![Page 1: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/1.jpg)
Intensity, Frequency and Relaxation time
in the CH stretch overtones
Brant Billinghurst
![Page 2: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/2.jpg)
Summary
CH Overtone Intensities: TMA and DMS Structural information from CH
overtones: Metallocenes ICL-PARPS: A instrument for
determining the V-T relaxation of Overtone Vibration
![Page 3: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/3.jpg)
CH-Stretch Overtone Study of Trimethyl amine and Dimethyl Sulfide
Lone pair trans effect on TMA and DMS: Different CH bond lengths in methyl
group Different CH stretching frequencies Different intensities
Project goals: Measure the experimental intensities Compare with prediction of the (HCAO)LM
model
![Page 4: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/4.jpg)
Geometries
Gauche: 1.0847 ÅTrans: 1.0956 Å
Gauche: 1.0823 ÅTrans: 1.0832 Å
![Page 5: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/5.jpg)
HCAO/LM Model
Calculations: H. G. Kjaergaard and G. Low The Hamiltonian: 3 Morse oscillators Dipole moment function from Grid LM parameters from Birge-Spöner plots No coupling between methyl groups
![Page 6: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/6.jpg)
Experimental
The 1st through 4th overtones of Trimethyl amine d0,d3,d6,d8,and d9 Dimethyl Sulfide
All spectra: Collected on a Nicolet 870 FT-IR With a 10 m Gas cell
Curve fit analysis was done for the second through fourth overtones Win-IR software was used for all curve fitting In all cases correlation (R2) better then .99 was achieved
![Page 7: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/7.jpg)
Second Overtone TMA d8
|3,0>|0> Fermi Resonance*
|3,0>|0> Fermi Resonance*
|0,0>|3> Fermi Resonance*
|0,0>|3> Fermi Resonance*
|0,0>|3> FermiResonance*
|3,0>|0> Fermi Resonance*
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
0.11
0.12
8000 8500
Arbitrary units
![Page 8: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/8.jpg)
Second Overtone TMA
|2,0>+|1>, |2,0>-|1>, |1,1>|1>, |2,1>+|0>*
|3,0>|0> Fermi Resonance*
|3,0>|0> Fermi Resonance*
|3,0>|0> Fermi Resonance*
|1,0>+2>
|0,0>|3> Fermi Resonance*
|0,0>|3> Fermi Resonance*
|0,0>|3> FermiResonance*
|2,1>-|0>
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
0.30
8000 8500
Wavenumbers (cm-1)
![Page 9: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/9.jpg)
Second Overtone TMA d6
|2,0>-|1>, |1,1>|1>, |2,1>+|0>*
|2,0>+|1>
|3,0>|0> Fermi Resonance*
|3,0>|0> Fermi Resonance*
|3,0>|0> Fermi Resonance*
|1,0>-|2>
|1,0>+|2>
|0,0>|3> Fermi Resonance*
|0,0>|3> FermiResonance*
|0,0>|3> FermiResonance*
|2,1>-|0>
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0.018
0.020
0.022
0.024
0.026
0.028
0.030
0.032
0.034
0.036
0.038 0.040
8000 8500
Arbitrary units
![Page 10: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/10.jpg)
Third Overtone TMA
|3,1>-|0>, |2,1>+|1>, |2,1>-|1>, |2,2>|0>
|3,0>+|1>, |3,0>-|1>, |1,1>|2>, |3,1>+|0>
Fermi*
|4,0>|0>
Fermi*
|0,0>|4>
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050
0.055
0.060
0.065
0.070
0.075
0.080
10500 11000 11500
Wavenumbers (cm-1)
![Page 11: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/11.jpg)
Fourth Overtone TMA
|0,0>|5>|4,0>|1>, |1,1>|3>, |3,0>-|2>
Fermi*
|5,0>|0>
Fermi*
|1,0>|4>
Fermi*
|4,1>|0>, |2,1>|2>, |3,1>|1>, |2,2>|1>
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
0.0040
0.0045
0.0050
0.0055
0.0060
0.0065
0.0070
13000 14000
Wavenumbers (cm-1)
