Notes on:

Molecular Physics

2004

Prof. W. Ubachs

Vrije Universiteit Amsterdam

1. Introduction

1.1 Textbooks

Therearea numbera textbooksto berecommendedfor thosewho wish to studymolecularspectroscopy; the best ones are:

1)The series of books by Gerhard HerzbergMolecular Spectra and Molecular StructureI. Spectra of Diatomic MoleculesII. Infrared and Raman Spectroscopy of Polyatomic MoleculesIII. Electronic Spectra of Polyatomic Molecules

2)Peter F. BernathSpectra of Atoms and Molecules

3)Philip R. Bunker and Per JensenMolecular Symmetry and Spectroscopy, 2nd edition

4)Hélène Lefebvre-Brion and Robert W. FieldPerturbations in the Spectra of Diatomic Molecules

A recent very good book is that of:5) John Brown and Alan CarringtonRotational Spectroscopy of Diatomic Molecules

1.2 Some general remarks on the spectra of molecules

Molecules are different from atoms:- Apart from electronictransitions,alwaysassociatedwith thespectraof atoms,alsopurelyvibrationalor rotationaltransitionscanoccur.Thesetransitionsarerelatedto radiationbymultipolemoments,similar to thecaseof atoms.While in atomsa redistributionof theelec-tronic chargeoccursin a moleculethe transitioncanoccurthrougha permanentdipolemo-ment related to the charges of the nuclei.- Superimposedon thespectrallinesrelatedto electronictransitions,thereis alwaysa rovi-brationalstructure,thatmakesthemolecularspectramuchricher.In thecaseof polyatomicsthreedifferentmomentsof inertiagiveriseto rotationalspectra,in diatomicsonly asinglero-tationalcomponent.Eachmoleculehas3n-6 vibrationaldegreesof freedomwheren is thenumber of atoms.- Atomscanionizeandionizationcontinuaarecontinuousquantumstatesthatneedto becon-sidered.In molecules,in addition,therearecontinuumstatesassociatedwith thedissociationof themolecule.Boundstatescancouple,throughsomeinteraction,to thecontinuaasaresultof which they (pre)-dissociate.

- 5 -

1.3 Some examples of Molecular Spectra

The first spectrum is that of iodine vapour. It shows resolved vibrational bands, recorded bythe classical photographic technique, in the so-called B3Π0u

+ - X1Σg+ system observed in

absorption; the light features signify intense absorption. The discrete lines are the resolvedvibrations in the excited state going over to the dissociative continuum at point C. Leftwardof point C the spectrum looks like a continuum but this is an effect of the poor resolution.This spectrum demonstrates that indeed absorption is possible (in this case strong) to thecontinuum quantum state.

Later the absorption spectrum was reinvestigated by Fourier-transform spectroscopy result-ing in the important iodine-atlas covering the range 500-800 nm. There is several lines ineach cm-1 interval and the numbers are well-documented and often used as a reference forwavelength calibration. Note that the resolution is determined by two effects: (1) Dopplerbroadening and (2) unresolved hyperfine structure. The figure shows only a small part of theiodine atlas of Gerstenkorn and Luc.

- 6 -

The hyperfine structure can be resolved when Doppler-free laser spectroscopic techniques areinvoked. The following spectrum is recorded with saturation spectroscopy. A single rotationalline of a certain band is shown to consist of 21 hyperfine components. These are related to theangular momentum of the two I=5/2 nuclei in the I2 molecule.

Usually molecular spectra appear as regular progressions of lines. In the vibrational bands ofdiatomic molecules the rotational lines are in first order at equal separation. If a quantum stateis perturbed that may be clearly visible in the spectrum. This is demonstrated in the spectra oftwo bands of the SiO molecule in the H1Σ+- X1Σ+ system. The upper spectrum pertains to the(0,0) band and is unperturbed; the lower one of the (1,0) band clearly shown perturbation of therotational structure.

-3000 -2000 -1000Frequency, MHz

R17(16-1)

*

- 7 -

2. Energy levels in molecules; the quantum structure

2.1. The Born-Oppenheimer approximation

The Hamiltonian for a system of nuclei and electrons can be written as:

wherethesummationi refersto theelectronsandA to thenuclei.Thefirst termon therightcorrespondsto thekinetic energyof theelectrons,thesecondtermto thekinetic energyofthenucleiandthethird termto theCoulombenergy,dueto theelectrostaticattractionandrepulsion between the electrons and nuclei. The potential energy term is equal to:

Thenegativetermsrepresentattraction,while thepositivetermsrepresentCoulomb-repul-sion.Notethata treatmentwith this Hamiltoniangivesa non-relativisticdescriptionof themolecule, in which also all spin-effects have been ignored.

Now assumethat thewavefunctionof theentiremolecularsystemis separableandcanbe written as:

whereψel representstheelectronicwavefunctionandχnuc thewavefunctionof thenuclearmotion.In thisdescriptionit is assumedthattheelectronicwavefunctioncanbecalculatedfor a particular nuclear distanceR. Then:

TheBorn-Oppenheimerapproximationnowentailsthatthederivativeof theelectronicwavefunctionwith respectto thenuclearcoordinatesis small,so is negligibly small. Inwordsthis meansthatthenucleicanbeconsideredstationary,andtheelectronsadapttheirpositionsinstantaneouslyto thepotentialfield of thenuclei.Thejustificationfor this origi-natesin thefactthatthemassof theelectronsis severalthousandtimessmallerthanthemassof thenuclei. IndeedtheBO-approximationis the leastappropriatefor the light H2-mole-cule.

If we insert the separable wave function in the wave equation:

then it follows:

H"

2

2m------- ∇i

2

i∑–

"2

2M A-----------∇A

2

A∑– V R r( , )+=

V R r( , )Z Ae

2

4πε0rAi-------------------

A i,∑–

Z AZBe2

4πε0 RA RB–-----------------------------------

A B>∑ e

2

4πε0rij-----------------

i j>∑+ +=

Ψmol ri RA( , ) ψel ri R;( )χnuc R( )=

∇i2ψelχnuc χnuc∇i

2ψel=

∇A2 ψelχnuc ψel∇A

2 χnuc 2 ∇Aψel( ) ∇Aχnuc( ) χnuc∇A2 ψel+ +=

∇Aψel

HΨ EΨ=

- 8 -

The wave equation for the electronic part can be written separately and solved:

for eachvalueof R. Theresultingelectronicenergycanthenbeinsertedin thewaveequationdescribing the nuclear motion:

Wehavenowin acertainsensetwo separateproblemsrelatedto two waveequations.Thefirstrelatesto theelectronicpart,wherethegoal is to find theelectronicwavefunctionandanenergy . This energyis relatedto theelectronicstructureof themoleculeanalo-gouslyto thatof atoms.Notethatherewedealwith an(infinite) seriesof energylevels,agroundstateandexcitedstates,dependentontheconfigurationsof all electrons.By searchingtheeigenvaluesof theelectronicwaveequationfor eachvalueof R wefind a functionfor theelectronicenergy, rather than a single value.

Solution of the nuclear part then gives the eigen functions and eigen energies:

In theBO-approximationthenucleiaretreatedasbeinginfinitely heavy.As aconsequencethepossibleisotopicspecies(HCl andDCl) havethesamepotentialin theBO-picture.Alsoall cou-plings between electronic and rotational motion is neglected (e.g.Λ-doubling).

2.2. Potential energy curves

The electrostatic repulsion between the positively charged nuclei:

is a functionof the internucleardistance(s)just astheelectronicenergy.Thesetwo termscanbe taken together in a single function representing the potential energy of the nuclear motion:

HΨmol χnuc"

2

2m------- ∇i

2

i∑– e

2

4πε0rij-----------------

i j>∑

Z Ae2

4πε0rAi-------------------

A i,∑–+

ψel +=

ψel

Z AZBe2

4πε0 RA RB–-----------------------------------

A B>∑ "

2

2M A-----------∇A

2

A∑–

χnuc+ EtotalΨmol=

"2

2m------- ∇i

2

i∑– e

2

4πε0rij-----------------

i j>∑

Z Ae2

4πε0rAi-------------------

A i,∑–+

ψel ri R;( ) Eel R( )ψel ri R;( )=

"2

2M A-----------∇A

2

A∑–

Z AZBe2

4πε0 RA RB–-----------------------------------

A B>∑+

χnuc R( ) Eel R( )χnuc R( )+ Etotalχnuc R( )=

ψel ri R;( )Eel R( )

χnuc R( )

Enuc Etotal Eel R( )– Evib Erot+= =

V N R( )Z AZBe

2

4πε0 RA RB–-----------------------------------

A B>∑=

V R( ) V nuc R( ) Eel R( )+=

- 9 -

In thecaseof a diatomthevector-charactercanberemoved;thereis only a singleinternu-clear distance between two atomic nuclei.

In thefigure belowa few potentialenergycurvesaredisplayed,for groundandexcitedstates. Note that:

- atsmallinternuclearseparationtheenergyis alwayslarge,dueto theedominantroleofthe nuclear repulsion

- it is not always so that de electronic ground state corresponds to a bound state- electronically excited states can be bound.

Electronictransitionscantakeplace,justasin theatom,if theelectronicconfigurationin themoleculechanges.In that casethereis a transitionform onepotentialenergycurvein themoleculeto anotherpotentialenergycurve.Sucha transitionis accompaniedby absorptionor emissionof radiation;it doesnotmakeadifferencewhetheror not thestateis bound.Thebinding (chemical binding) refers to the motion of the nuclei.

2.3. Rotational motion in a diatomic molecule

Staringpoint is dewaveequationfor thenuclearmotionin deBorn-Oppenheimerapproxi-mation:

where,justasin thecaseof thehydrogenatomtheproblemis transferredto oneof areducedmass. Note thatµ represents now the reduced mass of the nuclear motion:

Beforesearchingfor solutionsit is interestingto considerthesimilarity betweenthis waveequationandthatof thehydrogenatom.If a 1/R potentialis insertedthenthesolutions(ei-genvaluesandeigenfunctions)of thehydrogenatomwould follow. Only thewavefunction

hasa differentmeaning:it representsthemotionof thenucleiin a diatomicmole-

RR

VV

"2

2µ------∆

R– V R( )+ χnuc R( ) Eχnuc R( )=

µM AMB

M A MB+-----------------------=

χnuc R( )

- 10 -

cule.In generalwe do not know thepreciseform of thepotentialfunctionV(R) andalsoit isnot infinitely deep as in the hydrogen atom.

Analogouslyto thetreatmentof thehydrogenatomwecanproceedby writing theLaplacianin spherical coordinates:

Now a vector-operatorN can be defined with the properties of an angular momentum:

The Laplacian can then be written as:

The Hamiltonian can then be reduced to:

Becausethispotentialis only afunctionof internuclearseparationR, theonly operatorwith an-gulardependenceis theangularmomentumN2, analogouslyto L2 in thehydrogenatom.Theangular dependent part can again be separated and we know the solutions:

Theeigenfunctionsfor theseparatedangularpartarethusrepresentedby thewell-knownspher-ical harmonics:

and the wave function for the molecular Hamiltonian:

Inserting this function gives us an equation for the radial part:

∆R

1

R2

------R∂

∂R

2

R∂∂

1

R2 θsin

-----------------θ∂

∂ θsin θ∂∂

1

R2 θsin

-----------------φ2

2

∂∂

+ +=

N x i" yz∂

∂z

x∂∂

– – i" φsin θ∂

∂ θcot φcos φ∂∂

+ = =

N y i" zx∂

∂x

z∂∂

– – i" φcos–( ) θ∂

∂ θcot φsin φ∂∂

+ = =

N z i" xy∂

∂y

x∂∂

– – i" φ∂

∂= =

∆R

1

R2

------R∂

∂R

2

R∂∂

N

2

"2R

2------------–=

"2

2µR2

-------------R∂

∂R

2

R∂∂

–

1

2µR2

-------------N2

V R( )+ + χnuc R( ) Eχnuc R( )=

N2

N M,| ⟩ "2N N 1+( ) N M,| ⟩= with N 0 1 2 3 etc, , , ,=

N z N M,| ⟩ "M N M,| ⟩= with M N– N– 1 N,+,=

N M,| ⟩ Y NM θ φ,( )=

χnuc R( ) ℜ R( )Y NM θ φ,( )=

- 11 -

Now the wave equation has no partial derivatives, only one variableR is left.

2.4. The rigid rotor

Now assumethat themoleculeconsistsof two atomsrigidly connectedto eachother.Thatmeansthat the internuclearseparationremainsconstant,e.g.at a valueRe. Sincethe zeropointof apotentialenergycanbearbitrarilychosenwechooseV(Re)=0.Thewaveequationreduces to:

The eigenvalues follow immediately:

whereB is definedastherotational constant. Hencea ladderof rotationalenergylevelsap-pearsin a diatom.Notethattheseparationbetweenthelevelsis not constant,but increaseswith the rotational quantum numberN.ForanHCl moleculetheinternuclearseparationis Re=0.129nm;this follows from theanal-ysis of energy levels. Deduce that the rotational constants 10.34 cm-1.

This analysis gives also the isotopic scaling for the rotational levels of an isotope:

2.5 The elastic rotor; centrifugal distortion

In anelasticrotor R is no longerconstantbut increaseswith increasingamountof rotationasa resultof centrifugalforces.This effect is knownascentrifugal distortion. An estimateof this effectcanbeobtainedfrom a simpleclassicalpicture.As themoleculestretchesthe

"2

2µR2

-------------Rd

dR

2

Rdd

– N N 1+( ) V R( )+ + ℜ R( ) EvNℜ R( )=

1

2µRe2

---------------N2

χnuc Re( ) Erotχnuc Re( )=

EN"

2

2µRe2

---------------N N 1+( ) BN N 1+( )= =

N

B1µ---∝

- 12 -

centrifugalforceFc is, at somenewequilibriumdistanceRe’ , balancedby theelasticbindingforceFe, which is harmonic. The centripetal and elastic forces are:

By equatingFc=Fe and by assuming it follows:

The expression for the rotational energy including the centrifugal effect is obtained from:

Now useRe’ for theaboveequationsandexpandingthe first termof theenergyexpressionitfollows:

ThequantummechanicalHamiltonianis obtainedby replacingN by thequantummechanicaloperator . It is clearthatthesphericalharmonicsYNM(Ω) arealsosolutionsof thatHamilto-nian. the result for the rotational energy can be expressed as:

where:

is thecentrifugaldistortionconstant.Thisconstantis quitesmall,e.g.5.32x 10-4 cm-1 in HCl,but its effect can be quite large for high rotational angular momentum states (N4 dependence).Selection rules for the elastic rotor are the same as for the rigid rotor (see later).

2.6. Vibrational motion in a non-rotating diatomic molecule

If wesettheangularmomentumN equalto 0 in theSchrödingerequationfor theradialpartandintroduceafunctionQ(R)with thanasomewhatsimplerexpressionresults:

Thisequationcannotbesolvedstraightforwardlybecausetheexactshapeof thepotentialV(R)is notknown.Forboundstatesof amoleculethepotentialfunctioncanbeapproximatedwith aquadraticfunction.Particularlynearthebottomof thepotentialwell thatapproximationis valid(see figure).

