a practical procedure for ab initio determination of vibrational spectroscopic constants,...
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![Page 1: A Practical Procedure for ab initio Determination of Vibrational Spectroscopic Constants, Resonances, and Polyads William F. Polik Hope College, Holland,](https://reader035.vdocuments.us/reader035/viewer/2022062300/56649f4a5503460f94c6c46f/html5/thumbnails/1.jpg)
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
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Chemical Reactions Occur viaChemical Reactions Occur viaExcited Vibrational StatesExcited Vibrational States
Reaction Coordinate
Vibrational States
Reactants
Products
En
erg
y
Transition State
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HFCO Pure Vibrational SpectrumHFCO Pure Vibrational Spectrum
0 5000 10000 15000 20000
Inte
nsity
Frequency (cm-1)
31 HFCO
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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
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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
E
CCSD(T)/aug-cc-pVQZ
ωi xij K
POLYAD program
1 Δ
E
K
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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
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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
EE
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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 ,,
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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
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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)
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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
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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)
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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)
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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
,
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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
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• 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
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Compare to Manual MethodCompare to Manual Method
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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
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H2O Experimental Fits
-200
-100
0
100
200
0 5000 10000 15000
Observed Energy
Cal
c - O
bs E
nerg
y
HarmonicModel
AnharmonicModel
PolyadModel
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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
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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
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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