![Page 12: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/12.jpg)
Fourth Overtone DMS
See text
|0,0>|5>
|5,0>|0>
Fermi*
Fermi*
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
0.0040
0.0045
0.0050
0.0055
0.0060
0.0065
13500 14000
Wavenumbers (cm-1)
![Page 13: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/13.jpg)
Relative Intensities
Intensities: % of given overtone regionFor this discussion Intensities are reported on a per bond
basis L-L intensities given as a single value
![Page 14: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/14.jpg)
Comparison of the intensities of Trimethyl amine d8
0
5
10
15
20
25
30
35
40
Second OvertoneThird OvertoneFourth Overtone
Gauche ExperimentalGauche Calculated
Trans ExperimentalTrans Calculated
![Page 15: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/15.jpg)
Comparison of the Second Overtone intensities of Trimethyl
amine d0,d3,d6
0
5
10
15
20
25
30
N(CH3)3N(CH3)2(CD3)N(CD3)2(CH3)Calculated
Gauche Trans L-L
![Page 16: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/16.jpg)
Comparison of the Third Overtone intensities of Trimethyl amine
d0,d3,d6
0
10
20
30
40
N(CH3)3N(CH3)2(CD3)N(CD3)2(CH3)Calculated
Gauche
Trans
L-L
![Page 17: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/17.jpg)
Comparison of the Fourth Overtone intensities of Trimethyl amine
d0,d3,d6
0
10
20
30
40
N(CH3)3N(CH3)2(CD3)N(CD3)2(CH3)Calculated
Gauche Trans L-L
![Page 18: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/18.jpg)
Comparison of the intensities of Dimethyl Sulfide
10
20
30
40
Second OvertoneThird OvertoneFourth Overtone
Gauche Experimental
Gauche Calculated
Trans Experimental
Trans Calculated
L-L Experimental
L-L Calculated
![Page 19: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/19.jpg)
Summary
Spectra collected: 1st-4th overtones of TMA d0-d9 1st-4th overtones of DMS
Most peaks were assigned Predicted and experimental intensities
match well (HCAO)LM model showed bias towards
trans CH Possible evidence of coupling between the
methyl groups
![Page 20: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/20.jpg)
Metallocenes: Overtone Frequencies and C-H Bond length
Study 5 metallocenes 3 overtones observed rCH-CH correlation Goal: To determine
The effect of metal on CH bond length
Mg(C2H5)2 ionic ? If the combination bands are
brightened by metal
![Page 21: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/21.jpg)
Experimental
Spectra collected on an Nicolet Nexus 870Metallocenes in Carbon tetrachlorideSodium cyclopentadienyl in THF
The first and second overtones Metallocenes 1 cm path lengthSodium cyclopentadienyl 3mm path length
The third overtone Metallocenes 10 cm path lengthSodium cyclopentadienyl 3mm path length
Bond length: Gaussian 98 at the BLYP/hybrid level.
![Page 22: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/22.jpg)
First Overtone
NaCp-0.0
0.2
0.4
MgCp2 0.2
0.4
0.6
FeCp2 2
CoCp2-0.15
-0.10
NiCp2 1.0
RuCp2-0.8
-0.6
5500 6000
Wavenumbers (cm-1)
![Page 23: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/23.jpg)
Second Overtone
0.00
0.02
0.00
0.05
0.10
0.00
0.01
0.000
0.002
0.00
0.01
0.02
0.000
0.005
0.010
8500 9000 9500
Wavenumbers (cm-1)
NaCp
MgCp2
FeCp2
CoCp2
NiCp2
RuCp2
![Page 24: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/24.jpg)
Third Overtone
0.000
0.001
0.002
0.000
0.005
0.010
0.000
0.005
0.000
0.001
0.005
11200 11400 11600 11800 12000
Wavenumbers (cm-1)
NaCp
MgCp2
FeCp2
CoCp2
RuCp2
![Page 25: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/25.jpg)