Fc µω2Re′=

N2

µRe′3--------------≅ Fe k Re′ Re–( )=

Re′ Re≈

Re′ Re– N2

µkRe3

-------------=

EN

2

2µRe′2-----------------

12--- Re′ Re–( )+=

EN

2

2µRe2

------------- 1 N2

µkRe4

------------+ 2–

12--- N

4

µ2kRe

6---------------+ N

2

2µRe2

------------- N4

2µ2kRe

6------------------– +≅=

N

EN BN N 1+( ) DN2

N 1+( )2–=

D4Be

3

ωe2

---------=

ℜ R( ) Q R( ) R⁄=

"2

2µ------

R2

2

d

d– V R( )+ Q R( ) EvibQ R( )=

- 13 -

Near the minimumR=Re a Taylor-expansion can be made, where we useρ = R - Re:

and:

Hereagainthezerofor thepotentialenergycanbechosenat Re. Thefirst derivativeis 0 attheminimumandk is thespringconstantof thevibrationalmotion.Thewaveequationre-ducesto theknownproblemof the1-dimensionalquantummechanicalharmonicoscillator:

The solutions for the eigenfunctions are known:

whereHv are the Hermite polynomials; de energy eigenvalues are:

with the quantum numberv that runs over valuesv=0,1,2,3.

Fromthis we learnthatthevibrationallevelsin a moleculeareequidistantandthatthereisacontributionform azeropointvibration.Theaveragedinternucleardistancecanbecalcu-latedfor eachvibrationalquantumstatewith . Theseexpectationvaluesareplottedin thefigure.Notethatathighvibrationalquantumnumbersthelargestdensityis attheclas-sical turning points of the oscillator.

Re

V(R)

V R( ) V Re( )Rd

dVRe

ρ 12---

R2

2

d

d V

Re

ρ2+ + +=

V Re( ) 0=Rd

dVRe

0=R

2

2

d

d V

Re

k=

"2

2µ------

ρ2

2

d

d–

12---kρ2

+ Q ρ( ) EvibQ ρ( )=

Qv ρ( ) 2v 2⁄– α1 4⁄

v!π1 4⁄------------------------ 12---αρ2

exp Hv αρ( )= with αµωe

"----------= ωe

kµ---=

Evib "ωe v12---+

=

Qv ρ( ) 2

- 14 -

The isotopic scaling for the vibrational constant is

Note also that the zero point vibrational energy is different for the isotopes.

2.7. Anharmonicity in the vibrational motion

The anharmonic vibrator can be represented with a potential function:

On thebasisof energiesandwavefunctionsof theharmonicoscillator,that canbeusedasafirst approximation,quantummechanicalperturbationtheorycanbeappliedto find energylev-els for the anharmonic oscillator (with parametersk‘ andk‘‘):

In the usual spectroscopic practice an expansion is written (in cm-1),

with ωe, ωexe, ωeye andωeze to be consideredasspectroscopicconstants, that canbe deter-mined from experiment.Note that for theanharmonicoscillatortheseparationbetweenvibrationallevelsis no longerconstant.In the figure below the potentialandthe vibrational levelsfor the H2-moleculeareshown.

ωe1

µ-------∝

V ρ( ) 12---kρ2

k'ρ3k''ρ4

+ +=

Evib "ωe v12---+

154------ k'

2

"ωe---------- "

µωe----------

3v

2v

1130------+ +

– O k''( )+=

G v( ) ωe v12---+

ωexe v12---+

2– ωeye v

12---+

3ωeze v

12---+

4+ + +=

- 15 -

H2 has14 boundvibrationallevels.Theshadedareaabovethedissociationlimit containsacontinuumof states.The moleculecanoccupythis continuumstate!For D2 thereare17bound vibrational states.

A potentialenergyfunctionthatoftenresemblestheshapeof boundelectronicstatepoten-tials is theMorse Potential defined as:

wherethethreeparameterscanbeadjustedto thetruepotentialfor acertainmolecule.Onecanverify thatthis potentialis not sogoodat . By solvingtheSchrödingerequationwith this potential one can derive the spectroscopic constants:

The energies of the rovibrational levels then follow via the equation:

Anotherprocedurethatis oftenusedfor representingtherovibrationalenergylevelswithina certain electronic state of a molecule is that of Dunham, first proposed in 1932:

In this proceduretheparametersYkl arefit to theexperimentallydeterminedenergylevels;theparametersaretobeconsideredamathematicalrepresentation,ratherthanconstantswith

V R( ) De 1 ea R Re–( )–

–[ ]2

=

r ∞→

ωea

2π------

2De

µ---------= xe

"ωe

4De----------= αe 6

xeBe3

ωe------------- 6

Be2

ωe------–= Be

"2

2µRe2

-------------=

EvN ωe v12---+

xeωe v12---+

2– BeN N 1+( ) DeN

2N 1+( )2

– αe v12---+

N N 1+( )–+ +=

EvN Y kl v12---+

kN

lN 1+( )l

k l,∑=

- 16 -

aphysicalmeaning.neverthelessarelationcanbeestablishedbetweentheYkl andthemolecularparametersBe, De, etc. In approximation it holds:

2.8. Energy levels in a diatomic molecule: electronic, vibrational and rotational

In amoleculethereareelectronicenergylevels,justasin anatom,determinedby theconfigu-rationof orbitals.Superimposedonthatelectronicstructurethereexistsastructureof vibration-al and rotational levels as depicted in the figure.

Transitionsbetweenlevelscanoccur,e.g.via electricdipole transitions,accompaniedby ab-sorptionor emissionof photons.Justasin thecaseof atomsthereexistselectionrulesthatde-termine which transitions are allowed.

2.9. The RKR-procedure

Thequestionis if thereexistsaprocedureto deriveapotentialenergycurveform themeasure-mentson theenergylevelsfor acertainelectronicstate.Suchaprocedure,which is theinverseof aSchrödingerequationdoesexistandis calledtheRKR-procedure,afterRydberg,Klein andRees.

Y 10 ωe≈ Y 01 Be≈ Y 20 ωexe–≈ Y 11 αe≈ Y 30 ωeye≈

- 17 -

3. Transitions between quantum states

3.1. Radiative transitions in molecules

In asimplepictureamoleculeactsin thesamewayuponincidentelectromagneticradiationasanatom.Themultipolecomponentsof theelectromagneticfield interactswith thechargedistributionin thesystem.Again themostprominenteffect is theelectricdipoletransition.In amoleculewith transitionsin theinfraredandevenfar-infraredtheelectricdipoleapprox-imationis evenmorevalid, sinceit dependson theinequality.Thewavelengthλ of thera-diation is much longer than the size of the moleculed:

In the dipole approximation a dipole momentµ interacts with the electric field vector:

In aquantummechanicaldescriptionradiativetransitionsaretreatedwith a "transitionmo-ment"Mif defined as:

Thismatrixelementis relatedto thestrengthof a transitionthroughtheEinsteincoefficientfor absorption is:

Very generallythe Wigner-Eckarttheoremcanbe usedto makesomepredictionson al-lowedtransitionsandselectionrules.Thedipoleoperatoris an -vector,soa tensorof rank1. If thewavefunctionshavesomehowadependenceonaradialpartandanangularpartthetheorem shows how to separate these parts:

In thedescriptionthetensorof rank1 q cantakethevalues0, -1 and+1; this correspondswith x, y, andz directionsof thevector.In all casestheWigner-3jsymbolhasa valueune-qualto 0, if ∆J=0, -1 and+1.This is ageneralselectionrule following if J is anangularmo-mentum:

Therule ∆M=0 only holdsfor q=0, soif thepolarisationis alongtheprojectionof thefieldaxis.

2πλ

------d 1«

H int µ E⋅ er E⋅= =

M fi Ψ f µ E⋅ Ψi⟨ ⟩=

B fi ω( ) πe2

3ε0"2

-------------- Ψ f µ E⋅ Ψi⟨ ⟩2

=

r

γJM rq1( ) γ 'J'M'⟨ ⟩ 1–( )J M– J 1 J'

M– q M'γJ r

1( ) γ 'J'⟨ ⟩=

∆J J' J– 1– 0 1, ,= =

J J' 0= = forbidden

∆M M' M– 1– 0 1, ,= =

- 18 -

3.1. Two kinds of dipole moments: atoms and molecules

In atomsthereis nodipolemoment.Neverthelessradiativetransitionscanoccurvia atransitiondipolemoment;thiscanbeunderstoodasareorientationor relocationof electronsin thesystemasa resultof a radiativetransition.Moleculesaredifferent; theycanhavea permanentdipolemoment as well. The dipole moment can be written as:

WhereeandN referto theelectronsandthenuclei.In factdipolemomentscanalsobecreatedby the motion of the nuclei, particularly through the vibrational motion, giving rise to:

wherethefirst termis theelectronictransitiondipole,similar to theonein atoms,thesecondisthe permanent or rotating dipole moment and the third is the vibrating dipole moment.

3.2. The Franck-Condon principle

Herewe investigateif thereis a selectionrule for vibrationalquantumnumbersin electronictransitions in a diatom. If we neglect rotation the wave function can be written as:

The transition matrix element for an electronic dipole transition between statesΨ’ andΨ’’ is:

Notethaton theleft sidewithin theintegralthereappearsa complexconjugatedfunction.Thedipole moment contains an electronic part and a nuclear part (see above). Insertion yields:

If two differentelectronicstatesψ’el andψ’’ el areconcernedthenthesecondtermcancels,be-causeelectronicstatesareorthogonal.Note: it is thesecondtermthatgivesriseto purevibra-tional transitions(alsopurerotationaltransitions)within an electronicstateof the molecule.Here we are interested in electronic transitions. We write the electronic transition moment:

In first approximationthiscanbeconsideredindependentof internucleardistanceR. This is theFranck-Condonapproximation,or theFranck-Condonprinciple.As a resultthetransitionma-

µ µe µN+ er ii

∑– eZARAA∑+= =

µ µ0 Rdd µ

Re

ρ 12---

R2

2

d

d µ

ρ2+ + +=

Ψmol r i RA( , ) ψel r i R;( )ψvib R( )=

µi f Ψ'µΨ'' τd∫=

µi f ψ'elψ'vib µe µN+( )ψ''elψ''vib r Rdd∫= =

ψ'elµeψ''el rd∫( )ψ'vibψ''vib Rd∫= ψ'elψ''el r ψ'vibµNψ''vib∫d Rd∫+

Me R( ) ψ'elµeψ''el rd∫=

- 19 -

trix element of an electronic transition is then:

The intensityof a transitionis proportionalto the squareof the transitionmatrix element,hence:

SotheFranck-Condonprinciplegivesusaselectionrulefor vibrationalquantumnumbersinelectronictransitions.The intensityis equalto theoverlapintegralof thevibrationalwavefunctionof groundandexcitedstates.Thisoverlapintegralis calledtheFrank-Condonfac-tor. It is not a strict selection rule forbidding transitions!

3.3. Vibrational transitions: infrared spectra

In theanalysisof FC-factorsthesecondtermin theexpressionfor thedipolematrixelementwas not further considered. This term:

reduces,in caseof asingleelectronicstate(thefirst integralequals1 becauseof orthogonal-ity) it can be written as:

wherethefirst termrepresentsthepermanentdipolemomentof themolecule.In higheror-derapproximationin avibratingmoleculeinduceddipolemomentsplayarole,but thesearegenerally weaker.

µif Me R( ) ψ'vibψ''vib Rd∫=

I µif2

v' v''⟨ | ⟩ 2∝ ∝

µif ψ'elψ''el r ψ'vibµNψ''vib∫d Rd∫=

v'⟨ |µvib v''| ⟩ v'⟨ | aρ bρ2+ +( ) v''| ⟩=

- 20 -

An importantconsequenceis thatin ahomonuclearmoleculethereexistsnodipolemoment,, so there isno vibrational or infrared spectrum!

If we proceedwith theapproximationof a harmonicoscillatorthenwe canusetheknownwavefunctionsQv(ρ) to calculateintensitiesin transitionsbetweenstateswith quantumnum-bersvk andvn:

form which a selection rule follows for purely vibrational transitions:

In caseof ananharmonicoscillator,or in caseof aninduceddipolemomentso-calledovertonetransitions occur. Then:

Theseovertonetransitionsaregenerallyweakerbyafactorof 100thanthefundamentalinfraredbands.

Note that vibrational transitionsarenot transitionsinvolving a simplechangeof vibrationalquantumnumber.In vibrationaltransitionstheselectionrulesfor therotationalor angularpartmust be satisfied (see below).

3.4. Rotational transitions

Inducedby thepermanentdipolemomentradiativetransitionscanoccurfor whichtheelectron-ic aswell asthevibrationalquantumnumbersarenotaffected.Thetransitionmomentfor atran-sition between states and can be written as:

where the states represent wave functions:

Theprojectionof thedipolemomentontotheelectricfield vector(thequantizationaxis)canbewritten in vector form (in spherical coordinates) in the space-fixed coordinate frame:

Thefact thatthevectorcanbeexpressedin termsof a simplesphericalharmonicfunctionY1m

µvib 0=

n⟨ |ρ k| ⟩ Qn ρ( )ρQk ρ( ) ρd∫ "µω------- n

2---δk n 1–,

n 1+2

------------δk n 1+,+= =

∆v v' v– 1±= =

∆v 1± 2± etc, ,=

NM| ⟩ N′M′| ⟩

M fi ΨN′M′ µ E⋅ ΨNM⟨ ⟩=

NM| ⟩ ΨNM ρθφ( ) ψeR ρ( )Y NM Ω( )= =

µ µ0

θ φcossin

θ φsinsin

θcos

µ0Y 1m∝=

- 21 -

allows for a simple calculation of the transition moment integral:

This gives only a non-zero result if:

Sorotationaltransitionshaveto obeytheseselectionrules.Thesameholdsfor thevibration-al transitions.

3.5. Rotation spectra

The energy expression for rotational energy levels, including centrifugal distortion, is:

Hereweadopttheusualconventionthatgroundstatelevelsaredenotedwith N" andexcitedstatelevelswith N’. Thesubscriptν refersto thevibrationalquantumnumberof thestate.Then we can express rotational transition between ground and excited states as:

AssumeN’=N"+1 for absorption:

If the centrifugal absorption is neglected and an equally spaced sequence of lines is found:

The centrifugal distortion causes the slight deviation from equally separated lines.

Notethatin apurerotationspectrumthereareonly absorbingtransitionsfor which∆N=N’-N"= 1, so in the R-branch (see below).

