AppendixA9:ContactTransformationandPerturbationMethods335 Index 341
CHAPTER1
THEVIBRATION-ROTATIONPROBLEM
Contents
1.1 ClassicalKineticEnergy1
1.1.1 TheEckartConditions3
1.1.2 TransformationtoNormalCoordinates5
1.1.3 KineticEnergyinTermsofMomenta8
1.2 TheQuantumMechanicalHamiltonian10
1.2.1 LinearMolecules12
References 12
1.1ClassicalKineticEnergy
Inthischaptersymbolslike rj and _ r j areusedtorepresentcoordinateandvelocity vectorsofthe jthparticleinamolecule,andsymbolslike rjα and _ r j α ,with α 5 x, y, z, fortheircomponents,whenapplicable.
Thekineticenergyofamoleculeis
where N isthenumberofparticles,includingnucleiandelectrons,and r 0j and r0j are vectorsdeterminingthevelocityandpositionofthe jthparticle,ofmass mj,with respecttoareferential X0,Y0,Z0 fixedintheexternalspace.
Eachpositionvector r0j isthesumofthevectors R,whichdefinesthepositionof themolecularmasscenterrelativetotheoriginofthe X0,Y0,Z0 space-fixedsystem, and rj,whichdefinesthepositionofthe jthparticlewithrespecttothemasscenter (see Fig.1.1).Inanonrotatingvibratingmolecule,thevelocityofthe jthparticleis _ R 1 _ r j ,where _ r j isthevelocityofthe jthparticleduetoitsmotioninthemoleculefixedsystem(vibrationaldisplacementsofthenucleiandorbitalmotionsofthe electrons).
Ifthemoleculeundergoesanoverallrotation,atanangularvelocity ω aboutan axispassingthroughitsmasscenter,thevelocityofits jthparticleinthespace-fixed referencesystemis _ R 1 _ r j 1 ω 3 rj ,thereforethekineticenergyis:
Figure1.1 Positionandvelocityofaparticle Pj inamolecule.Theoriginandorientationofthe Cartesiansystem X0, Y0, Z0 arefixedintheexternalspace.WedefineanadditionalCartesiansystem X, Y, Z,withtheoriginfixedatthemasscenter O ofthemoleculeandtravelingwithit,butremainingalwaysparalleltothespace-fixedsystem,andamolecule-fixedsystem x, y, z,withitsoriginin O,whichtravelsandrotateswiththemolecule.Seetextforfurtherdetails.
where n isthenumberofnuclei, nel isthenumberofelectrons, mj isthemassofthe jthnucleus,and mel isthemassoftheelectrons.Oneobtainsbyexpansion:
where M isthemolecularmass(inclusiveofnucleiandelectrons), n isthenumberof nuclei, nel isthenumberofelectrons,and N isthetotalnumberofparticles n 1 nel
Thelasttwotermsin (1.3) vanish.Thefirstofthembecausethedefinitionofmass centerimpliesthatthequantity PN i51 mi ri vanishesatanytime,thereforeitschangesin timemustalsovanish,andthen PN i51 mi _ r i 5 0.Thesecondone,exploitingthepropertiesofthetriplescalarproduct(seeMargenauandMurphy [1],Chapter4:Symmetry ofwavefunctionsinvibration-rotationspectroscopy),canbewrittenintheequivalent form(PN i51 mi ri ) ( _ R 3 ω ),anditclearlyvanishesonaccountofthementioned propertyofthemasscenter.
contains R or R .Thustherotation-vibration-electronic(rovibronicforshort)kinetic energy Tevr canbewrittenasthesumofthetwoterms, Tvr and Ter,whichrepresent thekineticenergiesofthenucleiandelectrons,respectively,inarotatingmolecule:
Inmoleculeswithoneequilibriumgeometry,itisconvenienttodecomposethe positionvector rj ofanucleus,inthemolecule-fixedframe,asthesumofthevalue aj intheequilibriumpositionandthedisplacement dj causedbythevibrationalmotions (see Fig.1.1).Theequilibriumvaluesareconstant,therefore _ r j 5 _ d j .Weshallmake thissubstitutionin Eq.(1.5)
Fornonlinearmoleculestherearesixconstraints,becausetherearethreein Eq.(1.7),for x, y and z,andthreein Eq.(1.8).Forlinearmoleculestherearefive constraints,becausethe Eq.(1.8) relativetothe z-component(theinternuclearaxis)is trivial(the x and y componentsof aj arealwayszero,andthisthirdequationdoesnot imposeanyrestrictiontothepossiblevibrationalcoordinates).Thus,ina n-atomic moleculesthereare3n 6vibrationalmodesifthegeometryisnotlinear,and3n 5 ifthegeometryislinear.
