A Practical Procedure for ab initio Determination of Vibrational Spectroscopic Constants,...

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A Practical Procedure for A Practical Procedure for

ab initioab initio Determination of Determination of

Vibrational Spectroscopic Constants, Vibrational Spectroscopic Constants,

Resonances, and PolyadsResonances, and Polyads

William F. PolikHope College, Holland, MI

June 2006

Chemical Reactions Occur viaChemical Reactions Occur viaExcited Vibrational StatesExcited Vibrational States

Reaction Coordinate

Vibrational States

Reactants

Products

Transition State

HFCO Pure Vibrational SpectrumHFCO Pure Vibrational Spectrum

0 5000 10000 15000 20000

Frequency (cm-1)

31 HFCO

Vibrational State ModelsVibrational State Models

Harmonic Anharmonic Polyad

jiijii vvxvE 1 1

Calculation MethodCalculation Method

1. Compute equilibration geometry

2. Compute PES derivatives

3. Calculate spectroscopic constants

4. Identify important resonances

5. Compute excited vibrational states

CCSD(T)/aug-cc-pVQZ

ωi xij K

POLYAD program

1. Compute Equilibration Geometry1. Compute Equilibration Geometry

• Geometry of energy minimum needed for Taylor expansion of PES

• Key points in calculation– Correlated theory and high quality basis, e.g., CCSD(T) and

aug-cc-pVQZ

– Tight convergence of SCF wavefunction and optimized structure

• Program used– Molpro (Werner & Knowles)

lkjilkji

lkjikjikji

qqqqqqqq

2. Compute PES Derivatives 2. Compute PES Derivatives

• Taylor-series derivatives are molecular force constants

• Key points in calculation:– Symmetrized internal coordinates– Numerical derivatives

• Programs used– FE/BE (Martin): list of displaced geometries; assemble derivatives– Intder (Allen): coordinate transformations– Molpro (Werner & Knowles): energy points

Δq)E(Δq)E(

ijklijkij

lkjilkji

lkjikjikji

kjijiji

jiqqqq

3. Calculate Spectroscopic Constants3. Calculate Spectroscopic Constants

• Force field is defined in terms of displacements qi

but vibrational energy levels are quantized by vi

• Second order perturbation theory relate ijk and ijkl to xij

• Program used– Spectro (Handy)

lkjiijklkji

kjiijkii

iN qqqqqqqqqqqE,,,

632,1 ,,

iiN vvxvEvvvE 21

0632,1 ,,

Spectroscopic ConstantsSpectroscopic Constants

Refs: Nielsen (1959), Papousek & Aliev (1982)

iiiiiix 22

k kjikjikjikji

kjikijk

jjkiikiijjij Bx

4. Identify Important Resonances4. Identify Important Resonances

• Perturbation theory breaks down at resonances

• For each resonant interaction– Modify calculation of xij

– Determine resonance constant K

• Program used– Spectro-modified (Handy, Martin, Polik)

Spectroscopic ConstantsSpectroscopic Constants

Refs: Nielsen (1959), Papousek & Aliev (1982)

iiiiiix 22

k kjikjikjikji

kjikijk

jjkiikiijjij Bx

resonance denominator when 2ωi≈ωk

resonance denominator when ωi≈ωj+ ωj, ωj≈ωi+ ωk, or ωk≈ωi+ ωj

4. Identify Important Resonances4. Identify Important Resonances

• Perturbation theory breaks down at resonances

• For each resonant interaction– Modify calculation of xij

– Determine resonance constant K

• Program used– Spectro-modified (Handy, Martin, Polik)

Modified Spectroscopic ConstantsModified Spectroscopic Constants

Refs: Papousek & Aliev (1982), Martin & Taylor (1997)

k ikiikiiiik

iiiiiix

k kjikjikjikjiijk

jjkiikiijj

k kjikjikjikji

kjikijk

jjkiikiijjij

partial fraction expansion

drop resonance term(s)

Resonance ConstantsResonance Constants

Refs: Lehmann (1989), Martin & Taylor (1997)

ijkiijjiiijkijkkij kKkK 21

m mjjmjjmiimiikkmiim

m mkimkiikm

kiikiikk

m mkmimkkmiim

kiikiikkkkii

5. Compute Excited Vibrational States 5. Compute Excited Vibrational States

• Define model parameters (, x, K)

• Determine polyads

H2O CCSD(T): aug-cc-pVQZ/cc-pVTZw1o 1 0 0 1 0 0 3684.92372284 -1w2o 0 1 0 0 1 0 1610.20330698 -1w3o 0 0 1 0 0 1 3797.96450773 -1x11 2 0 0 2 0 0 -43.64165166 -1x12* 1 1 0 1 1 0 -37.99219452 -1x13 1 0 1 1 0 1 -167.62218355 -1x22* 0 2 0 0 2 0 -11.33265207 -1x23 0 1 1 0 1 1 -19.05478905 -1x33 0 0 2 0 0 2 -49.68139858 -1K22,1 0 2 0 1 0 0 -154.23334812 -1K11,33 2 0 0 0 0 2 -160.77598615 -1

2132K11,33

1221K1,22

1123K1,22

• Calculate matrix elements

• Diagonalize matrices; report energy & wavefunction

• Program used– Polyad (Polik)

Hamiltonian matrix: 8718.167 -188.897 0.000 -80.388 -188.897 8255.922 -243.864 0.000 0.000 -243.864 7767.700 0.000 -80.388 0.000 0.000 8957.964

Eigenvalues and vectors (columns): 7661.582 8286.153 8764.315 8987.704 0.071 -0.375 0.857 -0.345 0.398 -0.838 -0.361 0.095 0.915 0.394 0.088 -0.019 0.004 -0.045 0.356 0.933

Compare to Manual MethodCompare to Manual Method

Summary of MethodSummary of Method

1. Compute equilibration geometry

2. Compute PES derivatives

3. Calculate vibrational spectroscopic constants

4. Identify important resonances; modify constants & calculate resonance constants

5. Compute excited vibrational states using polyad model

H2O Experimental Fits

0 5000 10000 15000

Observed Energy

HarmonicModel

AnharmonicModel

PolyadModel

H2O Polyad Model Calculations

0 5000 10000 15000

Observed Energy

VTZ/VTZ

AVQZ/VTZ

PolyadModel Fit

Interpretations and ConclusionsInterpretations and Conclusions

• Polyad model is useful and practical

– Experimental fits are excellent for predicting excited vibrational states (± 10 cm-1)

– Ab initio computation of excited states is relatively accurate (± 20 cm-1)

• Appropriate basis sets are

– AVQZ for harmonic force field

– VTZ for anharmonic force field

• Include resonances when K*HO/E>0.1~0.3

AcknowledgementsAcknowledgements

• Ruud van Ommen (Netherlands)

• Ben Ellingson (Univ of Minnesota)

• John Davisson (Hope College)

• Bob Field (MIT)

• Peter Taylor (Univ of Warwick)

• Research Corporation, Dreyfus Foundation, NSF

A Practical Procedure for ab initio Determination of Vibrational Spectroscopic Constants,...

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