…234361+224351 9865.8 9865.4 0.3
214661 9875.4 9875.0 0.32531 9987.8 9990.8 -3.0
233143+122241 10066.3 10067.5 -1.2
DD22CO DF SpectrumCO DF Spectrum
0 5000 10000
DF
In
ten
sity
S0 Energy (cm-1)
41 D2CO
Dis
pers
ed F
luor
esce
nce
Energy (cm-1)
HFCO DF SpectrumHFCO DF Spectrum
31 HFCO
Dis
pe
rsed
Flu
ore
sce
nce
Energy (cm-1)
Summary of AssignmentsSummary 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
H2COH2+CO dissociation barrier 28,000 cm-1
HFCOHF+CO dissociation barrier 17,000 cm-1
MODELINGMODELING
Harmonic Oscillator ModelHarmonic Oscillator Model
• Equally spaced energy levels
vE i
2
1 i
0
Anharmonic ModelAnharmonic Model
• “Real” molecules deviate from the harmonic model
• Energy levels are lowered and are no longer equally spaced
i ji
jiijii vvxvE
HarmonicEnergy
AnharmonicCorrection
ResonancesResonances
• 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
2 + 6 = 1756 + 1249 = 3005 cm-1
5 = 2870 cm-1 (< 5% difference)
• Resonances cause energy level shifts, state mixing, and energy transfer
Classical ExampleClassical Example
• 1 2 2
• Resonant coupling by k1,22 results in energy transfer (121)
Classical ExampleClassical Example
• 1 2 2
• Resonant coupling by k1,22 results in energy transfer (121)
Quantum ExamplesQuantum Examples
• Molecular orbitals
• Molecular vibrations
A
B
A -B
A + B
2265
k26,5
215164
k26,5
5263
Polyad ModelPolyad Model
• Groups of vibrational states interacting through resonances are called polyads
• Polyad energy levels are calculated by solving the Schrödinger Equation
2265k26,5
215164k26,5
5263
k44,66 k44,66 k44,66
224263k26,5
21425162k26,5
425261
k44,66 k44,66
224461k26,5
214451
Diagonal Elements: Off-Diagonal Elements:
EH
HarmonicEnergy
AnharmonicCorrection
2/12/12/1
,22
1
2
1
kjikij
vvvk
Resonant Interactions
Matrix Form of SchrMatrix Form of Schröödinger Eqndinger Eqn
i
iiv ji
jiij vvx
1 1
2 2
3 3
12 13
21 23
31
11
22
3332
c cHH
H
H
H
H E
cH
c
c
H
H
c
HH22CO Anharmonic Polyad Model FitsCO 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
k11,55 140.5 137.4 129.3
k44,66 21.6 23.3
k25,35 18.5
Std Dev 23.4 4.34 3.34 2.80
Resonances Destroy Quantum NumbersResonances Destroy Quantum Numbers
• Resonances destroy bridges
• … and quantum numbers
What is v2? 3, 2, 1, 0 v6? 0, 2, 4, 6
23k2,66
2262k2,66
2164k2,66
66
Polyad Quantum NumbersPolyad Quantum Numbers
• Polyad quantum numbers are the conserved quantities after state mixing
• Example: k2,66
2
6
k2,66
Npolyad = 2v2 + v6 = 6
23k2,66
2262k2,66
2164k2,66
66
• Of the 3N-6 dim vibrational vector space, resonances couple a subspace, leaving the orthogonal subspace uncoupled
Determining Polyad Quantum NumbersDetermining Polyad Quantum Numbers
m-dim vibrational quantum number
vector space
n-dim
(m n)-dim
resonance vector subspace
polyad vector subspace
(1, 2)
(2,1)
resonance vector subspace
polyad vector subspace
k2,66
Npolyad=2v2+v6
v2
v6
• k2,66 Npolyad = 2 v2 + v6
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
HH22CO and DCO and D22CO Polyad Quantum NumbersCO Polyad Quantum Numbers
• H2CO
• D2CO
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
HH22CO and DCO and D22CO DF SpectraCO DF Spectra
Dis
pe
rsed
Flu
ore
sce
nce
Energy (cm-1)
HH22CO and HDCO DF Spectra – Symmetry!CO and HDCO DF Spectra – Symmetry!