SeeEqs.(3.5)and(3.6)fortheelementsofthetensor I Ifwereplacethenuclearvibrationaldisplacementcoordinates dαj (α 5 x, y, z)in themolecule-fixedframebytheirmass-weightedvalues qαj 5 mj 1/2dαj,thevibrationrotationkineticenergyexpression(1.9) canbewrittenreplacingthevectors dj and _ d j by qj and _ qj ,anddroppingoutthemasses mj.Theadoptionofmass-weightedcoordinatesissuggestedbythefactthatthevibrationalnormalmodesarealsomass-weighted coordinates.
1.1.2TransformationtoNormalCoordinates
Therelationbetweenmass-weightedCartesianandnormalcoordinatesislinear,and canbewritteninthematrixnotationas ~ Q tot 5 l~ q ,where ~ Q tot and ~ q arecolumnmatrices,and l isanorthonormalmatrix.Thevector ~ Q tot containsthevibrationalnormal coordinatesandtherigidmotionsofthemolecule,thatis,translationsandrotations. Thus,wecanwriteinmatrixform
where α 5 x, y, z identifiestherotationalcomponent Rα.Thisistherelationbetween mass-weightedCartesiandisplacementcoordinates,arrangedinacolumnandordered by x, y,and z components,andthecolumnincludingvibrationalnormalcoordinates ~ Q ,translationalcoordinates T , ,androtationalcoordinates ~ R .Assumingthatthetranslationalcoordinatestooaremassweighted(displacementsofthemasscentermultiplied bythesquarerootofthemolecularmass M),weobtain:
lxy;j 52 mj Ixx 1=2 azj ; lxz;j 5
=2 ayj andcyclicpermutations ð1 12Þ
Ixx
withreferencetoasystemofprincipalinertiaaxes.In Eq.(1.12) therotationalcoordinatestoohavedimensions m 1/2l,andaregivenbyanangulardisplacementmultiplied bythesquarerootoftheappropriatemomentofinertia.Theequilibriumcoordinates aj intheseequationsarenotmassweighted.SeeMealandPolo [6,7] andPapouˇ sek andAliev [3].
Fromrelations (1.11) and (1.12) itcanbeverifiedthatallthecomponentsof T , and ~ R vanish,iftheEckart Sayvetzconditionsin Eqs.(1.7) and (1.8) areobeyed. Thus,owingtotheorthonormalityofthematrix l (l ~ l 5 E ),wecanwrite
Thisequationsallowsonetoreplacein (1.9) the x, y, z componentsof dj and _ d j bythe Q’sand _ Q’s(rememberthat dja 5 mj 1/2 qja),andweobtain
Thecoefficients ζ α i;k arecalledCorioliscoefficientsaboutthemolecule-fixed α-axis,betweenthevibrationalnormalmodes Qi and Qk.Theirvaluesrepresentthe α-componentsofthevectorproductofthenormalmodes i and k.Obviously,the diagonalcoefficients(i 5 k)vanish,andthesecoefficientscanbearrangedin ζ α
Notethatour I correspondsto I I el ofPapouˇ sekandAliev [3].
1.1.3KineticEnergyinTermsofMomenta
Beforeapproachingthequantummechanicaltreatment,itisconvenienttoexpressthe kineticenergyintermsofmomenta,insteadofvelocities.Asoursystemisconservative,themomentaarethederivativesofthekineticenergy (1.20) or (1.22),with respecttothevelocities _ Qk , ω α,and qr .Oneobtains:
with α 5 x, y, z,where π0 α hasbeendefinedin (1.18),and
From (1.24) and (1.25) wecanseethatthefirstandlasttermsof Eq.(1.22) are sumsofthesquaresof Pk and pr,therefore
Nowweshouldreplacetheangularvelocitiesbytheirassociatedmomenta,but Eq.(1.26),with α 5 x, y, z arenotsatisfactoryforthispurpose,becauseoftheoccurrenceof π0 α and Π 0 α whichcontain _ Qk and _ qr ,insteadoftheconjugatedmomenta,see Eqs.(1.18)and(1.27)
From Eqs.(1.18)and(1.24) and (1.1),(1.25),and(1.27) wefind:
with
Intheabsenceofmolecularrotation, πα and Π α areequalto π0 α and Π 0 α
From (1.29) onefinds Jα
,andinmatrix notation
whereanarrowrepresentsavector(columnmatrix)and μ 5 I 0 1.Theelementsof μ, asthoseof I 0 ,arefunctionsofthenormalcoordinates,anddonotdependonthe coordinatesoftheelectrons.