Bond Length Frequency Correlations
5650 5750 5850 5950
1.074
1.079
1.084
1.089
Frequency (cm -1)
Bo
nd
len
gth
(Å
)
8200 8300 8400 8500 8600 8700 8800
1.074
1.079
1.084
1.089
Frequency (cm-1)
Bo
nd
len
gth
(Å
)
10600 10800 11000 11200 11400 11600
1.074
1.079
1.084
1.089
Frequency (cm -1)
Bo
nd
len
gth
(Å
)
5650 5750 5850 59501.090
1.095
1.100
1.105
Frequency (cm -1)
Bo
nd
len
gth
(Å
)
8200 8300 8400 8500 8600 8700 8800
1.094
1.099
1.104
Frequency (cm -1)
Bo
nd
len
gth
(Å
)
10600 10800 11000 11200 11400 11600
1.094
1.099
1.104
Frequency (cm -1)
Bo
nd
len
gth
(Å
)
First Overtone Second Overtone Third Overtone
HF
/6-3
11G
(d,p
)B
LY
P/6
-311
G(d
,p)
![Page 26: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/26.jpg)
Bond length Frequency Correlations
BasisBasisBasisCH SIR
I Error in I S Error in S R2 of Fit
HF/6-311G**
First Overtone 1.323 .01 4.16E-5 1.77E-06 0.9875
Second Overtone 1.287 .008 2.42E-5 8.81E-07 .09869
Third Overtone 1.273 .006 1.73E-5 5.59E-07 0.9876
BLYP
First Overtone 1.319 .01 3.58E-5 2.15E-06 0.9755
Second Overtone 1.270 .02 1.88E-5 2.1E-06 0.8891
Third Overtone 1.248 .02 1.23E-5 1.35E-06 0.8732
![Page 27: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/27.jpg)
Results
First Second Third BLYP
HF/6-311G**
BLYP HF/6-311G**
BLYP HF/6-311G**
BLYP Calc.
Mg(C5H5)2 1.071 1.085 1.071 1.085 1.072 1.083 1.087
Fe(C5H5)2 1.071 1.084 1.071 1.085 1.072 1.086 1.086
Co (C5H5)2 1.070 1.084 1.070 1.084 1.071 1.085 1.086
1.071 1.086 1.085
1.072 1.087 1.086
Ni (C5H5)2 1.071 1.084 1.070 1.084 1.085
Ru (C5H5)2 1.071 1.084 1.070 1.084 1.072 1.086 1.086
Na (C5H5)2 1.073 1.088 1.074 1.088 1.075 1.090 1.090
±.002 ±.002 ±.002 ±.003 ±.002 ±.003
![Page 28: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/28.jpg)
Summary
Combination bands:Not due to metalLikely due to aromatic character of Na(Cp)
Mg(Cp)2 is likely not ionic
The nature of metal has little effect on rCH
![Page 29: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/29.jpg)
V-T relaxation of Overtones
The phase shift of a PA signal can determine V-T relaxation timesLittle work on V-T relaxation of overtone vibrations.V-T relaxation is of interest because: Lazing of gases Chemical kinetics Transport properties
![Page 30: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/30.jpg)
Dealing with variables
Previous studies have been hampered by many variables that effect V-T relaxation. These include: Pressure Incident radiation intensity Presence of a buffer gas Cell design Electronics causing lag times Heat relaxation of the gas
The use of a wire as a reference to eliminate problems with many of these variables
![Page 31: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/31.jpg)
Cell design
![Page 32: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/32.jpg)
Experimental setup
![Page 33: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/33.jpg)
Flow Chart of ICL-PARPS
![Page 34: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/34.jpg)
ICL-PARPS Signal
![Page 35: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/35.jpg)
Possible Interpretations
Case 1: The wire takes longer to relax than V-T relaxationCase 2: V-T relaxation causes a phase shift > 180ºCase 3: Resonance causes “Inversion of phase shift”
![Page 36: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/36.jpg)
Test for Case 1
Signal of the heated wire with a 50 khz frequency
In theory the relaxation of the wire cannot takelonger than 0.00002 sec