M fi µ0 YN′M′°θ φcossin

θ φsinsin

θcos

YNM ΩdΩ∫ YN′M′°Y1mYNM Ωd

0

π∫0

2π∫∝=

14π------ 2N′ 1+( )3 2N 1+( ) N′ 1 N

0 0 0

N′ 1 N

M′ m M=

∆N N′ N– 1±= =

∆M M′ M– 0 1±,= =

Fv BvN N 1+( ) DvN2

N 1+( )2–=

ν Fν N′( ) Fν N′( )–=

BvN′ N′ 1+( ) DvN′2 N′ 1+( )2– BvN′ N′ 1+( ) DvN′2 N′ 1+( )2

–[ ]–( )=

νabs Bv N″ 1+( ) N″ 2+( ) N″ N″ 1+( )–[ ] Dv N″ 1+( )2N″ 2+( )2

N″2N″ 1+( )2

–[ ]–=

2Bν N″ 1+( ) 4Dν N″ 1+( )3–=

νabs N″( ) νabs N″ 1–( )– 2Bν=

- 22 -

3.6. Rovibrational spectra

Now theterm values, or the energies, are defined as:

For transitions one finds the transition energies:

Here is the so-calledbandorigin, the rotationlesstransition.Note thatthere is no line at this origin. So:

Now thedifferentbranchesof atransitioncanbedefined.TheR-branchrelatesto transitionforwhich∆N=1.Notethatthisdefinitionmeansthattherotationalquantumnumberof theexcitedstateis alwayshigherby 1 quantum,irrespectiveof thefact thatthetransitioncanrelateto ab-sorptionor emission.With neglectof thecentrifugaldistortiononefinds thetransitionsin theR-branch:

Similarly transitionsin theP-branch,definedas∆N=-1transitions,canbecalculated,againwithneglect of centrifugal distortion:

Now the spacing between the lines is roughly 2B; more precisely:

wherethestatementon theright holdsif Bv’ < Bv". Hencethespacingin theP-branchis largerin theusualcasethat therotationalconstantin thegroundstateis larger.Thereis a pile up oflines in the R-branchthat caneventuallylead to the formationof a bandhead, i.e. the pointwhere a reversal occurs.An energylevel diagramfor rovibrationaltransitionsis shownin thefollowing figure.Wherethespacingbetweenlinesis 2B thespacingbetweentheR(0)andP(1)linesis 4B. Hencethereis aband gap at the origin.

T G ν( ) Fν N( )+=

Fν N( ) BvN N 1+( ) DvN2

N 1+( )2–=

G ν( ) ωe ν 12---+

ωexe ν 12---+

2–=

ν″ ν′→

σ ν′ ν″–( ) Fν′ N′( ) Fν″ N″( )– G ν′( ) G ν″( )–+=

σ0 G ν′( ) G ν″( )–=

σ ν′ ν″–( ) σ0 Fν′ N′( ) Fν″ N″( )–+=

σR σ0 Bν′ N 1+( ) N 2+( ) Bν″N N 1+( )–+=

σ0 2Bν′ 3Bν′ Bν″–( )N Bν′ Bν″–( )N2

+ + +=

σP σ0 Bν′ Bν″+( )N– Bν′ Bν″–( )N2

+=

σR N 1+( ) σR N( )– 3Bν′ Bν″–∼ so 2Bν′<

σP N 1+( ) σP N( )– Bν′ Bν″+∼ so 2Bν′>

3.7. Rovibronic spectra

If therearetwo differentelectronicstatesinvolvedrovibronictransitionscanoccur,i.e.tran-sitionswheretheelectronicconfiguration,thevibrationalaswell astherotationalquantumnumberschange.Transitionsbetweena lower electronicstateA anda higherexcitedstateB as in the following scheme can take place:

Possibletransitionsbetweenthelowerandexcitedstatehaveto obeytheselectionrules,in-cluding the Franck-Condon principle. Transitions can be calculated:

012

3

012

3

σ0

R0

R1

R2

P1P2

P3

ν"

ν'

R-branch P-branch

N"

N’

TA

TB

T’term valueof excited state

T"term valueof lower state

- 24 -

AgainRandP branchescanbedefinedin thesamewayasfor vibrationaltransitionswith tran-sition energies:

But now the constantshavea slightly different meaning:σ0 is the bandorigin including theelectronicandvibrationalenergies,andtherotationalconstantsBv’ andBv" pertainto electron-ically excited and lower states. If now we substitute:

Thenwe obtainan equationthat is fulfilled by the lines in the R branchaswell asin the P-branch:

This is a quadratic function in m; if we assume thatB’ < B", as is usually the case, then:

aparabolaresultsthatrepresentstheenergyrepresentationsof RandP branches.Suchaparab-ola is calleda Fortrat diagramor a Fortrat Parabola. Thefigure showsonefor a singlerovi-bronic band in the CN radical at 388.3 nm.

ν T′ T″–=

T′ TB G′ v′( )– F′ N′( )+=

T″ TA G″ v″( )– F″ N″( )+=

σR σ0 2Bν′ 3Bν′ Bν″–( )N Bν′ Bν″–( )N2

+ + +=

σP σ0 Bν′ Bν″+( )N– Bν′ Bν″–( )N2

+=

m N 1+= for R branch–

m N–= for P branch–

σ σ0 Bν′ Bν″+( )m Bν′ Bν″–( )m2+ +=

σ σ0 αm βm2

–+=

- 25 -

Notethatthereis noline for m=0; this impliesthatagainthereis aband gap. Fromsuchfig-ureswe candeducethat therealwaysis a bandheadformation,eitherin theR-branchor intheP-branch.In thecaseof CN in thespectrumabovethebandheadformsin theP-branch.The bandhead can easily be calculated, assuming that it is in theP-branch:

It follows that the bandhead is formed at:

3.8. Population distribution

If line intensitiesin bandsare to be calculatedthe populationdistribution over quantumstateshasto beaccountedfor. Fromstatisticalthermodynamicsapartitionfunctionfollowsfor populationof statesat certainenergiesundertheconditionof thermodynamicequilibri-um. In caseof Maxwell-BoltzmannstatisticstheprobabilityP(v) of finding a moleculeinquantum state with vibrational quantum numberv is:

When filling in the vibrational energy it follows:

whereN is theZustandssumme,andkT is expressedin cm-1. As oftenin statisticalphysics(ergodictheorem)P(v) canbeinterpretedasa probabilityor a distribution.As anexampleP(v) is plotted as a function ofv in the following figure.

At eachtemperaturetheratioof moleculesin thefirst excitedstateoverthosein theground

Nd

dσP B′ B″+( )– 2N B′ B″–( )+ 0= =

NB′ B″+B′ B″–------------------=

P v( ) eE v( )( ) kT( )⁄–

eE v′( )( ) kT( )⁄–

v′∑-------------------------------------=

P v( ) 1N----e

ωe v12---+

–

kT---------------------------

ωexe v32---+

kT-----------------------------+

=

- 26 -

statecanbecalculated.P(v=1)/P(v=0)is listedin theTablefor severalmoleculesfor 300Kand1000K.

In caseof thedistributionoverrotationalstatesthedegeneracyof therotationalstatesneedstobeconsidered.Everystate has(2J+1) substates . Hencethepartitionfunctionbecomes:

In the figure the rotationalpopulationdistributionof theHCl moleculeis plotted.Note that itdoesnotpeakatJ=0. Thepeakvalueis temperaturedependentandcanbefoundthroughsetting:

J| ⟩ JM| ⟩

P J( ) 2J 1+( )eErot kT( )⁄–

2J′ 1+( )eErot kT( )⁄–

J′∑-----------------------------------------------------

1N rot---------- 2J 1+( )e

BJ J 1+( )– DJ2 J 1+( )2+= =

Jdd

P J( ) 0=

Herzberg fig 59

- 27 -

4. High vibrational levels in the WKB-approximation

4.1. The Wentzel Kramers Brillouin approximation

The WentzelKramersBrillouin (WKB) approximationis a tool to solve the one-dimensionalSchrödingerequation.Both, wavefunctionsandenergylevelscanbedeterminedfor a givenpo-tential.Beforecomputerswerein commonuse,thisapproximationbelongedto thestandardtopicsin basicquantummechaniccourses,but afterthe1960’sthis approximationreceivedlessandlessattention.Recently,however,theWKB approximationgainedinterestagain,dueto developmentsin thefield of coldatoms.TheWKB is wieldedto determinethebindingenergyof theuppervibra-tional levelsin diatomicmoleculeswhich is importantfor thevalueof thescatteringlengthfor s-wave collisions; an important parameter for experiments with cold atoms.In molecularphysicsthis approximationcanaid in theinvestigationof vibrationallevelsclosetoadissociationlimit andcangivetunnelingrateconstantsin thecaseof, for instance,auto-ionizationor pre-dissociation.Besidesthis, theWKB approximationis widely applicablefor a lot of quantummechanicalprob-lems,provideda potentialcurveis known,andcanhelpto interpretphysicalphenomenain somedetail. Therefore, this approximation is presented in this chapter.

Thisandthefollowing threeparagraphsarebasedon thebook:Quantum Mechanics, by E. Merz-bacher (1964).

Thederivationstartswith theone-dimensionalSchrödingerequation.Thevariableis chosento beR rather thanx, as this is the symbol for the internuclear distance in diatomic molecules.

with µ as the reduced mass.In the case thatV = constant, the equation is easily solved and the solutions are

with

In the following only k will beused,whetherE > V or E < V. In the lattercasea purelypositiveimaginary number is assumed fork.If V = V(R) the solutions can be written in the form

The problem is now to obtain the functionu(R). Substitute this wave function and also

d2Ψ

dR2

----------2µ"

2------ E V–[ ]Ψ+ 0=

Ψ eikR±

= for E V>

Ψ eκR±

= for E V<

k2µ"

2------ E V–[ ]

1 2⁄

= κ 2µ"

2------ V E–[ ]

1 2⁄

=

Ψ R( ) eiu R( )±

=

k R( ) 2µ"

2------ E V R( )–[ ]

1 2⁄

=

- 28 -

in the Schrödinger equation to get

In the case ofV(R) = V = constant,k(R) = k = constant, the solution is

The second derivative (the first term) is then zero.Omit the first term to obtain a zeroth order approximation

which has the solution

This solution will be starting point in an iteration procedure

And u1 will then become

At this point onelikes to stoptheprocedure.This is only valid, however,if u1 ≅ u0, that is if thefirst order solution is almost the exact one. The conditionu1 ≅ u0 is fulfilled when

If this is the case, the square root may be rewritten andu1 becomes

Using this, the wave functionΨ can be determined.

Theconstantof integrationC1 affectsonly thenormalizationof Ψ andwill not beconsideredinthe following.

4.2. Breakdown of the approximation criterion

The criterion for the approximation was

This breaks down whenR is close to a classical turning point whereE = V and hencek = 0. The

dudR-------

2k R( )[ ]2

id

2u

dR2

---------+=

u R( ) kR=

du0

dR---------

2

k R( )[ ]2=

u0 k R( ) R C0+d∫±=

dun 1+

dR----------------

2

k R( )[ ]2id

2un

dR2

-----------+=

u1 k2

R( ) i± k′ R( ) R C1+d∫±=

k′ k2

«

u1 k R( ) 1i2---± k′ R( )

k2

R( )--------------

R C1+d∫±=

k R( ) Rd∫ i2--- k R( ) C1+log+±=

Ψ iu R( )[ ]exp∼ 1

k R( )------------------ i k R( ) Rd∫±[ ]exp=

k′ k2

«

- 29 -

strategyto find asolutionfor thewholerangeof R consistsof twoparts.First,find theWKB-solutionswherevertheapproximationis valid (i.e. far from theturningpoints).Second,findawavefunctionwhich is valid aroundtheturningpointsandis alsovalid in acertainregionwhereonecanalsousetheWKB solutions.Now, we havesolutionsoverthewholerange,andall wehaveto do is to tie thedifferentsolutionstogetherto find thetotalwavefunction.To do this, it is necessaryto havea certainregionwhereboth, the WKB solutionandthesolution around the turning points, are valid. This is schematically depicted below.WKB standsfor theregionswheretheapproximationis valid andCF for theregionswheretheso-called‘connectionformulas’haveto befound.R1 andR 2 arerespectivelytheinnerand outer turning point.

4.3. The connection formulas

To obtainthewavefunctionsaroundtheclassicalturningpointstheSchrödingerequationwill beexpressedin two newparameters.Theequationwill bereducedto averysimpleoneafterthelinearizationof thepotentialaroundtheturningpoints.Thetwo newparameters,vandy are related toΨ andR in the following way

which implies

The Schrödinger equation expressed in the new parameters will be

E

V(R)

R R2R1

WKB WKB WKBCFCF

v k Ψ=

y k Rd∫=

v iy±[ ]exp=

d2v

dy2

--------1

4k2

-------- dkdy------

2 12k------d

2k

dy2

--------– 1+ v+ 0=

- 30 -

Now, the potential is replaced by a straight line

with Rt as one of the turning points. Note that V and E do not appearexplicitly in theSchrödinger equation, but are hidden ink.When this approximation is applied the Schrödinger equations takes on a simple form

This equationgivesthe correctwavefunctionsaroundthe turning pointsandhopefully oversucha rangethatit hassomeoverlapwith therangeswheretheWKB solutionsarevalid to beable to connect the two solutions.In theremainingof this paragraphit will beshownthat it is possibleto find a solutionfor thisdifferential equation.Suppose that the solution can be written in the form

with, λ, f(t) andthepathof integrationnot specifiedyet.After substitutionthefollowing mustbe valid

If it canbeproventhatthis is equalto zerofor all y, thentheproposedsolutionis theright one.Chooseλ suchthatthefirst termcancelstheconstant5/36,i.e. λ = 1/6 or 5/6.Theintegralbe-comes

Integrationby partsgivestwo termswhich shouldgive zerowhenadded,but if we canprovethat both terms themselves are zero, this condition is automatically fulfilled. The two parts are

V R( ) E α R Rt–( )≈–

d2v

dy2

-------- 1 5

36y2

-----------+ v+ 0=

v y( ) yλ

eyt

f t( ) td∫=

λ λ 1–( ) 2λyt y2t2

y2 5

36------+ + + + e

ytf t( ) td∫ 0=

f t( ) 2λt 1 t2

+( )td

d+ e

yttd∫ 0=

2λtf t( ) ddt----- 1 t

2+( ) f t( )–

eyt

td∫ 0=

ddt-----∫ 1 t

2+( ) f t( )e

yt[ ]dt 0=

- 31 -

The first one gives an expression forf(t)

This expression forf(t) is inserted in the second equation. This leads to the integral

If the limits arechosensuchthat the function is zero,thenof coursealsothedifferenceiszero. This leads to the following solutions

With this choiceof t, f(t) andλ, we havefound a non-trivial solutionof the Schrödingerequation.Thismeansthatwenowhaveanexpressionfor thewavefunctionaroundtheturn-ingpoints.All thewavefunctionsin thedifferentregionsshouldbematchedtogether.Doingso(somecomplexfunctiontheoryis involved),it canbeshownthatthedifferentpartsof theWKB solution should be connected as follows:Around the inner turning point (R = R1)

Around the outer turning point (R = R2)

These formulas are known as the connection formulas.