Rα Rβ ,wherethepure rotationalangularmomentum R isequaltothetotalangularmomentum J minus thecontributions π and Π ,generatedbythevibrationalandelectronicmotion, respectively.
therotationalangularmomentumonlyinthelimitedcasethatthementionedcontributionscanbedisregarded.Thisisconsistentwiththefactthat,inamolecule-fixed frame,therotational-typeoperator J isindependentoftheinternalorbitalandvibrationaloperators Π and π,thentheuncoupledrepresentation ψeψvψr,where ψe includestheelectronspin,doesactuallyexist.Thiswouldnotbeallowedforthetrue rotationaloperator R,seeSectionsA3.1andA3.2.Thusacommonvibronicbasis, inclusiveofnuclearspin,is jeijv ij J ; k; M i ni j ,wherethelasttermisanuclearspin function,seeSection5,Nuclearspinstatisticalweights.
1.2.1LinearMolecules
Forlinearmolecules,where z istheinternuclearaxis, ω z vanishesand Eq.(1.29) appliesonlytothexand y components. I 0 isa2 3 2matrix,whichhasbeenfoundto bediagonal,with I 0 xx 5 I 0 yy,evenaccountingforthedependenceofitselementson thenormalcoordinates(seeRefs. [15,16]).Therefore ω α 5 (Jα
Watson [15] hasshownthatthequantummechanicalrovibronicHamiltonian H evr canbecastinthesameformastheclassicalexpression (1.39) ,replacingthe momentabythecorrespondingoperators,byintroducingafictitiousangleofrotation χ aboutthe z -axis.Contrarytononlinearmolecules,thisangleisnotavariable ofmotion,butisakindofphaseangledeterminingtheorientationinspaceofthe x and y axes.
wherethewavefunctionsappearascombinationsofstationarywavefunctions,with coefficientsparametricallydependingontime.Thus,applicationofthetimedependentSchrödinger Eq.(2.1),with H 5 H(0) 1 H0 ,yields H 0 ðÞ X n cn ðt ÞΨ ð0Þ n ð~ r ; t Þ 1 H
Thefirstandlasttermsofthisequationareequalandcancelout,becausethetimedependentSchrödingerequationholdsevenintheabsenceoftheperturbation H0 ,for any n-state,thus
:3Þ
Theprobabilityamplitudethatthesystemisinthestate Ψ ð0Þ n ð~ r ; t Þ attime t isequal to| cn ðt Þ |2.
Multiplyingbothsidesof (2.3) by Ψ ð0Þ m ð~ r ; t Þ ontheleft,integratingoverthespatial coordinatesandapplyingtheorthonormalityrelationoftheeigenfunctionsof H(0) , Ð ~ r ψ
0Þ l ð~ r Þ ψð0Þ n ð~ r Þd~ r 5 δ l ;n ,weobtain: dcm ðt Þ dt 52 i h X n cn ðt Þ
Theindex m specifiesaneigenfunctionof H(0),andwecanwriteasmanyequationsasthenumberofsucheigenfunctionsusedintheexpansions (2.2).Thusthe problemisinprincipledetermined,andthevaluesofthecoefficients cn(t)canbe determinedatanyvalueof t,forgiveninitialconditions(thatis,knowingthatthesystemisinagivenstate Ψ ð0Þ m ð~ r ; t Þ whentheperturbation H0 isswitchedon),thatis cm(t 5 0) 5 1, cn¼m(t 5 0) 5 0.SeeRefs. [1,2]
F 5 eE 1 1 c _ r 3 H ð2:5Þ where r isthepositionvectorofthecharge, _ r isitsvelocity(accordingtotheconventionthatadotonasymbolrepresentsitstimederivative),and c isthespeedofthe lightinthevacuum.Wecall e theelectricchargebecausewearemainlyinterestedin theelectroncharge,butthetreatmentisgeneral.
Figure2.1 Vectors E, H, A,and k ofapropagatingelectromagneticfield.Seetextforfurtherdetails.
Eq.(2.5) includestheeffectsoftheelectricfield E andtheLorentzforce.
Asanexample,weshowin Fig.2.1 avectorpotentialassociatedwitharadiation propagatinginthe z-directionandoscillatingalong x.Theassociatedelectricand magneticfieldscanbedeterminedfromthetwoequationsabove.Theelectricfield oscillatesinthesamedirectionof A,whereasthemagneticfieldoscillatesinadirection normaltothoseof A and E andofthewavepropagationvector k.Thefields E and H havethesamephase,andadephasing π/2withrespectto A.Theamplitudesof E and H arerelatedtothatof A,asindicatedinthefigure.
Thus,using (2.6) and (2.7),theforceactingonthechargecanberewritteninthe form
.Similarexpansionsholdfor thetermsin y and z,andtheHamiltonianbecomes:
Δ
2.3ASystemofChargedParticlesinaRadiationField
Ifweconsidertheelectromagneticfieldassociatedwitharadiationfield,thenthescalarpotential φ iszeroandthedivergenceof A vanishes(seeRef. [4]).Moreover,normallytheradiationfieldrepresentsaweakperturbation,andthesquareterm A A can bedisregarded.
Thereforewecanwriteforasystemofchargedparticles
where Aj isthevector A atthepositionofthe jthcharge.
