vibrational spectroscopy basics
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Deriva'onoftheFundamental
Equa'onsofVibra'onalSpectroscopy
RobertKalescky
SouthernMethodistUniversity
CATCO
March18,2011
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Overview
Lagrangian Overview CartesianandInternalCoordinates DisplacementCoordinates Rela'onshipBetweenInternalandCartesianCoordinates Kine'cEnergyinInternalCoordinates Poten'alEnergyinInternalCoordinates
Euler-LagrangeEqua'on Overview NewtonianMechanicsExample Vibra'onalEuler-LagrangeEqua'on PossibleSolu'ons NormalModeVectors NormalCoordinate BasicEqua'onofVibra'onalSpectroscopy
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LagrangianforVibra'onal
Spectroscopy Lagrangian
Differencebetweenkine'candpoten'alenergydescrip'ons.
Kine'cEnergy 3KCartesiandisplacement
coordinatevelocityelements.
Misa3Ksymmetricsquarematrixofatomicmasses.
Poten'alEnergy 3KCartesiandisplacement
coordinatexelements.
fisa3Ksymmetricsquarematrixofforceconstants. Thedotindicates
differen'a'onwithrespectto'me.
L(x, !x) =T( !x) !V(x)
=
1
2!xM!x ! 1
2xfx
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CartesianandInternalCoordinates
ExternalReferences Theposi'onoftheatoms
iswithrespecttoexternalreferencepointssuchas
thegridofCartesianspace. InternalReferences
Theposi'onofatomsarewithrespecttootheratomsinthemolecule.
Atomicposi'onsaredescribedusingbondlengthsandangles.
ExampleofanExternalReference
O -1.9 1.5 0.0
H -0.9 1.5 0.0
H -2.2 2.4 0.0
ExampleofanInternalReference
O
H 1 B1
H 1 B1 2 A2
B1 0.96
A2 104.5
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DisplacementCoordinates
DisplacementCoordinates Cartesiandisplacement
coordinatesarethe
differencebetweenacertainposi'onandthe
equilibriumposi'on.
Internaldisplacementcoordinatesarethe
differencebetweena
certaininternalcoordinate
anditsequilibriumvalue.
!x = x " xe# x
!r = r " re# r
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Poten'alEnergyofDisplacement
Poten'alEnergy Describesthepoten'al
energyofasystemconnectedwithsprings.
HookesLaw Analogoustotheintegrated
HookesLawequa'onwithrespecttox.
Displacement
Thepoten'alenergyiszerowhentheatomsareattheirequilibriumdistancefromeachotherandgreaterthanzerootherwise.
V(x) = 12
xfx
F= !kx"V=1
2kx
2
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Kine'cEnergyofDisplacement
Kine'cEnergy Afunc'ondescribingthe
kine'cenergyofa
vibra'ngmolecule. AtomicMo'on
Thevibra'ngmoleculesatomshaveakine'cenergypropor'onalto
thefrequencyoftheiroscilla'ons.
Analogoustomv2
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T(!x) =1
2!x M !x
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Rela'onshipBetweenInternaland
CartesianCoordinates TheBMatrix
Providesarela'onshipbetweeninternaland
Cartesiancoordinates. 3KLrinternal
displacementcoordinates.
3KxCartesiancoordinates.
Bisarectangular3Kby3KLmatrix.
Ithasnoinverse.
r = Bx
Bni=
!rn(x)
!xi
"#$
%&'x
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Kine'cEnergyinInternalCoordinates
Kine'cEnergyDescrip'on Becausethereisnoinverseof
B,thereisnodirectwaytoconvertkine'cenergydescrip'onusingMinto
internalcoordinates. TheGMatrix
TheGmatrixisthemassmatrixininternalcoordinates.
Itisa3KLsymmetricsquarematrix.
TheKMatrix TheKmatrixistheinverseof
theGmatrix.
Itisa3KLsymmetricsquarematrix.
T( x) = 1
2rK r
K=G1= BM
1B
1
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Poten'alEnergyinInternal
Coordinates Poten'alEnergy
Descrip'on
3KLrelements. TheFMatrix
Theforceconstantmatrixininternalcoordinates.