2λtf t( ) ddt----- 1 t

2+( ) f t( )=

2λtf t( ) 2tf t( )– 1 t2

+( ) ddt----- f t( )=

λ 1–( ) 2t

1 t2

+------------- td

0

t

∫ 1f t( )---------- f t( )d

t=0

t=t

∫=

λ 1–( ) 1 t2

+( )lnf t( )f 0( )-----------

ln=

f t( ) f 0( ) 1 t2

+( )λ 1–

=

ddt-----∫ 1 t

2+( )

λe

yt[ ]dt 0=

1 t2

+( )λe

ytlower limit

upper limit0=

t +i=

t i–=

t +∞= y 0<t ∞–= y 0>

+1

k--------- k

R

R1

∫– dR 2

k------ k

R1

R

∫ dR14---π–

cos↔exp

1

k---------– + k

R

R1

∫ dR 2

k------ k

R1

R

∫ dR14---π–

sin↔exp

2

k------ k

R

R2

∫ dR14---π–

+1

k--------- k

R2

R

∫– dR exp↔cos

2

k------ k

R

R2

∫ dR14---π–

1

k---------– + k

R2

R

∫ dR exp↔sin

- 32 -

4.4. Bound states and the WKB approximation

In this paragraph,anexpressionwill bederivedto determinetheenergiesof boundlevelsin apotentialwell. If the potentialexhibitsonly onewell, threeregionscanbe distinguished:theclassicallyallowedregion(RegionII) andtwice a classicallyforbiddenregion(RegionsI andIII).In Region I, the WKB wave function is

with A andB normalization constants.ΨI must vanish rigorously whenR < R1 and thusB = 0. By applying the connection formulas

ΨII can be found

which can be rewritten into

Only thesecondtermgivesrise to a decreasingexponentialfunction in RegionIII (thewavefunction should vanish for largeR) and therefore the first term should be equal to zero

This term should be zero for everyR, which means that the cosine should vanish or

with v a positive integer.

ΨI R( ) A1

k--------- k

R

R1

∫– dR B

1

k--------- + k

R

R1

∫ dR exp+exp=

ΨII R( ) 2

k------ k

R1

R

∫ dR14---π–

cos=

ΨII R( ) 2

k------ k

R1

R2

∫ dR k

R

R2

∫ dR14---π–

sincos–2

k------ k

R1

R2

∫ dR k

R

R2

∫ dR14---π–

cossin+=

2

k------ k

R1

R2

∫ dR k

R

R2

∫ dR14---π–

sincos 0=

R1 R2

V(R)

E

D

Region I Region IIIRegion II

kR1

R2

∫ dR v12---+

π=

- 33 -

This leads to the formula known as the WKB-condition for bound levels in a potential well

All oneneedsfor a calculationof theenergylevelsis thepotentialanda numericalprocedurefor the integral.

4.5. Levels close to the dissociation limit

This paragraphis basedon an article by RobertJ. Leroy and RichardBernstein(JCP 52,3869-3879(1970)).In thisarticleenergylevelsareinvestigatedcloseto adissociationlimit. Anexpressionis derivedfor thedependenceof thebindingenergyasa functionof thevibrationalquantumnumber.Experimentallyit usuallybecomesharderto probethehigherlevelsin a po-tentialandwith thisformulathehighestlevelscanbepredictedonthebasisof lowerones.Alsoan estimation of the value of the dissociation limit can be given.Thestartingpoint is theWKB-conditionfor boundlevelsaswasderivedin thepreviouspara-graph

Differentiating this expression with respect toE(v) gives

The outer part of a molecular potential close to the dissociation limit can be described by

with Cn aconstantandn is relatedto thedominantelectrostaticinteractionbetweenthetwo at-oms(for instancen = 1 in thecaseof ionsandn = 6 in thecaseof avanderWaalsinteraction).D is thedissociationlimit. Insertthis potentialin the formulaandalsochangethevariableofintegration tox = R2/R

This integralis knownif theupperlimit is ∞ i.e. if R1 = 0.Onthenextpageit is shownthatthisis avalid approximation.Theintegralof thelower figure is finite thoughthevalueof thefunc-tion itself goesto infinity at the turningpoints.Thesurfacebeneaththe functionat the right-handsidein thelowerfigure,is biggerthanattheleft-handsideandthisdifferencewill increasewhenonegetscloserto thedissociationlimit. Theerrormadeby usingtheapproximatepoten-tial becomessmallercloseto thelimit. And theerrormadeby taking0 aslower limit in steadof R1 is alsonegligible.This meansthat theintegralderivedabovemaybeevaluatedbetween1 and∞.With thoselimits, the integral is analyticallysolvable(seefor instanceI. S.GradshteynandI. M. Ryzhik, Table of Integrals, Series and Products, AcademicPressInc., New York, 1965,

v12---+

2µ( )π"

-----------1 2⁄

E v( ) V R( )–[ ]1 2⁄

R1

R2

∫ dR=

v12---+

2µ( )π"

-----------1 2⁄

E v( ) V R( )–[ ]1 2⁄

R1

R2

∫ dR=

dvdE v( )---------------

µ 2⁄( )π"

---------------1 2⁄

E v( ) V R( )–[ ] 1– 2⁄

R1

R2

∫ dR=

V R( ) DCn

Rn

------–=

dvdE-------

µ 2⁄( )π"

---------------1 2⁄ Cn

1 n⁄

D E–[ ]1 2 1 n⁄+⁄--------------------------------------- x2–

1

R2 R1⁄

∫ xn

1–( )1 2⁄–

dx=

- 34 -

Sec. 3.251, p. 295)

with in our caseµ = −1, p = n andν = 1/2, this becomes

Theβ-function is defined as follows

and theΓ-function

xµ 1–

1

∞∫ x

p1–( )

ν 1–dx

1p---β 1 ν–

µp--- ν,–

=

x2–

1

∞∫ x

n1–( )

1 2⁄–dx

1n---β 1

2---

1n--- 1

2---,+

=

β p q,( ) Γ p( )Γ q( )Γ p q+( )------------------------=

Γ z( ) et–

0

∞∫ t

z 1–dt=

R1 R2

R

V(R)

D - Cn/Rn

E

[E(v

) -V

(R)]

-1/2

∞

∞

- 35 -

Using those definitions and results in the following equation

with Kn a constant depending on bothCn andµ, the reduced mass.Rearrange the terms and integrate from the dissociation limitD to E(v)

PutE(v) to one side

with vD the‘effective’ vibrationalquantumnumberatthedissociationlimit. It indicateshowclosethehighestlevel is to thedissociationlimit. If for instancevD = 13.01thenv = 13 isjust bound.If, however,vD = 12.99thenv = 13 is just not boundin thepotential.Thede-nominatorof theexponentbecomes0 for n = 2, andtheexpressionis only valid for n > 2.In thecasen = 1, theSchrödingerequationcanbesolvedanalyticallyandfor n = 2, adiffer-ent expression can be found (described in the same article, but not treated here).The formula can be rewritten in terms of the binding energyε

with an a constant.An applicationof this formulais shownin thenextfigure.Withoutgoinginto toomuchde-tail, anelectronicstatein H2 with a long rangedependenceof R-3, waspopulatedin amultiphotonlaserexperiment.Boththebindingenergyandthevibrationalquantumnumberwereknownandusingtheformula,apredictioncouldbemadefor thehighervibronic levels(in-dicatedwith arrows).If thedissociationlimit wasnot known,thenalsothebindingenergywould havebeenunknown.In thatcase,theexpressioncouldbeusedin a fitting routineto

Γ 12---

π=

dEdv------- "

2πµ------

1 2⁄ Γ 1 1 n⁄+( )Γ 1 2⁄ 1 n⁄+( )----------------------------------- n

Cn1 n⁄----------- D E–[ ]

n 2+2n

------------

=

Kn D E–[ ]n 2+2n

------------

=

1Kn------ D E v( )–[ ]

2– n–2n

----------------

E v( )dD

E v( )∫ vd

vD

v

∫=

1Kn------–

2nn 2–------------ D E v( )–[ ]

n 2–2n

------------

v vD–=

E v( ) D vD v–( )n 2–2n

------------Kn

2nn 2–------------

–=

v vD anεv

n 2–2n

------------

–=

- 36 -

determine the value of the dissociation limitD.

Fortheso-calledI’ potentialin H2 n = 3andC3 = 0.554929atomicunits.Thisnumberis isotopeindependent,buta3 is not.ForH2 a3 = 3.2343cm1/6 andfor D2 a3 = 4.5722cm1/6. Plottingthevibrational quantum number vs.ε1/6, a straight line is to be expected.Finally, a list is presented with the interpretation of the differentn-values

n = 1 2 charged atoms (Coulomb)n = 2 1 charged atom and 1 atom with a permanent dipole momentn = 3 2 atoms with permanent dipole momentsn = 3 identicalunchargedatomsin electronicstateswhosetotalangularmomentadiffer

by one (i.e.∆L = 1)n = 4 1 charged atom and one neutral atomn = 4 1 atom with a permanent dipole moment and 1 atom with a permanent quadrupole momentn = 5 2 atoms with permanent quadrupole momentsn = 6 induced dipole - induced dipole interaction

ThelastR dependenceis alsoknownasthevanderWaalsinteractionandthis termwill alwaysbe present.

4.6. The harmonic oscillator

TheWKB approximationgivestheright energylevelsin thecaseof aharmonicoscillator.Thisremarkable result will be derived below.Theenergylevelsof apotential canbefoundin anyelementarybookonquan-

10-4 0.01 0.1 1 2 5 10 20 50 100 2000

binding energy (cm-1)

0

2

4

6

v (H

2)

0

2

4

6

8

10

v(D

2 )

D2H2

V R( ) 12---cR

2=

- 37 -

tum mechanics

Start with the WKB condition for bound levels

with R1 andR2 the two turning points. For a certain energyE(v), the turning points are

The equation becomes

By applying the following standard integral

the condition becomes

which can be rearranged into the first formula of this paragraph

Notehowever,thatthecorrespondingwavefunctionsΨ arenot exactlythesameastheanalyticalsolutions.In the analyticalcasethe solutionsareHermitepolynomialfunctions,but in the WKBcase, the functions are somewhat different.

E v( ) "cµ--- v

12---+

=

v12---+

2µ( )π"

-----------1 2⁄

E v( ) 12---cR

2–

1 2⁄

R1

R2

∫ dR=

R2 R1– 2Ec

-------= =

v12---+

2µ( )π"

-----------1 2⁄

E v( ) 12---cR

2–

1 2⁄

R1

R2

∫ dR=

cµ( )π"

-----------1 2⁄ E v( )

c----------- R

2–

1 2⁄

R1

R2

∫ dR=

a2

x2

–( ) xd∫ 12--- x a

2x

2–( ) a

2 xa-----asin+=

v12---+

cµ( )"

-----------1 2⁄ E v( )

c-----------=

E v( ) "cµ--- v

12---+

=

- 38 -

5. Electronic states

5.1. Symmetry operations

Symmetryplaysanimportantrole in molecularspectroscopy.Quantumstatesof themolecularHamiltonianareclassifiedwith quantumnumbersthatrelateto symmetriesof theproblem;theinvarianceof the Hamiltonianundera symmetryoperationof the moleculein its body fixedframe is connected to a quantum number. For a diatomic molecule the symmetries are:

The HamiltonianH0:

is invariant under the symmetry operations:- Rφ rotation over every angleφ about the molecular axis- σv reflection in a molecular plane containing the molecular axis- i in version in the molecular centre

Theseoperatorsnot only leavethemolecularHamiltonianinvariant,theyarealsocommutingobservables.In thelanguageof quantummechanicsthismeansthattheseoperatorscangeneratea set of simultaneous eigenfunctions of the system.

Note that the operatori only appliesin a diatomicmoleculewith inversionsymmetry,i.e. ahomonuclearmolecule.Theseoperatorsform groups, for a homonuclearmoleculesthe ,for the heteronuclear molecules the point group.

5.2. Classification of states

Theelectronicstatesof themoleculesareclassifiedaccordingto theeigenvaluesunderthesym-metry operations.

The reflectionoperatorσv (later we will seethat this operatoris connectedto the conceptofparity for a molecular eigen state) acts has two eigenvalues:

TheoperatorRφ is connectedto anotherconstantof themotion,Lz. Assumethatin a moleculetheelectronicangularmomentaarecoupledto a resultingvector . In anatom is a

φi

σv

H0"

2

2m------- ∇i

2

i∑– V ri R,( )+=

D∞hC∞v

σvψe ψe±= eigenvalues –,+

L lii

∑= L

- 39 -

constant of the motion, since there is overall rotational symmetry. Here is the distinct differencebetween atoms and molecules; the overall rotational symmetry is broken. In a diatom there isonly axial symmetry around the internuclear axis of the molecule. Hence only Lz is a constantof the motion. The eigen value equation is:

In the nomenclature of diatomic molecules the electronic states are called:

Σ for Λ = 0Π for Λ = ∆ for Λ = Φ for Λ = ,etc.

The energy of the molecule depends on Λ2; states with Λ and -Λ are degenerate.

For the inversion operator there are two eigenvalues:

The g (gerade) and u (ungerade) symbols are chosen for a distinction with the eigenvalues ofthe σv operator.Hence we find simultaneous eigenvalues, under the three symmetry operations, resulting in pos-sible quantum states:

Homonuclear Heteronuclear

Λ = 0 Σg+ Σu

+ Σg- Σu

- Σ+ Σ-

Λ = 1 Πg+ Πu

+ Πg- Πu

- Π+ Π-

Λ = 2 ∆g+ ∆u

+ ∆g- ∆u

- ∆+ ∆-

etc

Remarks.- There is a double degeneracy under the σv operator for states . Therefore the +/- signsare usually omitted for .- There is no degeneracy under the i operator for u and g states. So u and g states have differentenergies.

The electron spins are added in the molecule in the same way as in atoms: . In theclassification of states the multiplicity (2S+1) due the electron spin is given in the same way asin atoms. Hence we identify states as:

1Σg+ for the ground state of the H2 molecule

3Σg- for the ground state of the O2 molecule

2Π3/2 for the ground state of the OH molecule; here spin-orbit coupling is included (see later)

Additional identifiers usually chosen are the symbols X, A, B, C, ..., a, b, c, ... These just relate

Lzψe"i---

φ∂∂ψe Λ"ψe= = eigenvalues Λ 0 1 2 3,±,±,±,=

1±2±3±

iψe ψe±= eigenvalues g u,

Λ 0≠Λ 0≠

S sii

∑=

- 40 -

to a way of sortingthe states.The electronicgroundstateis referredto with X. The excitedstatesof thesamemultiplicity getA, B, C, etc,whereasa,b, c arereservedfor electronicstatesof differentmultiplicity. Forhistoricalreasonsfor somemoleculesthesymbolsX, A, B, C, ...,a, b, c, ... are used differently, e.g. in the case of the N2 molecule.

