vibrational spectroscopy basics

8/7/2019 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



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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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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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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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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

2"


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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



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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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vibrational spectroscopy basics

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