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