5.3. Interchange of identical nuclei; the operator P

In molecularphysicsusuallytwo different framesof referencearechosenthat shouldnot beconfused.As theoriginsof thebody fixed frame andthespace fixed frame thecentreof gravityof themoleculeis chosen.Thecoordinatesin thespacefixed framearedenotedwith capitals(X, Y, Z) andthosein thebodyfixed framewith (x, y, z). By makinguseof Euler-anglesthetworeferenceframescanbetransformedinto oneanother.Thez-axisis by definition theline con-nectingnucleus1 with nucleaus2 andthis definestheEuler-anglesθ andφ. By definition χ =0 and this ties thex- andy-axis (see figure). For an Euler-transformation withχ = 0:

If R is theseparationbetweenthenuclei,thenR, θ andφ canbeexpressedin thepositionsofthe nuclei in the space fixed frame (see also figure below):

Where (X1, Y1, Z1) is the position of nucleus 1 in the space fixed frame.If the operator interchanging the two nuclei is calledP then:

x X θ φcoscos Y θcos φsin Z θsin–+=

y X– φsin Y φcos+=

z X θsin φcos Y θsin φsin Z θcos+ +=

Euler-transformationwith χ = 0. First (x, y, z) rotatedaroundthez-axisoverangleφ. Thenthex- andy-axisstayin theXY-plane.Subsequently(x, y, z) is rotatedaroundthey-axisoverangleθ. They-axisstaysin theXY-plane by doing so. The grey plane in the drawing is thexz-plane.

φ φ

θφ θ

X

Y

Z

x

y

z

XX

YY

ZZ

xx

yyzz

θZ1

X12

Y 12

Z12

+ +------------------------------------

acos=

φX1

X12

Y 12

+-----------------------

acos= and R 2 X12

Y 12

Z12

+ +=

- 41 -

Or in R, θ andφ:

Becausethez-axisby definitionrunsfrom nucleus1 to 2, it will beturnedaround.Fromtheequationsit follows thatthey-axisalsoturnsaround.If theith electronhasaposition(xi, yi,zi), thentheposititionsof all particlesof themoleculerepresentedby (R, θ, φ; xi, yi, zi) andso:

The inversion-operator in the space-fixed frameISF, is then defined as:

It can be deduced that:

For the inversion-operator in the body-fixed frame iBF, it holds that:

P X1 Y 1 Z1 X2 Y 2 Z2, , , , ,( ) X2 Y 2 Z2 X1 Y 1 Z1, , , , ,( )=

X– 1 Y– 1 Z– 1 X– 2 Y– 2 Z– 2, , , , ,( )=

Fig: Underthe inversion-operationP not only theanglesθ andφ change,butalso thez-axis.

θ

φ1

2

z

Y

X

Z

R

P R θ φ, ,( ) R π, θ– φ π+,( )=

P R θ φ xi; yi zi, ,, ,( ) R π, θ– φ π+ xi; y– i z– i, , ,( )=

ISF

X Y Z, ,( ) X– Y– Z–, ,( )=

ISF

R θ φ xi; yi zi, ,, ,( ) R π, θ– φ π+ x– i; yi zi, , ,( )=

θ

φ1

2

z

Y

X

Z

1

2z

Y

X

Z

π + φ

π − θP

Fig: Under the interchange operator P not only the anglesθ enφ change, but also thez-as.

- 42 -

By combining the last two equations it follows:

Hence the important relationship for the inversion operators is proven:

5.4. The parity operator

Parityis definedastheinversionin a space-fixedframe,denotedby theoperatorISF. We wishto proveherethat this operatorISF is equivalentto a reflectionthrougha planecontainingthenuclearaxis (z-axis).For this planewe takexz, but the sameproof would hold for any planecontaining thez-axis. One can write:

with σv(xz) a reflectionthroughthexz-plane.A rotationof 180˚aroundtheaxisperpendicularto the chosen plane (so they-axis), gives in the body-fixed frame:

with R180(y) the rotation-operatoraroundthe y-axis. In sometextbooksR180(y) is written asC2(y). The nuclei exchange position:

and thexyz-frame then rotates. The total rotation is:

By combining equations one gets:

This is the prove that:

or in general:

wheretheaxisof R180 mustbeperpendicularto theplaneof σv. In isotropicspacethestateofamoleculeis independentof theorientation;henceamoleculecanundergoanarbitraryrotationwithout change of state. Hence it is proven thatσv signifies the parity operation:

5.5 Parity of molecular wave functions; total (+/-) parity

Parity playsan importantrole in molecularphysics,particularly in determiningthe selectionrulesfor allowedtransitionsin thesystem.Quantummechanicsdictatesthatall quantumstateshavea definiteparity (+) or (-). As discussedaboveparity is connectedto theoperatorISF de-fined in thespace-fixedframe,but mostmolecularpropertiesarecalculatedin thebody-fixed

iBF

R θ φ xi; yi zi, ,, ,( ) R θ φ x– i; y– i z– i, , , ,( )=

iBF

ISF

R θ φ xi; yi zi, ,, ,( ) R π, θ– φ π+ xi; y– i z– i, , ,( )=

P iBF

ISF

=

σv xz( ) R θ φ xi; yi zi, ,, ,( ) R θ φ xi; y– i zi, , , ,( )=

R180 y( ) x y z, ,( ) x– y z–, ,( )=

R180 y( ) R θ φ, ,( ) R π, θ– φ π+,( )=

R180 y( ) R θ φ xi; yi zi, ,, ,( ) R π, θ– φ π+ x– i; y– i zi, , ,( )=

σv xz( )R180 y( ) R θ φ xi; yi zi, ,, ,( ) R π, θ– φ π+ x– i; yi zi, , ,( )=

ISF σv xz( )R180 y( )=

ISF σvR180=

ISF σv=

- 43 -

frame.Henceweusuallyreferto σv astheparityoperator.Thetotalwavefunctionof amo-lecular system can be written:

and hence the parity operator should be applied to all products.In diatomicmoleculesthevibrationalwavefunctionis only dependenton theparameterR,the internuclear separation and therefore:

Note that this is not generally the case for polyatomic molecules.TherotationalwavefunctionscanbeexpressedasregularYJM functionsfor whichtheparityis:

where J is the rotational angularmomentum,previously defined as N. More generallywavefunctionscanbeused,in similarity to symmetrictop wavefunctions ,

in which J is theangularmomentumandΩ is theprojectionontothemolecularaxis in thebody-fixedframe,while M is the projectionin the body-fixedframe.In fact Ω is alsothetotal electronic angular momentum. The effect of the parity operator is:

whereJ takes the role of the total angular momentum.Soin generalthewavefunctionsfor rotationalmotionaresomewhatmorecomplicatedthanthesphericalharmonics , whicharethepropereigenfunctionsfor amoleculeina1Σ state.Thesituationis differentwhen and/or aredifferentfrom zero.Then is notperpendicular to the molecular axis. It can be shown that the wave functions are:

whereD standsfor theWignerD-functions.Thephasefactordependson thechoiceof thephaseconvention;theaboveequationis in accordnacewith theCondon-Shortlyconvention.Note that other conventions are in use in the literature.Thisis relatedto theeffectof theparityoperatoronthespinpartof theelectronicwavefunc-tion:

NotethathereΣ hasthemeaningof theprojectionof thespinS ontothemolecularaxis;thatis acompletelydifferentmeaningof Σ thanfor thestatesin caseΛ=0.Fortheorbitalangularmomentum of the electrons:

Ψmol ψelψvibψrot=

σvψvib ψvib+=

σvY JM 1–( ) J–Y JM=

ΩJM| ⟩ JKM| ⟩

σv ΩJM| ⟩ 1–( )J Ω– Ω– J M, ,| ⟩=

Y NM θ φ,( )L S J

ΩJM| ⟩ –( )M Ω– 2J 1+

8π2---------------DMΩ

J( ) αβγ( )=

σv SΣ| ⟩ 1–( )S Σ–S Σ–,| ⟩=

σv Λ| ⟩ 1–( )± Λ Λ–| ⟩=

- 44 -

So remember forΛ=0 states there are indeed two solutions:

because the statesΣ+ andΣ- are entirely different states with different energies.The effect of the parity operator on the total wave function is then:

whereσ=0 for all states except forΣ- states, for whichσ=1.Sincetheσv operationchangesthesignsof Λ, Σ, andΩ thetrueparityeigenfunctionsarelinearcombinations of the basis functions, namely:

for which the parity operator acts as:

Thesesymmetrizedwavefunctionscanbeusedto derivetheselectionrulesin electricdipoletransitions.With theseequationstheparity of thevariouslevelsin a diatomcanbededuced.In a Σ- statetheparity is (-)N+1, with N thepurerotation.For a Σ+ statetheparity is (-)N. Stateswith Λ>0aredoubledegenerateandbothpositiveandnegativerotationallevelsoccurfor eachvalueofN. NotethatwehavejumpedbackfromtheangularmomentumJ (whichincludesΩ) to N whichrefers to pure rotation.

σv Σ±| ⟩ Σ±| ⟩±=

σv ψelψvibψrot( ) σv nΛΣΩ| ⟩ v| ⟩ ΩJM| ⟩( )=

1–( )J 2Σ– S σ+ +n Λ– S Σ–, , ,| ⟩ v| ⟩ Ω– J M, ,| ⟩=

Λ2S 1+Ω ±| ⟩

Λ2S 1+Ω| ⟩ 1–( )J 2Σ– S σ+ + Λ2S 1+

Ω–| ⟩±

2-----------------------------------------------------------------------------------------=

σv Λ2S 1+Ω ±| ⟩ Λ2S 1+

Ω ±| ⟩±=

01

2

3

4

N

+

+

+

-

-

Σ+01

2

3

4

N

-

-

-

+

+

Σ-

0

1

2

3

N

-/+

-/+

+/-

+/-

Π

0

1

2

N

-/+

-/+

+/-

∆

- 45 -

Thelowestenergylevelsin theΠ and∆ statesarepurposelydepictedhigher.Thosearethelevelsfor whichthepurerotationalangularmomentumis N=0. Notethatin astateof Π elec-tronicsymmetrythereis 1 quantumof angularmomentumin theelectrons;hencethelowestquantum state isJ=1. In a∆ stateJ=2 is the lowest state.

5.6 Rotationless parity (e/f)

Becauseof theJ-dependentphasefactor the total parity changessign for eachJ-level in arotationalladder.Thereforeanotherparityconceptwasestablishedwherethisalternationisdivided out. (e) and (f) parity is defined in the following way (for integer values ofJ):

For half-integer values ofJ the following definitions are used:

It canbeverified thatall levelsin aΣ+ statehave(e)parity.Similarly, all levelsin aΣ- statehave (f) parity. ForΠ states all levels occur in e/f pairs with opposing parity.Theuseof e/f suppressesthephasefactorin thedefinitionof theparityeigenfunctions.Nowit is found, for example in the evaluation of symmetrized basis functions for2Π states:

5.7 g/u and s/a symmetries in homonuclear molecules

Forhomonuclearmoleculesthepointgroup containstheinversionoperationi definedin thebody-fixedframe.Theoperationi leavesthevibrational,rotationalandelectronspinpartsof thewavefunctionunchanged;it onlyactsontheelectronicpartof thewavefunction.Theimportantpoint to realizeis thatthetransitiondipolemomentoperator is of u-parityand hence the selection rules for electric dipole transitions are .

σvψ 1–( )Jψ+= for e

σvψ 1–( )Jψ–= for f

σvψ 1–( )J 1 2⁄– ψ+= for e

σvψ 1–( )–J 1 2⁄– ψ= for f

Π23 2⁄ e f⁄,| ⟩

Π23 2⁄| ⟩ Π2

3 2⁄–| ⟩±

2-----------------------------------------=

Π21 2⁄ e f⁄,| ⟩

Π21 2⁄| ⟩ Π2

1 2⁄–| ⟩±

2-----------------------------------------=

Σ2 +1 2⁄ e f⁄,| ⟩

Σ2 +1 2⁄| ⟩ Σ2 +

1 2⁄–| ⟩±

2-----------------------------------------=

Σ2 –1 2⁄ e f⁄,| ⟩

Σ2 –1 2⁄| ⟩ Σ2 –

1 2⁄–| ⟩±

2---------------------------------------=

D∞h

µg u↔

- 46 -

In the above the interchange operator P was defined and it was proven that:

States which remain unchanged under the P operator are called symmetric (s), while thosechanging sign are called anti-symmetric (a). Under the operation ISF or σv the levels get their(+/-) symmetry, while the operation iBF introduces the g/u symmetry. Thus it follows when theelectronic state is:

gerade + levels are symmetric- levels are anti-symmetric

ungerade + levels are anti-symmetric- levels are symmetric

This gives the following ordering:

For Λ>0 states the Πg- states are ordered as Σg

-, etc.

5.8 The effect of nuclear spin

The magnetic moment of the nuclei interact with the other angular momenta in the molecularsystem. When all the angular momenta due to rotation, electronic orbital and spin angular mo-mentum are added to then the spin of the nucleus can be added:

If both nuclei have a spin they can both be added following the rules for addition of angular mo-menta.

The additions of angular momenta play a role in heteronuclear as well as homonuclear mole-cules. Of course the degeneracy of the levels should be taken into account: (2I1+1)(2I2+1).

In a homonuclear molecule the symmetry of the nuclear spin wave functions play a role. Fordiatomic homonuclear molecules we must distinguish between nuclei with:

P iBF

ISF

iBFσv= =

→

→

01

2

3

4

N

+

+

+

-

-

Σ+g

-

-

-

+

+

Σ-g

s

s

s

a

aa

a

a

s

s+

+

+

-

-

Σ+u

-

-

-

+

+

Σ-u

s

s

s

a

aa

a

a

s

s

J I

F J I+=

F J I1 I2+ +=

- 47 -

- integral spin, which obey the Bose-Einstein statistics- half-integral spin which obey the Fermi-Dirac statistics

The symmetrization postulate of quantum mechanics tells us that all wave functions are ei-therunchanged or change sign under permutation of two particles. The total wave function Ψmust be symmetric for integral spin particles, anti-symmetric for half-integral spin particles.This gives rise to symmetry restrictions that can be viewed in various ways; the textbooksgive also various arguments, starting from different perspectives. One view is to start fromthe interchange operator P, consider electronic states with g-symmetry (under iBF) and +-symmetry (under σv) and neglect the vibrational part (always a positive parity in diatomics).Then the rotational parts of the wave function and the nuclear spin wave functions remain.The product of the exchange properties of these wave functions should follow the proper sta-tistics. For the rotational levels (here we restrict ourselves to pure rotation; in case of angularmomentum coupling between electronic and rotational motion it also applies) the parity is(-)N. Hence it follows:

FD-nuclei even N require ψnuc anti-symmetricodd N require ψnuc symmetric

BE-nuclei even N require ψnuc symmetricodd N require ψnuc anti-symmetric

If the symmetry of the wave function is considered then the rules change for -parity statesand for u-states. One can derive:

FD-nuclei s levels require ψnuc anti-symmetrica levels require ψnuc symmetric

BE-nuclei s levels require ψnuc symmetrica levels require ψnuc anti-symmetric

The nuclear spin weight is (2I+1)2 where I is the spin. Of the (2I+1)2 possible states:

(2I+1)(I+1) are symmetric(2I+1)I are anti-symmetric

5.9 Para and ortho hydrogen

In the hydrogen molecule with two spin IH=1/2 the total nuclear spin is:

There exist (2I+1)2=4 possible quantum states of which 3 are symmetric and one anti-sym-

I tot IH IH+ 0 1,= =

- 48 -

metric under interchange of the two particles (note that α means spin up,β spin down):

and:

Note that the three states forI=1 haveMI=+1, -1 and 0.For thehydrogennucleiFD-statisticsapplies,hencethesymmetricwavefunctionψnuccoupleswith a levelsandin theelectronicgroundstate,of 1Σg

+ symmetry,with theoddN levels.Thisform of hydrogenis calledortho-hydrogen; theotheris para-hydrogen. Thereis a 3:1 ratio oflevelsin orthovspara.It is noteasyfor themoleculeto undergoatransitionfrom orthoto para;in electromagnetictransitionsthis doesnot occur,sincetheelectricdipoledoesnot affect theordering of nuclear spins.