3KLsymmetricsquarematrix.
Eachelementisthe2ndderiva'veofthepoten'al
energy.
V(r) =1
2 rFr
Fij =
2V(r)rirj
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Overview
Lagrangian Overview CartesianandInternalCoordinates DisplacementCoordinates Rela'onshipBetweenInternalandCartesianCoordinates Kine'cEnergyinInternalCoordinates Poten'alEnergyinInternalCoordinates
Euler-LagrangeEqua'on Overview NewtonianMechanicsExample Vibra'onalEuler-LagrangeEqua'on PossibleSolu'ons NormalModeVectors NormalCoordinate BasicEqua'onofVibra'onalSpectroscopy
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TheEuler-LagrangeEqua'on
Lagrangian Thedifferenceofthekine'c
andpoten'alenergies.
TheLagrangianisamorekine'canddynamicdescrip'onversusthemorepoten'alandsta'cbasedHamiltoniandescrip'on.
Euler-Lagrange Thedynamicsofthevibra'ng
atomsinamoleculecanbefoundbysolvingthesystemofEuler-Lagrangeequa'onsfori=1,,3K
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L(x, x) =T( x) V(x)d
dt
L(x, x)dx
i
L(x, x)dx
i
= 0
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Euler-LagrangeExample
NewtonsLawsofMo5on
1. Abodyinmo'onstaysinmo'onun'lacteduponby
anexternalforce.
2. Abodyacteduponbyaforceaccelerates
propor'onally,F=ma.
3.
Forcesbetweenbodiesareequalandopposite,
Fa-b=-Fb-a
LagrangianMechanics
Lagrangianmechanicsisadifferentwayofmathema'callyexpressingNewtonianmechanics,
butthephysicsstaysthesame. Theprimaryadvantageofusing
theLagrangianisthatitisnotcoordinatesystemdependent. ChangingfromCartesian
coordinatestopolarcoordinatesforsomeNewtonianproblemscanbetedious.
AstheLagrangianisnotcoordinatesystemdependent,changingcoordinatesystemsforapar'culartypeofproblemaretrivial.
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Euler-LagrangeExample
PrincipleofLeastAc'on Thepaththrough
configura'onalspaceas
afunc'onof'meissuchthatac'onis
minimized.
TheLagrangianischosensuchthatthepathtaken
isthepathofleast
ac'onaccordingto
NewtonsLaws.
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L(x, !x) =T(!x) !V(x)
=
1
2m!x 2 !V(x)
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Euler-LagrangeExample
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L(x, x) =1
2mx2 V(x)
d
dt
L(x, x)dx
i
L(x, x)dx
i
= 0
d
dt
1
2mx2 V(x)
dxi
1
2mx2 V(x)
dxi
= 0
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Euler-LagrangeExample
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d
dt
1
2mx 2 V(x)
dxi
1
2mx 2 V(x)
dxi
= 0
d
dtmx V(x)
dxi
= 0
md
dtx V(x)
dxi
= 0
mxi+F
Vi= F
Ti+F
Vi= 0
FTi= F
Vi
2ndLaw:F=ma
3rdLaw:Fa-b=-Fb-a
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Euler-LagrangeEqua'on
Euler-Lagrange Thedynamicsofthe
vibra'ngatomsina
moleculecanbefoundbysolvingthesystemofEuler-
Lagrangeequa'onsfor
i=1,,3KL.
Thevibra'onalEuler-Lagrangeequa'onisfound
bysubs'tu'ngthe
vibra'onalLagrangianinto
theequa'on.
L(r,r) =T(r) V(r) = 12
rKr 12rFr
d
dt
L(r,r)r
i
L(r,r)r
i
= 0
d
dt
T(r) +V(r)r
i
T(r) +V(r)
ri
= 0
d
dt
T(r)
ri
V
(r)
ri
= 0
Kr +Fr = 0
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PossibleSolu'ons
Possiblesolu'onstheEuler-LagrangeEqua'on Systemof3KLsolu'ons. kandareappropriately
chosenconstants.
karevibra'onaleigenvaluesfromwhichtheharmonicfrequenciescanbedetermined.