![Page 37: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/37.jpg)
Analysis for Case 1
Negative apparent relaxations
|0,0>|6> < |6,0>|0>|0,0>|7> < |0,0>|6>All values < -0.00002 sec
0.0 2.5 5.0
-0.002
-0.001
-0.000
Relaxation time (sec)
TMA |0,0>|6> -0.0002+/- 0.0001TMA |6,0>|0> -0.00009 +/- 0.00002TMA |0,0>|7> -0.00023 +/- 0.00003Methane -0.0008 +/- 0.0004
1/Pressure (ATM)
Re
lax
ati
on
tim
e (
se
c)
tan
p
![Page 38: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/38.jpg)
Analysis for Case 2
0.0 2.5 5.0
0.002
0.003
Relaxation time (sec)
TMA |0,0>|6> -0.00005+/- 0.00003TMA |6,0>|0> -0.00025 +/- 0.000004TMA |0,0>|7> -0.000055 +/- 0.000004Methane 0.00011 +/- 0.00002
1/Pressure (ATM)
Re
lax
ati
on
tim
e (
se
c)
All relaxation times for TMA are negative
Positive relaxation time for Methane
3 6 0 oL H
p
![Page 39: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/39.jpg)
Analysis for Case 3
0 1 2 3 4 5 6 70.0000
0.0025
TMA |0,0>|6> 0.0002 +/- 0.0001TMA |6,0>|0> 0.00009 +/- 0.00002TMA |0,0>|7> 0.00023 +/- 0.00003Methane 0.0009 +/- 0.0004
Relaxation time (sec)
1/Pressure (ATM)
Re
lax
ati
on
tim
e (
se
c)
All relaxation times are positive|0,0>|6> > |6,0>|0>
|0,0>|6> < |0,0>|7> |6,0>|0> < |0,0>|7> Methane 450 Times
greater then what has been observed for the fundamental mode
tan ( )
p
![Page 40: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/40.jpg)
Conclusions and Future Work
Case 3 seems to be the correctMore experimentationError unacceptably high Replace resonance with a lock-in
amplifier Collect both signals simultaneously
Overall the system shows promise
![Page 41: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/41.jpg)
Acknowledgements
Supervisor:– Dr. K. M. Gough
Committee Dr. A. SeccoDr. TabiszDr. HenryDr. Wallace
My Family & FriendsMy fellow Graduate studentsThe Faculty and staff at the University of Manitoba
CollaboratorsDr. H. G. KjaergaardDr. G. LowDr. FedorovDr. SnavelyDr. T. Gough
Funding NSERCUMGFBrock award for Physical ChemistryMedicure
![Page 42: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/42.jpg)
(HCAO)LM Model Theory
The oscillator strength between the ground state g and excited state e is given by:
f cmDeg eg eg 4 7 0 2 22
.
Where:
eg eg e g
Is the frequency of the transition in wavenumbers
Is the dipole momment function
|e> and |g> are the vibrational wavefunctions
![Page 43: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/43.jpg)
LM Parameters
LM FrequencyAnharmonicityMazannares et.alFang et al.Mazannares et.alFang et al.Trimethyl Amine3074 4 cm-13085 26 cm-13069 5 cm-163 1 cm-164 6 cm-162 1 cm-1Gauche2892 8 cm-12877 15 cm-12915 8 cm-166 2 cm-165 3 cm-169 2 cm-1Trans
Dimethyl Sulfide3060 32 cm-13062 7 cm-154 5 cm-155 1 cm-1Gauche3038 34 cm-13070 4 cm-158 6 cm-162 2 cm-1Trans
The values shown here a larger difference in anharmonicity By using more values the previous work lower error was achievedAgreement with previous work is generally within experimental errorIn all cases the presence of Fermi resonance contributed to the error
![Page 44: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/44.jpg)
(HCAO)LM Model Theory
For a methyl group the Hamiltonian is that of three Morse oscillators
333233333
112122
21121
00\00|
0
~)(~~
~)(~)(/)(
hcEH
Where:
E0
0|00|
i
i~
ii~
Is the energy at the ground vibrational state
Is the vibrational quantum number
Is the LM frequency
Is the anharmonicity
![Page 45: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/45.jpg)
(HCAO)LM Model Theory