5.10 Missing levels in the oxygen molecule

Thenuclearspinof 16O nucleiis I=0. As aconsequencethenuclearspinwavefunctioncanbeleft outof theproblem,or in otherwords,it shouldbeconsideredasahavingpositivesymmetry.Theelectronicgroundstateof O2 hasa3Σg

- symmetry,hencehasanegativeparity for theelec-tronicwavefunction.16O nucleifollow Bose-Einsteinstatistics,sothetotalwavefunctionmustbesymmetricundertheinterchangeoperator.Thesymmetricstates(ssymmetry,seethefigurefor Σ- states)aretheoneswith oddN quantumnumbersfor purerotation.Theses-statescom-binewith symmetricnuclearspinwavefunctions,the(a)-stateswouldcombinewith anti-sym-metricnuclearspinwavefunctions,but these do not exist. As a consequencethe(a) states,orthe states with even rotational angular momentum do not exist.

I 1=

α 1( )α 2( )β 1( )β 2( )

1

2------- α 1( )β 2( ) β 1( )α 2( )+[ ]

I 0=1

2------- α 1( )β 2( ) β 1( )α 2( )–[ ]

- 49 -

The energy levels of the oxygen molecule in its ground state are depicted in the figure.

Note that this analysis only holds for the 16O2 molecule, and for the 18O2 molecule, becausethe 18O nucleus also has I=0. The heteronuclear species (isotopomers) 16O18O, 16O17O and17O18O do not follow this peculiar behaviour since the additional inversion symmetry is lift-ed. In the 17O2 isotopomer the situation is also different, because the nuclear spin is I=5/2.This gives rise to an intensity alternation (which one?), but not to a disappearing of lines. Sofor all isotopomers except 16O2 and 18O2 the level scheme depicted on the right is appropri-ate.In the figure the rotational levels of the electronic ground state (3Σg

-) are split into three com-ponent as a result of the triplet structure. The electron spins of the two paired outer electronslign up to a triplet giving molecular oxygen a paramagnetic character. The interaction be-tween the resulting spin vector and the rotational angular momentum ,causes a lifting of the degeneracy and a splitting into three components, wherever possible(not for N=0 obviously).In the electronically excited state of 1Σg

+ symmetry, the situation is similar. Because theelectronic parity is positive here the odd N-levels are missing; also there is no triplet split-ting, since we deal with a singlet state.The transitions depicted in the figure are also anomalous. Since both the ground and excitedstates are of g-symmetry electric dipole transitions are not allowed. A second reason is that1Σg

+ - 3Σg- transitions are not allowed for electric dipole. The thick lines are the allowed but

very weak magnetic dipole transitions, while the thinner lines refer to the branches of the

N = 3

N = 1

N = 2

N = 0

N = 2

N = 0

N = 1

N = 3

J =0J =2J =1

J =0

J =1

J =1J =3J =2J =2

J =4J =3

J =3

J =1

J =2

RR RQ

PP PQ

N = 4 J =4

PO

NO

RS

TS

RQ

PP PQ

RRΣ1 +

g

Σ3 -g

Figure: The allowed and forbidden states and transitions in the 16O2 molecule.

S N J N S+=

- 50 -

electric quadrupole transitions (again weaker by a factor 106). The fact that the transition is be-tween a triplet state and a singlet state is also a reason for its weakness.

5.11 The 3:1 ratio in N2

Herzberg measured, in the 1930s, a spectrum (the Raman spectrum in the electronic groundstate of 1Σg

+ symmetry) for the nitrogen molecule and observed a 3:1 ratio between lines. Thisphenomenon could only be explained by assuming that the nitrogen (14N) nucleus has a nuclearspin of I=1. In those days nuclei were considered to be built from protons and electrons; theneutron was not yet observed, postulated however. The 14N nucleus was considered to be builtform 14 protons and 7 electrons giving rise to a charge of 7+ and a mass of 14 amu. But 21 par-ticles of half-integer spin should build a nucleus of total half-integer spin and should thereforeobey Fermi-Dirac statistics. This paradox gave support to the neutron hypothesis.

6. Open Shell Molecules

6.1 Introduction

In our discussion of rotational energies we have assumed (tacitly) that was the only angularmomentum. This assumption is very good for in which all electronic spins are paired off andthe orbital angular momentum, although in principle not necessarily zero, manifests itself only insecond order. The situation is drastically different in states other than in which both andcan be effectively different from zero. On the other hand in the discussion of electronic energiesthe molecule was considered as non-rotating . In actual molecules all these angular mo-menta may be present and coupled in a complicated way by gyroscopic and magnetic forces. In-dividual angular momenta then lose their identity and only certain sums resulting from effectivecouplings are constants of motion which can be determined from the observed spectra.The presence of the various angular momenta introduces a number of new phenomena and prob-lems:

(1) coupling schemes,(2) interactions which may not only shift but also split electronic energy levels,(3) breakdown of certain rules and approximations.

In the table all the angular momenta are collected which appear in calculations of molecular en-ergies, with their projections in SF-Z axis and the BF-z axis (the molecular axis), and associated

ANGULAR MOMENTA DEFINITION QUANTUMNUMBER

Electronic orbital

SF projection

BF projection

Electronic spin

SF projection

BF projection

Rotational

SF projection

BF projection -

Total orbital

BF projection

Total molecular

SF projection

BF projection

Total electronic -

BF projection -

NΣ1

Σ1L S

N 0=

L li∑= L

LZ ML

Lz Λ

S si∑= S

SZ MS

Sz Σ

R R

RZ M

Rz

N R L+= N

N z Λ

J N S+= J

JZ MJ

J Ω Λ Σ+=

J ji∑=

Ω

- 52 -

quantumnumbers.Thecouplingof ’s to and ’s to correspondsto theatomicRussel-Saunderscoupling,while thecouplingto representsananalogonof jj-coupling.Wehavedisregardedthepossibilityof avibrationalangularmomentum.Variouspossiblewaysof couplingtheangularmomentaintroducedby Hundin 1926(knownasHund’scases)aredis-cussed in the following section.Thenewinteractionswhich haveto beconsideredin thepresenceof unpairedelectronicspinand non-zero orbital momenta are:spin-spin,spin-orbit (and also spin other orbit),spin-rotation.The microscopic hamiltonian for the spin-spin and spin-orbit interaction is:

and:

In these expressions ge is the electronic g factor, the Bohr magneton, and stands for the velocity of the particle.

Actually this expression, first derived by Van Vleck (1951), is a sum:

where

Thehamiltonian is themostgeneralform of thefine structureinteraction.It containstheusualspin-orbitinteraction,thespin-other-orbitinteractionandthecrosstermsbetweenthevar-ious and . The most common forms of are:

Thelastform of canonly beusedwhen and arewell (or almostwell) defined.In el-ementary text books this form is written as:

Thisexpressioncanonly beusedfor thediagonalcontributionof andonly when andaregood(or almostgood)quantumnumbers.In this expression is the spin-orbitcoupling

li L si SJa ji li si+=( )

Hss( ) Hsor( )

Hss

µ0

4π------

ge2µB

2 si s j⋅( )rij2

3 rij si⋅( ) rij s j⋅( )–[ ]

rij5

---------------------------------------------------------------------------j i>∑=

Hsor

µ0

4π------

2mgeµB2 Zα

r jα3

-------

r jα12---v j vα–

× s j⋅

α j,∑ –=

µ0

4π------

2mgeµB2 1

rij3

-----

rij12---v j vi–

× s j⋅j i≠∑–

µβrµκ rµ rκ rµκ rµκ=( )–= vµ µ

Hsor Hso Hsr+=

Hso

µ0

4π------

2mgeµB2 Zα

r jα3

-------

r jα12---v j× s j⋅

1

rij3

-----

r ji12---v j vi–

× s j⋅

j i≠∑–

α j,∑

=

Hsr

µ0

4π------

2mgeµB2 Zα

r jα3

-------

r jα vα×[ ] s j⋅( )α j,∑–=

Hso

r ′s v′s, s′s Hso

Hso′ ς j r j( ) l jα s j⋅( )j α,∑= or Hso″ A L S⋅( )=

Hso L S

Hso″ AΛΣ=

Hso Λ ΣA

- 53 -

constant( for normaland for invertedfine structure)and is theorbitalangularmomentumof the j-th electronwith respectto theα nucleus.Thespinrotationhamiltonianisusually written in the form:

(often with replacing and replacing ).

6.2 Hund’s Coupling Cases

Case (a)

This case occurs when(1) all ’s are coupled to and all ’s to ,(2) the coupling of and to the axial internuclearfield (sometimescalledthe and

coupling,respectively)is muchstrongerthanthespin-orbit( ) or anyotherpossiblecoupling (e.q. ) i.e.:

A gyroscopicdiagramof this couplingis shownin the figure below.Both and precessindependentlyabouttheinternuclearaxisandonly their components( and , respectively)and their sum:

areconstantsof motion.This sum,written asa vector coupleswith to a total molecularangular momentum:

It’s quantum number can take the values:

Consequently,the levelswith cannotoccur.Hund’s case(a) is quite commonin theground state of molecules, especially the light ones.

Case (b)

In thiscase is still coupledto internuclearaxisbut is decoupledfrom it, moreor less.Put

A 0> A 0< l jα

Hsr γ R S⋅( )=

N R λ γ

l L si SL S L A⋅

S A⋅ L S⋅R L⋅

L A⋅ L S⋅» and L A⋅ R L⋅»

S A⋅ L S⋅» and S A⋅ N S⋅»

L SΛ Σ

Ω Λ Σ+=

Ω R

R

J

LS

Λ Σ R

L

S

Λ

Ν

J

Hunds case (a) Hunds case (b)

J Ω Ω 1+ Ω 2 …,+, ,=

J Ω<Σ1

L S

- 54 -

differently, is coupled more strongly to than to , i.e.

The couplings of the various angular momenta and their precessions are shown in the figure.Explicitly the couplings are:

The quantum numbersJ andN can take the values:

Precessionof around is slow comparedto rotationbecausetheinteractionwhich couplesthesevectors is relativelyweak.Thecase(b) couplingis especiallyimportantwhen

but (CN, H2+, HgH, NH, O2,...), but canalsooccurfor otherelectronicstates,

particularly when there are relatively few electrons.

Case (c)

Thecouplingdiagramfor this caseis shownin thefigure below.It occurswhenthespin-orbitcoupling is much stronger than the coupling to the internuclear axis:

This is usuallythecasein heavymolecules,like Br2, I2. Thecouplingwhichproduces maybe of Russel-Saunders or of jj type:

Case (d)

In cases(a)and(b) the couplingis assumedto bestrong.However,in someexcitedstatesof H2, He2 andothermolecules,theelectronicorbit is solargethat this couplingbecomestooweak.Thegyroscopiceffectsuncouple from theinternuclearaxis.Thecouplingdiagramap-

S N R Λ+= A

L A⋅ L S⋅» and L A⋅ R L⋅»

N S⋅ S A⋅>

R Λ+ N=

N S+ J=

J N S+ N S 1–+ N S–, , , ,=

N Λ Λ 1+ Λ 2+ Λ 3+ …, , , ,=

S Nγ N S⋅( )

Λ 0= S 0≠

L S⋅ L A⋅» and L S⋅ S A⋅»

R

J

L

S

Ω

Ja

R

L S

ΝJ

Hund’s case (c) Hund’s case (d)

Ja

Ja L S+= Ja ji∑= ji li si+=( )

L A⋅

L

- 55 -

propriate for this situation, shown in the figure above, corresponds to the coupling scheme:

When S = 0 we expect a splitting of a level into 2L + 1 components characterized by:

Splitting betweentheJ componentswhen is determinedby the interactionwhichis usually very weak.The Hund’s couplingschemesare idealizations,at their bestactualmoleculescanonly ap-proachthem.Nevertheless,theyareusefulasaclassificationandasanindicationof the“best”basis for calculations of molecular energies.

6.3 Calculation of rotational energies

6.3.1 Case (a)

An expressionfor therotationalenergycanin thiscasebeobtainedfrom the“pure” rotationalhamiltonian

assuminga simple“rotational” basissetof correspondingto eigenvaluesofand . The result for the energy, or rotational term value, is:

In thesecondline theterm is disregardedbecauseit is a constantin a givenelectronicstate.Rotationallevelsof a multipletareshownschematicallyin thefigure.Thelowestlev-elsin this figurecorrespondto . For a state,apartfrom J = 1/2,eachvalueof J oc-curs for each of the two multiplet components which have .

Thequestionis, how goodis theexpressionfor therotationalenergy.First of all we notethatR is NOT a good quantum number. Its value is fixed by the condition:

Strictly speakingthe only goodquantumnumbersfor a rotatingmoleculeareJ, MJ and .Then should be consideredmore properly as the perpendicularcomponentof (i.e.

). But this doesnot resultfrom nuclearrotationalone,but hasalsoa contri-

R L+ N=

N S+ J=

N R L+ R L 1–+ … … R L–, , , ,=

S 0> N S⋅

Hr BR2

B J2

Ω2

–( )= =

JΩM > J2

Jz,JZ

Fv J( ) B J J 1+( ) Ω2–[ ] BJ J 1+( )= =

BΩ2–Π3

J Ω= Π2

Ω 3 2⁄ 1 2⁄,=

01

2

3

4

1

2

3

4

2

3

4

J J J

3Π03Π1

3Π2

Rotationalenergylevels in a 3Πstate

R2

J J 1+( ) Ω2–=

ΩR J

J⊥ Ω; J ||= J⊥

- 56 -

bution from the rapidly precessing perpendicular component

of . The“pure” nuclearrotationcoupleswith to produce . Theoperator of thepure rotational angular momentum and the rotational hamiltonian are:

Hence:

The last term in this expression may be approximated by:

Theterm hasaconstantvaluein agivenelectronicstateandcanberemovedasa part of electronic energy. The rotational energy can then be written as:

representscontributionof thelasttermof in theequation.As will beshownlaterthisterm is responsible for the-doubling.Spinorbit coupling splitstheelectronicstateinto 2S+ 1 multipletcomponentsthathavedifferent values of and .