ThelVector Contains3KLnormalmode
vectors.
Thepossiblesolu'onsaresubs'tutedintothedifferen'atedformoftheEuler-Lagrangeequa'on.
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Kr + Fr = 0ri= l
ikcos(2
k+ )
ri=
klikri
F kK[ ]lik = 0
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NormalModeVectors
NormalModeVectors Describethemo'onof
thevibra'onalnormal
modes.
Example:NormalModesofWater
Symmetricstretch. Bending. Asymmetricstretch.
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NormalCoordinate
Normalcoordinatesrefertothedisplacementofnucleifromtheirequilibriumposi'onsduringanormalmodevibra'on.
Anormalcoordinateisalinearcombina'onofmassweightedinternalorCartesiancoordinatedisplacements.
Thereisasinglenormalcoordinateforeachvibra'onalnormalmode.
Normalcoordinatesarerequiredforaquantummechanicalversusclassicaldescrip'onofmolecularvibra'ons.
Thekine'candpoten'alenergiesaresummedoveri=3KL.
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T(Q) = 12
!i !Qi2"V(Q) = 1
2Q
i
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BasicEqua'onofVibra'onal
Spectroscopy BasicEqua'onofVibra'onal
Spectroscopy Providesconnec'onbetweenthe
3K-Lnormalmodevectorsliandtheirfrequenciesvia.
isamatrixofwhichthediagonalelementsare3K-Lvibra'onaleigenvaluesfromwhichthevibra'onalharmonicfrequenciescanbedetermined.
Eisaunitmatrix. FinalEqua'on
Mul'plyfromthelebyK-1whichisG.
Thebracketedequa'onsareequivalent.
TheLmatrixcontainsthenormalmodeeigenvectors.
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F kK[ ]lik = 0
GF kE[ ]lik = 0
GFL = L
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Overview
Lagrangian Overview CartesianandInternalCoordinates DisplacementCoordinates Rela'onshipBetweenInternalandCartesianCoordinates Kine'cEnergyinInternalCoordinates Poten'alEnergyinInternalCoordinates
Euler-LagrangeEqua'on Overview NewtonianMechanicsExample Vibra'onalEuler-LagrangeEqua'on PossibleSolu'ons NormalModeVectors NormalCoordinate BasicEqua'onofVibra'onalSpectroscopy
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References
1. Kraka,E.;Cremer,D.Characteriza'onofCFBondswithMul'ple-BondCharacter:BondLengths,StretchingForceConstants,andBondDissocia'onEnergies.ChemPhysChem2009,10,686-698.
2. Ochterski,J.Vibra'onalanalysisinGaussian.GaussianInc.2000.3. Konkoli,Z.;Cremer,D.Anewwayofanalyzingvibra'onalspectra.I.Deriva'onofadiaba'c
internalmodes.Interna'onalJournalOfQuantumChemistry1998,67,1-9.
4. Cremer,D.;Larsson,J.Newdevelopmentsintheanalysisofvibra'onalspectraOntheuseofadiaba'cinternalvibra'onalmodes.Theore'calOrganicChemistry1998,5,259-327.
5. McQuarrie,D.A.;Simon,J.D.PhysicalChemistry;AMolecularApproach;UniversityScienceBooks:Sausalito,1997.
6. Atkins,P.W.;Friedman,R.S.MolecularQuantumMechanics;3rded.;OxfordUniversityPress:Oxford,1997.
7. Woodward,L.A.Introduc'ontotheTheoryofMolecularVibra'onsandVibra'onalTheory;OxfordUniversityPress:London,1972.
8. Gans,P.Vibra'ngMolecules;AnIntroduc'ontotheInterpreta'onofInfraredandRamanSpectra;ChapmanandHall:London,1971.
9. Wilson,E.B.;Decius,J.C.;Cross,P.C.MolecularVibra'ons;McGraw-HillBookCompany,Inc.:NewYork,19.Images
1. hp://www.phy.cuhk.edu.hk/contextual/heat/tep/trans/solid_state_model.gif2. hp://disc.sci.gsfc.nasa.gov/oceancolor/addi'onal/science-focus/ocean-color/images/47Z.jpg
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Ques'ons?
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