H hc a a a a a a a a a a a arain t, ,/ ( ) ( )11 2 1 2 1 2 1 3 1 3 1 3 2 3 2 3
a and a+ are annihilation and creation operators, with approximately step down and step up properties
The remaining terms are the coupling parameters
![Page 46: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/46.jpg)
(HCAO)LM Model Theory
The coupling parameters are
12 12 12 1, ( ) ~
1 3 1 3 1 3 1 3, ( ) ~ ~
Where ij
ij
ii jj
G
G G
1
2
0
0 0 ij
ij
ii jj
F
F F
1
2
0ijG
ijFAre elements of the G matrix
Are elements of the force matrix
![Page 47: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/47.jpg)
(HCAO)LM Model Theory
ijki j k
ijk
q q q1 2 3
ijk Is the derivative of the dipole moment multiplied by (1/i!j!k!), obtained from 2D grids of the dipole moment as a function of both (q1,q2) and (q1,q3)
q coordinates are displacements from equilibrium bond length
![Page 48: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/48.jpg)
Fermi Resonance
W W dn i n i 0 0
W is the perturbation function given by the anharmonic terms in the potential energy
E E W
W E Eno
n i
in i
0 0
W E Wn i n i 1
24
2 2
![Page 49: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/49.jpg)
Fermi Resonance
n n i
i n i
a bb a
0 0
0 0
aW
W
bW
W
n i
n i
n i
n i
4
2 4
4
2 4
2 2
2 2
1 2
2 2
2 2
1 2
/
/
=0 then 50/50 as increases approaches unperturbed
![Page 50: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/50.jpg)
First Overtone
|1,0>+|1>|2,0>+|0>|2,0>-|0>
|1,1>|0>
TMA d0
0.05
|1,0>+|1>
|2,0>+|0>|2,0>-|0>
|1,1>|0>
TMA d3
0.05
|1,0>+|1>
|2,0>|0>
|1,1>|0>
TMA D6
0.05
|2,0>|0>
TMA d8
0.0
0.1
|2,0>|0>|0,0>|2>|1,1>|0>
DMS
2
4
5600 5800 6000
Wavenumbers (cm-1)
![Page 51: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/51.jpg)
Second Overtone TMA d3
|2,1>-|0>
|2,0>+|1>, |2,0>-|1>, |1,1>|1>, |2,1>+|0>*
|3,0>|0> Fermi Resonance*
|3,0>|0> Fermi Resonance*
|3,0>|0> Fermi Resonance*
|1,0>+|2>
|0,0>|3> Fermi Resonance*
|0,0>|3> Fermi Resonance*
|0,0>|3> FermiResonance*
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
0.20
8000 8500
Wavenumbers (cm-1)
![Page 52: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/52.jpg)
Third Overtone TMA d8
|4,0>|0>
|0,0>|4>
0.0000
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0.0035
0.0040
0.0045
0.0050
0.0055
0.0060
0.0065
0.0070
0.0075
0.0080
0.0085
0.0090
10500 11000 11500
Wavenumbers (cm-1)
![Page 53: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/53.jpg)
Third Overtone TMA d3
|3,1>-|0>, |2,1>+|1>, |2,1>-|1>, |2,2>|0>
|3,0>+|1>, |3,0>-|1>, |1,1>|2>, |3,1>+|0>
|2,0>+|2>, |2,0>-|2>
|4,0>|0>
Fermi* |0,0>|4>
0.0000
0.0002
0.0004
0.0006
0.0008
0.0010
0.0012
0.0014
0.0016
0.0018
0.0020
0.0022
0.0024
10500 11000 11500 Wavenumbers (cm-1)
![Page 54: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/54.jpg)
Third Overtone TMA d6
|3,1>-|0>, |2,1>+|1>
|3,0>+|1>, |3,0>-|1>, |1,1>|2>, |3,1>+|0>
Fermi*
|4,0>|0>
Fermi*
Fermi*
|0,0>|4>
|2,1>-|0>, |2,2>|0>
0.000
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.010
0.011
0.012
10500 11000 11500
Wavenumbers (cm-1)
![Page 55: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/55.jpg)
Fourth Overtone TMA d3
Fermi*
|5,0>|0>
|1,0>|4>
|0,0>|5>
Fermi*
Fermi*
|4,1>-|0> +
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0.0008
0.0009
0.0010
0.0011
0.0012
0.0013
0.0014
13000 14000 Wavenumbers (cm-1)
![Page 56: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/56.jpg)
Fourth Overtone TMA d8
|5,0>|0>
|0,0>|5>
10 -4
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
13000 14000
Wavenumbers (cm-1)