6.3.2 Case (b)

In the pure case (b) with an expression for the rotational energy follow from:

i.e.

which reducesto because is independentof the rotationalquantum number (R or N).A (somewhat) better approach is analogous to that followed in case (a). By writing:

we get:

i.e.:

Thelast termrepresentsa couplingbetweenelectronicandrotationalmotion,in fact thesame

J⊥

R

L⊥

L⊥ L L⊥ J⊥ R

R J⊥ L⊥–=

Hr BR2

J⊥ L⊥–( )2J⊥

2L⊥

22 J⊥ L⊥⋅( )–+= = =

Fv J( ) Bv J J 1+( ) Ω2–[ ] Bv L⊥

2⟨ ⟩ 2Bv J⊥ L⊥⋅⟨ ⟩–+=

2Bv R L⊥⋅⟨ ⟩–

Bv L⊥2⟨ ⟩ Ω2

–( )

Fv J( ) Bv J J 1+( )[ ] f v Ω J,( )+=

f v Ω J,( )Λ

A Λ ΣΩ f v Ω J,( )

S 0= Fv N( )

R2

N2

Λ2

–=

Fv N( ) Bv N N 1+( ) Λ2–[ ]=

Fv N( ) Bv N N 1+( )[ ]= BvΛ2

R N⊥ L⊥–=

Hr BvR2

Bv N⊥2

L⊥2

2N⊥L⊥–+( )= =

Bv N2

N ||2

– L⊥2

2 N⊥ L⊥⋅( )–+( )=

Fv N( ) Bv N N 1+( ) Λ2– L⊥

2⟨ ⟩+[ ] f v Λ N,( )+=

- 57 -

asabove.If this couplingis neglectedandtheterms areincludedin thero-vibrational energy, then reduces to:

Thetreatmentsfollowed aboveneglectnot only otherelectroniccouplings,but alsofor theHund’scase(b) essential couplingto obtain . Thiscouplingcan,in goodapproxima-tion, be written as:

with asthespin-rotationconstant.It canbeinterpretedasaninteractionof thespinmag-neticmomentwith themagneticfield producedby molecularrotation.Thisfield canbepro-duced by:(1) simple rotation of the nuclear frame and(2) by excitation of electrons to states with non-zero angular momentum.Thesecondeffectis generallythemostimportantone,dependingontheseparationbetweentheelectronicstates.We considernow two examplesof case(b) couplingassuminga basisset of .

(1) states

Forthesestatesthespin-rotationinteractionis theonly onecontributingto therotationalen-ergy.

i.e.

Energylevels for and a statesare shownschematicallyin the figure. Statesforand areoftendesignatedasF1andF2 following Herzberg.The

spinrotationsplitting in thefigure is greatlyexaggerated.Eachvalueof occursin associ-ation with two possible values ofN (e.g. from N = 1 andN = 2).It shouldbenotedthat in the statea spin-orbittype interaction (see states)maycontributesignificantlyto . Thesplitting of a statedueto thespin-rotationinter-action is known asρ-doubling.

2Bv Λ2<– L⊥

2>( )–

Fv N( )Fv N( ) BvN N 1+( )=

N S⋅ J

HSN γ S N⋅( )=

γ

NSJM| ⟩

Σ2

HSN⟨ ⟩ 12--- J J 1+( ) S S 1+( )– N N 1+( )–[ ] 1

2---γ

N for J N 1 2⁄+=

N 1+( )– for J N 1 2⁄–== =

Fv N( )BvN N 1+( ) 1

2---γN+ for J N 1 2⁄+=

BvN N 1+( ) 12---γ N 1+( )– for J N 1 2⁄–=

=

Σ2 Π2

J N 1 2⁄+= J N 1 2⁄–=J

J 3 2⁄=Σ2

S li⋅∑ Σ3

γ Σ2

- 58 -

2) states

Thebestknownexampleof thisstateandcouplingis O2 in its groundstate.Sincetherearetwounpaired electronic spins the following interactions have to be considered:(1) spin-rotation,(2) spin-spin,(3) polarization, or induced spin-orbitSpin-rotationinteractionis, in principle,thesameasin thecaseof states.Thecontributionis givenby above,which for a state(S= 1) yieldsa triplet (thissplitting is sometimescalledρ-tripling):

It wasshownby Kramers(1939)thatin thecaseof two parallelspinsthis interaction,whenav-eraged over the molecular rotation, is equivalent to:

where is theanglebetween andthemolecularaxis.Sothespin-spininteractionis equivalentto . Calculationof thespin-spincontributionis ratherstraightforwardif usecanbemadeofangular momentum techniques. The result is:

where:

and is the spin-spin coupling constant. For S=1 we obtain from this expression:

Thepolarizationeffect involvesexcitationof electronsto stateswith non-zeroorbital momen-tum ; this momentum(or momenta)interact then with via a spin-orbit type coupling

. Hebb(1936)consideredacouplingof thetype . Anotherpossiblemechanismisamagneticspinexcitationof orbitalangularmomentum.Thisexcitationis especiallyimportant

2Σ

0

1

2

3

N1/2

1/23/2

3/25/2

7/2

5/2

2Π

1

2

3

N

1/23/2

3/25/2

7/2

5/2

Σ3

Σ2

Σ3

HSN⟨ ⟩γN for J N 1+=

γ– for J N=

γ N 1+( )– for J N 1–=

=

Hss λ 3 χ2cos 1–( ) λ 3Sz

2S

2–( )= =

χ SS A⋅

Hss⟨ ⟩ NSJ Hss NSJ⟨ ⟩ λ 3X X 1+( ) 4S S 1+( )N N 1+( )–2 2N 1–( ) 2N 3+( )

--------------------------------------------------------------------------------–= =

X J J 1+( ) S S 1+( )– N N 1+( )–=

λ

Hss⟨ ⟩

λN2N 3+----------------- for J N 1+=

λ for J N=

λ N 1+2N 1–---------------- for J N 1–=

=

li SS li⋅

i∑ S L⊥⋅

- 59 -

whenthereis a low lying state.Both mechanismsinvolve perturbationof thegroundstateby a state.Effectivehamiltoniansin bothmechanismshavethesameangulardependenceasthespin-spininteractionsand,exceptfor adifferentcouplingconstant,givethesamecontribu-tion to theenergy.Theeffectsof spin-spinandof theotherinteractioncannotbeseparatedfromobservedspectra.In the following we assumethat containsall theeffects.Theconstantcan be large; for O2.

= 39 667 MHz = -252.7 MHz B = 43 102 MHzThe resulting rotational energies are:

6.3.3 Case (c) and (d)

Case(c) occursquiteoften in heavydiatomicmolecules,(d) is howeverquiterare.Thelattercaserequiresa weakcouplingof to theinternuclearaxisanda strong (or ) cou-pling, conditions which are difficult to fulfil, simultaneously.If wedisregardall theweakereffectstherotationalenergyin case(c) is givenby thesameex-pression as in case (a):

Similar arguments yield for case (d):

Whenspin-orbitinteractionis largetheLS manifoldmayappearasasetof distinctenergylev-els.In case(d) themanifoldmaycontain levelsbecauseN cantakevaluesfrom to , exceptwhenR<L. SplittingbetweentheL sublevelsmaybelargebutbe-tweentheJ componentswhen is normallynegligiblebecausethe couplingis usu-ally very weak.

6.4. Intermediate Cases

6.4.1 Background

The calculationof rotationalenergiesin the preceedingsectionare not very accuratefor anumberof reasons.First of all, moleculesnevercomply to pureHund’scouplingcases.In amolecule canbe coupledto or to the molecularaxis in low rotationalstates,but astherotationalfrequencyincreasesandbecomeslargerthantheprecessionfrequencyof about ,

decouplesfrom andcouplesto instead.We thengeta transitionfrom case(a) to case(b), alsoknownasspin-uncoupling. Manymolecules,e.g.OH belongto anINTERMEDIATECASE.SimilarL-uncoupling occurswhenrotationbecomesmuchfasterthanprecessionofaboutthemolecularaxis.Wethengeta transitionfrom case(b) to case(d), which is ratherun-common.Anotherreasonliesin energiesinvolvedin thevariouscouplings.In case(a) is stronglycou-pledto andstatesdiffering (becausee.g.of spin-orbitcoupling)havelargeenergydiffer-ences.In case(b) is weaklycoupledto , andstatesdiffering in orientationof (i.e. in J)showonly asmallenergydifference.Consequentlyaneffectwhich is consideredasrelatively

ΠΠ

λ λ

λ γ

Fv N( ) BvN N 1+( ) γ N 1+( ) 3λ N 1+2N 3+-----------------–+=

Fv N( ) BvN N 1+( )=

Fv N( ) BvN N 1+( ) γN– 3λ N2N 1–----------------–=

L N S⋅ R S⋅

Fv J( ) BvJ J 1+( )=

Fv R( ) BvR R 1+( )=

2L 1+( ) 2S 1+( )R L+ R L–

S 0> N S⋅

S ΛS Λ

S Λ N

L

SA Σ

S N S

- 60 -

small in onecaseis not necessarilyso in another.A practicalconsequenceof this is thatoff-diagonalcontributionsof someinteractionsgenerallycannotbedisregardedapriori. In thissit-uation energies are solutions of (often large) secular equations.

6.4.2. Hamiltonians and representations

It shouldbe obviousby now that electroniceffectsuponrotationalenergieshaveto be takeninto accountright from thebeginning.Weshallconsiderexplicitly only therotationalenergiesin acertainelectronicstate,thevarioussmallereffectswill beincludedin alaterstage.Themo-lecular hamiltonian can, in this approximation be written as a sum:

is thevib-electronichamiltonianof anon-rotatingmoleculeand is therotationalham-iltonian. When is the molecular axis can be written as:

For calculations an equivalent hamiltonian is:

It is seenfrom thisexpressionthat equals whenboth and canbeignored(case(a) and(c), when canbut cannotbe ignored,and whennone of them can be ignored case (d).When spin-orbit interaction is considered explicitly can be written as:

The simplest basis functions for the calculation are:

where is therotationaland theelectronicwavefunction.Both haveto bespecifiedformally asfar aspossible.To this endwe haveto look which quantumnumbersaregoodorbad.QuantumnumberswhichMUST begoodin non-rotatingmoleculeanddegeneraciesof thestates in question are:

case (a) Ω (Ω=Λ+Σ) 2 or 1case (b) same 2(2S+1) or 2S+1case (c) Ω 2 or 1case (d) L,Λ,S,Σ (2L+1)(2S+1)

Whenthespin-orbitinteractionis large is agoodquantumnumberand is agoodbasisbecause the operator associated with satisfies:

However,onthisbasis or isNOTagoodquantumnumber.Whenthespin-orbitinteractionis small , and arealmostgoodquantumnumbersand is thebestbasiswe canchoose.Finally, whenbothspin-orbitandtheaxis interactionarenot strong(e.g.electronsinRydbergorbitals) , , and arealmostgoodand is agoodbasis.Thisbasismaycorrespondto case(d) whenenergies of the multipletsarelessthenthe rotationalseparationBJ; we have case (d) when all , case (b) is between (a) and (d).In thefollowing we shalladoptthe basisfor thenon-rotatingmolecule.But it shouldbe kept in mind that thesearenot exacteigenfunctionsof the hamiltonianbecausespin-orbit

H Hev Hr+=

Hev Hrz Hr

Hr B Rx2

Ry2

+( ) B J x Lx– Sx–( )2J y Ly– Sy–( )2

+[ ]= =

Hr B J2

Jz2

–( ) B L2

Lz2

–( ) B S2

Sz2

–( ) B L+S- L-S++( ) –+ + +=

B J+L- J -L++( )– B J+S- J -S++( )–

Hr >< BJ J 1+( ) L SBN N 1+( ) L S BR R 1+( )

Hev

Hev Hev0

A L S⋅( )+ Hev0 ς ri( ) li si⋅( )

i∑+= =

ev r,| ⟩ ev| ⟩ r| ⟩=

r> ev>

Ω Ω>Ω

Lz Sz+( ) Ω| ⟩ Ω Ω| ⟩=

Λ ΣΛ S Σ ΛSΣ>

Λ S Σ L ΛSΣ>E L

2S 1+( )E L

2S 1+( ) BJ»ΛSΣ>

- 61 -

mixesstateswith different , and . In calculationsit is oftenmoreconvenientto assumethat aretheexacteigenfunctionsbut thequantumnumbers , and arenot per-fectly good, i.e.

and are small functions which may be considered as due to mixing effects.The rotationalfunctions canbe specifiedas when and is the quantumnumberassociatedwith and , respectively.In theabsenceof externalfields and areperfectquantumnumbersbut is NOT a quantumnumberfor rotationbecauseit is not aneigenvalueof arotationaloperator.It entersin theproblembecauseof thepresenceof operator

. In the rotational function , should be considered as a parameter.The basis set for a rotating molecule is chosen to be

It is seenfrom theexpressionfor thattheoperators and contributeto matrixelementsvia products and . Selectionrules for matrix elementsof and

are:

Matrix elements of the relevant operators can be obtained from well known expressions, forexample:

For the operatorreplace by , by and by , but for the operatorshould be replaced by .Therearecomplicationsin actualcalculations(1) becausesomequantumnumbersmaynotbegoodor only approximatelygoodand(2) becausesomeof thequantumnumbersmaynot bedefinedatall, e.g.L. In thelattercasesomeof theexpressionscannotbeusedandthesematrixelements have to be represented by parameters.

6.4.3 State

Forthisstate . Thebasisset comprisesfour func-tions:

Diagonal matrix elements of are:

The matrix of these elements corresponds to two degenerate states with .In the representationchosen the matrix elements of and

Λ S ΣΛSΣ> Λ S Σ

Lz ΛSΣ| ⟩ Λ ΛSΣ| ⟩ δΛ| ⟩+=

Sz ΛSΣ| ⟩ Σ ΛSΣ| ⟩ δΣ| ⟩+=

δΛ> δΣ>r> ΩJM> J M

J2

JZ J MΩ

Lz Sz+ ΩJM> ΩJ>≡ Ω

ev r,| ⟩ ΛSΣ ΩJ,| ⟩ ΛSΣ| ⟩ ΩJ| ⟩= =

Hr L SJ xLx J yLy J xSx,, J ySy Lx Ly,

Sx Sy,∆Λ 1±= ∆Σ 1±=

SΣ S2

SΣ⟨ ⟩ S S 1+( )=

SΣ Sz SΣ⟨ ⟩ Σ=

S Σ 1±( ) S± SΣ⟨ ⟩ S Σ±( ) S Σ± 1+( )=

L S L Σ Λ SΣ> LΛ> J S±J+−

Π2

Λ 1 S,± 1 2 Σ,⁄ 1 2⁄±= = = ΛSΣ ΩJ, >

1| ⟩ 112--- 1

2--- 3

2--- J, , , ,| ⟩= 3| ⟩ 1–

12--- 1

2---– 3

2---– J, , , ,| ⟩=

2| ⟩ 112--- 1

2---–

1 3( )2

----------- J, , , ,| ⟩= 4| ⟩ 1–12--- 1

2--- 1

2---– J, , , ,| ⟩=

Hev

1 Hev 1⟨ ⟩ 3 Hev 3⟨ ⟩ E12---A+= =

2 Hev 2⟨ ⟩ 4 Hev 4⟨ ⟩ E12---A–= =

Ω 3 2 1 2⁄±,⁄±=B J

2Jz

2–( ) B S

2Sz

2–( ),

- 62 -

are all diagonal. The (off-diagonal) matrix elements of:

areall zero(theseoperatorsdo not contributeon thediagonal).Thereasonis thatthebasissetcontainsonly functionswith andtheoperators cannotproducenon-zeromatrixelementscorrespondingto . Sotheonlyoff-diagonalcontributioncomesfromtheop-erator:

whosematrixelementscanbeobtainedfrom thegeneralexpressions.Thefinal form of theen-ergy matrix with the present approximation is given in the table (see below).Eigenvalueof thismatrixaremosteasilyobtainedby subtracting1/2 tracefrom each2 x 2 ma-trix. The resulting eigenvalues are:

Thisexpression,with ignored,wasfirst derivedby Hill andVanVleck (1932)andis knownastheHill-Van Vleck expressionfor states.It representsenergyof two doublydegenerate levels.