0 2000 4000 6000
46
214
4
224
2
44
214
2
42
23
222
1
00
"46"
"214
4""2
24
2"
"44"
"214
2"
"42"
23
22
21
00
Energy (cm-1)
Dis
pers
ed F
luor
esce
nce
41 HDCO
41 H2CO
COMPUTATIONCOMPUTATION
Model Fits to Experimental DataModel Fits to Experimental Data
H2O Experimental Fits
-120
-80
-40
0
40
80
120
0 5000 10000 15000
Observed Energy
Ca
lc -
Ob
s E
ne
rgy
HarmonicModel
AnharmonicModel
PolyadModel
Ab InitioAb Initio Calculations Calculations
1. Compute force constants via numerical differentiation for Taylor expansion of PES with MOLPRO
2. Calculate xij via perturbation theory and identify important kijk, kijkl with SPECTRO
3. Compute excited vibrational states from , x, k with POLYAD
lkjilkji
lkjikjikji
kji
jiji
jii
i
qqqqqqqq
Eqqq
qqq
E
qqqq
Eq
q
EEE
,,, 0
4
,, 0
3
, 0
2
0
0
!4
1
!3
1
!2
1
k ikiikii
iikiiii
iix411
32
1
162
1 1
2 2
3 3
12 13
21 23
31
11
22
3332
c cHH
H
H
H
H E
cH
c
c
H
H
c
Van Vleck Perturbation TheoryVan Vleck Perturbation Theory
~0
~0
HHHH o ˆ
2/1; )2)(1)(2)(1(
4
1...2,...
~...,2... llkkllkklklk nnnnKnnHnn
rlkDrlkDhcrllDrllDrkkDrkkDhcBhcKr
klrr
llrkkrlk
lkklkkllllkk ,,,,
4
1,,,,,,,,
16
1)(
42
22
;
)(
1),,(
mlk
mlkD
Parallel ComputingParallel 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 VibrationsComputation of PES and Vibrations
H2O Multi-Resonant Anharmonic Calculations
-120
-80
-40
0
40
80
120
0 5000 10000 15000
Observed Energy
Ca
lc -
Ob
s E
ne
rgy VTZ/VTZ
AVQZ/VTZ
PolyadModel Fit
Ab InitioAb Initio Computation of Molecular PES Computation of Molecular PES’’ss
Molecule
Average Absolute Difference
EnergyRange
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 - 0 - 9,500 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 0 - 9,500 12.0
HFCO 9.4 2.7 5.8 0 - 22,500 42.8
DFCO 2.7 1.8 7.7 0 - 9,500 6.2
SCCl2 3.9 1.8 - 0 - 20,000 18.6
Average 4.3 2.6 14.9 20.5
APPLICATIONSAPPLICATIONS
Application: Quantum NumbersApplication: 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)
A+B C
AB‡
Tk
E
BA
B BeQQ
Q
h
Tk
BA
ABfk
0‡‡
‡ABfBAkRate
Application: KineticsApplication: Kinetics
• Anharmonicity increases QA and QB
• Polyad quantum numbers decrease the accessibility of QA and QB
Application: Computational ChemistryApplication: 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
ConclusionsConclusions
• 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 (Nenergy is always conserved)
• High level quartic PES calculations and the multi-resonant anharmonic model accurately predict excited vibrational states and potential energy surfaces
AcknowledgementsAcknowledgements• H2CO
Rychard Bouwens (UC Berkeley - Physics), Jon Hammerschmidt (U Minn - Chemistry), Martha Grzeskowiak (Mich St - Med School), Tineke Stegink (Netherlands - Industry), Patrick Yorba (Med School)
• D2COGregory Martin (Dow Chemical), Todd Chassee (U Mich - Med School), Tyson Friday (Industry)
• HFCOKatie Horsman (U Va - Chemistry), Karen Hahn (U Mich - Med School), Ron Heemstra (Pfizer - Industry)
• HDCOKristin Ellsworth (Univ Mich – Dental School), Brian Lajiness (Indiana Univ– Med School), Jamie Lajiness (Scripps – Chemistry)
• TheoryRuud van Ommen (Netherlands – Physics), Ben Ellingson (U Minn – Chemistry), John Davisson (Indiana Univ – Med School), Andreana Rosnik (Hope College ‘13)
• FundingNSF, Beckman Foundation, ACS-PRF, Research Corporation, Dreyfus Foundation
Polik GroupPolik Group