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

En

erg

y

Transition State

HFCO Pure Vibrational SpectrumHFCO Pure Vibrational Spectrum

0 5000 10000 15000 20000

Inte

nsity

Frequency (cm-1)

31 HFCO

Vibrational State ModelsVibrational State Models

Harmonic Anharmonic Polyad

iivE

i ji

jiijii vvxvE 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

Calculation MethodCalculation Method

1. Compute equilibration geometry

2. Compute PES derivatives

3. Calculate spectroscopic constants

4. Identify important resonances

5. Compute excited vibrational states

lkji

4

kji

3

ji

2

qqqq

E

qqq

E

qq

E

CCSD(T)/aug-cc-pVQZ

ωi xij K

POLYAD program

1 Δ

E

K

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)

...!4

1

!3

1

!2

1

,,, 0

4

,, 0

3

, 0

2

0

0

lkjilkji

lkjikjikji

kji

jiji

jiii

i

qqqqqqqq

Eqqq

qqq

E

qqqq

Eq

q

EEE

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

q q 0

E( q)

E(q)

Δq

Δq)E(Δq)E(

q

E

2

ijklijkij

!4

1

!3

1

!2

1

,,, 0

4

,, 0

3

, 0

2

lkjilkji

lkjikjikji

kjijiji

jiqqqq

qqqq

Eqqq

qqq

Eqq

qq

EE

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)

lkji

lkjiijklkji

kjiijkii

iN qqqqqqqqqqqE,,,

241

,,612

21

632,1 ,,

ji

jiiji

iiN vvxvEvvvE 21

21

21

0632,1 ,,

Spectroscopic ConstantsSpectroscopic Constants

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

k kil

kiiik

iiiiiix 22

222

416

38

16

k kjikjikjikji

kjikijk

k i

j

j

iij

k

jjkiikiijjij Bx

2222

2

2

44

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)

k kik

kiiik

iiiiiix 22

222

416

38

16

k kjikjikjikji

kjikijk

k i

j

j

iij

k

jjkiikiijjij Bx

2222

2

2

44

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

iiii

k kik

kiiik

iiiiiix

411

32

1

16

416

38

16

2

22

222

k kjikjikjikjiijk

k i

j

j

iij

k

jjkiikiijj

k kjikjikjikji

kjikijk

k i

j

j

iij

k

jjkiikiijjij

B

Bx

1111

8

1

44

2

44

2

2

2222

2

partial fraction expansion

drop resonance term(s)

Resonance ConstantsResonance Constants

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

ijkiijjiiijkijkkij kKkK 21

,,

m mjjmjjmiimiikkmiim

m mkimkiikm

ki

kiikiikk

m mkmimkkmiim

m kim

mikm

ki

kiikiikkkkii

B

BK

1111

16

1

11

4

1

4

4

1

4

1

8

1

2

1

4

222

2222

22

222

,

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

25

• 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

-200

-100

0

100

200

0 5000 10000 15000

Observed Energy

Cal

c - O

bs E

nerg

y

HarmonicModel

AnharmonicModel

PolyadModel

H2O Polyad Model Calculations

-40

-20

0

20

40

60

80

100

0 5000 10000 15000

Observed Energy

Cal

c - O

bs E

nerg

y

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