![Page 57: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/57.jpg)
Fourth Overtone TMA d6
Fermi*
|5,0>|0>
Fermi*
|0,0>|5>
Fermi*
Fermi*
|4,1>-|0> +
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0.0008
0.0009
0.0010
0.0011
0.0012
13000 14000
Wavenumbers (cm-1)
![Page 58: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/58.jpg)
Second Overtone DMS
|1,0>-|2>
|2,0>-|1>
|2,0>+|1>
|2,1>-|0>,|1,0>-|2>
|2,1>+|0>
|0,0>|3>
|3,0>|0>
Fermi*
Fermi*
-0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
8400 8600 8800 Wavenumbers (cm-1)
![Page 59: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/59.jpg)
Third Overtone DMS
|1,0>+|3>*
|3,1>|0>, |3,0>|1>
|0,0>|4>
|4,0>|0>
Combination*
Combination*
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
0.045
0.050
0.055
0.060
10500 11000 11500
Wavenumbers (cm-1)
![Page 60: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/60.jpg)
Density Functional Theory
HF energy has the form:
E V hP PJ P PK PHF 1 2 1 2/ ( ) / ( )
V is the nuclear repulsion energy P is the density matrix<hP> is the one-electron energy1/2<PJ(P)> is the classical coulomb repulsion of the electrons-1/2<PK(P)> is the exchange energy
DFT energy has the form:
E V hP PJ P E P E PKS X C 1 2/ ( ) [ ] [ ]EX[P] is the exchange functionalEC[P] is the correlation functional
![Page 61: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/61.jpg)
Comparison of the intensities of Trimethyl amine d0
0
10
20
30
40
Second OvertoneThird OvertoneFourth Overtone
Gauche Experimental
Gauche Calculated
Trans Experimental
Trans Calculated
L-L Experimental
L-L Calculated
![Page 62: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/62.jpg)
Excitation of the acoustic wave
q
k E NC
B Ie
v c
i t
12 2
0
2 2 1 222
1
/( / )
2 01B I For the most common case
![Page 63: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/63.jpg)
Energy Transfer Physics
The collisional deexcitation rate is given by
c Z Pi j A B i j
ZA Bnumber of kinetic collisions per cm3 persecond.
P i j probability of energy transfer
P ei j
u E
k Tc
2
2
1 3
/
u = relative velocity
![Page 64: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/64.jpg)
Helmholtz Resonator Cell
c
A
lV Ar
1 2 2 21 2
4/
/
c =speed of sound l =length of the channel
Vr=VVc/V+Vc V =tube volume Vc =cavity volume
=viscosity of the gas =mass density of the gas
A = area of the channel
![Page 65: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/65.jpg)
Test: Equivalence of Resonance
There is some difference between the sides The difference is not significantThe difference also varies and is likely not due to a lack of symmetry
Side #1 Side #2 Phase
Freq. Amp. Phase Amp. Phase Diff.
560.036 11.536 266.061 11.792 266.829 0.768
560.035 11.516 265.943 11.935 266.650 0.707
560.037 11.543 265.978 11.671 266.907 0.929
560.036 11.482 265.689 11.743 266.819 1.13
![Page 66: Intensity, Frequency and Relaxation time in the CH stretch overtones Brant Billinghurst](https://reader033.vdocument.in/reader033/viewer/2022061305/55144785550346494e8b4c80/html5/thumbnails/66.jpg)
Effect of Voltage
Amplitude increases with voltageIncrease is not linearNo systematic change of phase with voltagePhases do differ between trialsThe difference is less for the phase differences
Heated Reference Wire Laser Induced PhaseDiff.Volt. Freq. Amp. Phase Freq. Amp. Phase
10 345.02 59.31 24.63 345.05 14.74 -26.46 -51.08
1 345.02 40.99 22.94 345.02 18.24 -29.45 -52.40
0.7 345.02 21.36 19.79 345.02 19.34 -32.30 -52.10
0.5 345.02 11.10 20.79 345.02 18.86 -30.49 -51.29