WhenJ=1/2 the functions and do not exist in both matrices.Thehamiltonianmatrix factorstheninto two identical1x1 matrices.Theresultis a doublyde-generate level with energy:

Fromthegeneralexpression thelimiting (a)and(b) casecanbeobtainedby expandingthesquare root.

Case (a)

with replaced by and keeping only terms of order B/A we obtain:Apart from this expression agrees with .

Case (b)

B L2

Lz2

–( ) BL⊥2

=

B L + S– L– S ++( ) B J + L– J– L ++( )–

Λ 1±= L±∆Λ 1±=

B J + S– J– S ++( )–

EΠ E B L ⊥2⟨ ⟩ B J 1 2⁄+( )2

1–[ ] 12--- A A 4B–( ) 4B

2J 1 2⁄+( )2

+±+ +=

E B L⊥2⟨ ⟩+

Π2

B[J(J+1)-7/4+<L⊥2>]

-B[(J-1/2)(J+3/2)]1/2

0

0

-B[(J-1/2)(J+3/2)]1/2 0 0+A/2

B[J(J+1)+1/4+<L⊥2>]

-A/20 0

0

0

B[J(J+1)-7/4+<L⊥2>]

+A/2

B[J(J+1)+1/4+<L⊥2>]

-A/2

-B[(J-1/2)(J+3/2)]1/2

-B[(J-1/2)(J+3/2)]1/2

|1> |2> |3> |4>

|1>

|2>

|3>

|4>

1> 3>

EΠ1 2⁄E

12---A– B L⊥

2⟨ ⟩ B J J 1+( ) 14---++ +=

EΠ

sqrt A 1 2BA---–

12--- 4B

2

A2

---------

J 1 2⁄+( )21–[ ]+

=

A A12---B B S⊥

2⟨ ⟩= B J J 1+( ) Ω2–[ ]

- 63 -

For a pure Hund’s case (b) . With this value of A we obtain

These expressions are consistent with for .Thebasissetconsistingof functions is alsoquiteconvenientto handle statesinintermediatecoupling.This setcorrespondsto case(b), while theonewe haveusedis clearlyan(a) set.It is alsointerestingto notethatmolecularenergiesfollow approximatelycase(b),not only when but alsowhen . his quantityis a measureof thecouplingto the internuclearaxis. Generally has the samevalue for and just

. When therearetwo possiblevaluesof (onepositiveandonenegative)whichgive termswith thesameseparations.Thiscorrespondsto regular andinverted

doublets.In thefigure is shownacorrelationdiagrambetweencase(b) andcase(a)bothfor regularandinverted fine structure.Light molecules (large B, small A) approach case (b) already at low J-values, heavier mole-cules are closer to case (a). OH radical is a good example of this situation.

EΠ3 2⁄

12---A B J J 1+( ) 7

4---–+=

EΠ1 2⁄

12---A– B J J 1+( ) 1

4---++=

A 0=

EB J 1 2⁄+( ) J 3 2⁄+( ) 1–[ ]B J 1 2⁄–( ) J 1 2⁄+( ) 1–[ ]

=

BN (N+1) N J 1 2⁄±=ΛSNJ> Π2

BJ AΛ» A B⁄ 4=A A 4B–( ) A B⁄ x=

A B⁄ 4 x–= x 4> AA 0>( )

A 0<( )

1/2

3/2

3/2

5/2

5/2

7/2

1/23/2

5/2

7/2

3/2

5/2

J

2Π1/2(a)

2Π3/2(a)

F23/2

5/2

7/2

3/2

5/2

J

1/2

F1

2Π3/2(a)

2Π1/2(a)

2Π(b)

6.5 -doubling

6.5.1 Qualitative Features

As hasbeenshownelectronicenergydependson . Consequentlyall stateswith aredoublydegenerate.Similarly, in caseof electronicmultiplets,theenergydependsonly onandsothestateswith arealsodoublydegenerate.The -degeneracy,for examplein ,

statescanbelifted by spin-rotationandspin-spininteraction.It wasdiscoveredalreadyinthe1930’sthatalsostateswith weresplit into doublets.Theorigin of this splitting,for examplein and , mustobviouslybea lifting of the -degeneracyin thesestates.The splitting was consequently called-doubling.Interactionsresponsiblefor -doublingmustbe capableto reversenot only (e.g. from

or ) but also to . Theinteractionmustinvolve opera-tors and becauseonly thesefollow theselectionrule andsocanreversein oneormoresteps.An obviouscandidateis interactionbetweenmolecularrotationandelec-tronicorbitalmotion.It maybeseenasanincipientdecouplingof from themolecularaxisandhenceatransitionfrom case(a)or (b) to case(d). In thevectormodelpossibleinteractionsarecontainedin or i.e.of thetype or . However,theseeffectsmayalsobeseenasapartialbreakdownof theBorn-Oppenheimerapproximation.As acon-sequenceof thisbreakdownelectronsfollow, with aconsiderableamountof slippage,thenu-clear motion. The coupling operator is of the type .Fromaquantummechanicalpointof view -doublingmayalsobeseenasaresultof pertur-bationof thestatein questionby otherstates.Sincethisperturbationmustchange theper-turbatingstatemusthavedifferentvaluesof (so or statein caseof -doublingof a

-state),but thesamemultiplicity. Althougha perturbingstatemaylie morethan104 cm-1

awayfromthestatein questionit mayproducestrong,easilymeasurable,effects.Forexamplein OH the -splitting is:

1.6 GHz in , J= 3/26.0 GHz in , J = 5/24.7 GHz in , J= 1/2

althoughtheperturbing stateis 32000cm-1 away.Generallythesplittingis greatestforsmallest , andin statesmuchsmallerthanin states.A (very) essentialpoint in theperturbationapproachto -doublingis thattheeffectarisesfrom thedifferencein theinter-actionbetween(+) and(-) parity rovibrationalsublevelsof the interactingelectronicstates,for example and . Consequentlythetwo componentsof a -doublethavedifferentsym-metries,one(+), theother(-). The(+) statecanhaveloweror higherenergythanthe(-) state,bothcasesoccur.Lambdasplitting in a is shownschematicallyin thefigure belowwiththe correct parity of the -doublets (case of OH).Althoughelectroninteractionmayberegardedasthemainagentof -doublingspin-orbitin-teractioncannotbeneglectedin stateswith . In thesestatesthelatterinteractionis muchstronger than -doubling and may have a profound perturbing effect.

Λ

Λ2 Λ >0Ω2

Ω± Ω Σ2

Σ3

Ω 0=Π1 Π3

0 ΛΛ

Λ ΩΣ2

1 2⁄ Σ21 2⁄–→ Π2

1 2⁄ Π21– 2⁄→ Λ Λ–

Lx Ly ∆Λ 1±= Λ

L

f v Ω J,( ) f v Λ N,( ) BN L⋅ BJ L⋅

BR L⋅Λ

ΛΛ Σ ∆ Λ

Π

ΛΠ2

3 2⁄Π2

3 2⁄Π2

1 2⁄Σ2

1 2⁄Ω ∆ Π

Λ

Π Σ Λ

Π2

ΛΛ

Λ 0>Λ

- 65 -

The -splittings in can be described by simple empirical expressions:1Π qJ(J+1)2Π case(b) qN(N+1)2Π1/2 case(a) a(J+1/2)2Π3/2 case(a) b(J-1/2)(J+1/2)(J+3/2)3Π case(b) qN(N+1)

In the so-called pure precession approximation (introduced by Van Vleck) rough values of theconstants are ( is the separation between the and a perturbing state):

In the following section we shall discuss -doubling in a state in some detail and derive amore complete expression for and ; the limiting case (b) will yield the value for the -cou-pling constant q.Since the two -doublets have opposite symmetries electric dipole (E1) transitions are possiblebetween them. The transitions produce well known -doubling spectra in microwave and EBRspectroscopy. Microwave spectra of interstellar OH radical originate from these transitions.

6.5.2. Λ-doubling in 2Π states

Lambda-doubling in these states is most common and best known. Celebrated examples areOH, CH, NO. These states are split by spin-orbit interaction into and multiplets.The next higher electronic states are , and these may be regarded as the only perturbers. Inessence, -doubling in states arises because interaction between even parity stateswith, for example states is slightly different from the interaction between odd parity and

3/2+

-

5/2-

+

7/2+

-

1/2 -

+

3/2 +

-

5/2 -

+

J

J

2Π1/22Π3/2

Λ Π

∆ Π Σ,( ) Π Σ

q 2Bv2L L 1+( )

∆ ΠΣ( )---------------------

4Bv2

∆ ΠΣ( )-----------------= =

a4ABv

∆ ΠΣ( )-----------------=

b8Bv

3

A∆ ΠΣ( )---------------------=

Λ Π2

a b Λ

ΛΛ

Π21 2⁄ Π2

3 2⁄Σ2

Λ Π2 Π2

Σ2 Π2

- 66 -

states.Sincein thespin-doubletstatesspin-orbitinteractionscannotbedisregarded,a fairly completehamiltonian for a state which can show all the relevant effects, is:

(z is themolecularaxis).With this hamiltoniana case(a) basisset is theobviouschoice.Operatorswhichconnectstateswith different are and . Whenonly theseoper-ators are present the selection rules for matrix elements are:

Similarly different statesareconnectedby and andtheselectionruleswith only theseoperators are:

In thematrix elements and do not changethestates.Obviouslythreesuchoperatorsareneededto change into . Theseoperatorschangealso and , asrequiredfor -doubling.Consequentlywe expectthat -splitting in a stateshouldbeproportionalto if thissplittingis determinedonly by agyroscopicinteraction.But therea-soningfollowed aboveappliesalsoto states;we needthree and operatorstochangeboth and . However,it is knownfrom experimentsthat -doublingin statesis largerthanin statesandthat it is proportionalto , andsomustbegeneratedby asingleoperator.This is possiblethanksto thespin-orbitinteractionof theperpendicularcom-ponentsof and . Matrix elementsof theseoperatorswith and

connectthe stateswith , and , in a singletransition.It is interestingto notethat three matrix elementcanreverse

the signs not only of but also of and for a state.It shouldbeclearfrom theseremarksthata correctpictureof -doublingcannotbeobtainedby takingonly e.g.the interaction,insteadthecompletehamiltonianshouldbeconsideredin a secular matrix equation.To startwith weconsiderthetwo degenerate statesandasingle(alsodegenerate) stateperturbing the states. So we have a basis set of six states:

In this basis the non-zero matrix elements of the hamiltonian are:

Σ2

Π2

H Hr Hso+ BR2

A L S⋅( )+= =

B J x Lx– Sx–( )2J y Ly– Sy–( )2

+[ ]= ALzSz A LxSx LySy+( )+ +

B J2

Lz2

–( )= S2

ALzSz 2B J S⋅( )– B Lx2

Ly2

+( )+ + + +

2B A+( ) LxSx LySy+( ) 2B J xLx J yLy+( )–+

ΛSΣ ΩJ>,Λ Lx Ly

∆Λ 1±= ∆Σ 0=

Σ Sx Sy

∆Λ 0= ∆Σ 1±=

Sz LzΠ2

3 2⁄± Π23 2⁄+− Λ Λ–→ Σ Σ–→

Λ Λ Π23 2⁄

B3J

3

Π21 2⁄ ∆Λ ΛΣ

Λ Σ Λ Π21 2⁄

Π23 2⁄ Bv

L S LxSx LySy+( ) ∆Σ +1=∆Λ -2= Σ 1 2⁄–= Λ +1= Σ 1 2⁄= Λ -1=∆Ω -1= ∆Ω 1±=

Ω Λ Σ Π23 2⁄

ΛJ L⋅

Π2 Σ2

Π2 ΛSΣ ΩJ,| ⟩ ΛΣΩ| ⟩≡

012--- 1

2---, ,| ⟩ Σ2

1 2⁄| ⟩= 0 12---– 1

2---–, ,| ⟩ Σ2

1 2⁄–| ⟩= 1 12---–

12---, ,| ⟩ Π2

1 2⁄| ⟩=

1–12--- 1

2---–, ,| ⟩ Π2

1 2⁄–| ⟩= 112--- 1

2---, ,| ⟩ Π2

3 2⁄| ⟩= 1– 12---– 3 1( )

2-----------–, ,| ⟩ Π2

3 2⁄–| ⟩=

- 67 -

Energyof the stateis assumedto bezeroandso andthesubscript or is introduced to indicate that B and A can be different in the two states.

The diagonal matrix elements , , follow form the first four terms of H with. The “purely spin” matrix elements and follow from the

part of using the following phase conventions:

The “purely orbital” matrix elements and follow from the last term, with:

SinceL is notdefinedmatrixelementsof (and ) haveto beconsideredasparameters.The matrix elements follow from the last but oneterm.The term yields asmall contribution independent ofJ and is left out:

Σ21 2⁄ H Σ2

1 2⁄⟨ ⟩ Σ21 2⁄– H Σ2

1 2⁄–⟨ ⟩ α≡ BΣ J J 1+( ) 14---+ E+= =

Π21 2⁄ H Π2

1 2⁄⟨ ⟩ Π21 2⁄– H Π2

1 2⁄–⟨ ⟩ β≡ BΠ J J 1+( ) 14---+

12---AΠ–= =

Π23 2⁄ H Π2

3 2⁄⟨ ⟩ Π23 2⁄– H Π2

3 2⁄–⟨ ⟩ γ≡ BΠ J J 1+( ) 74---–

12---AΠ+= =

Σ21 2⁄ H Σ2

1 2⁄–⟨ ⟩ Σ21 2⁄– H Σ2

1 2⁄⟨ ⟩ δ≡ BΣ J12---+

= =

Π23 2⁄ H Π2

1 2⁄⟨ ⟩ Π23 2⁄– H Π2

1 2⁄–⟨ ⟩ ε≡ BΠ J12---–

J32---+

= =

Π21 2⁄ H Σ2

1 2⁄–⟨ ⟩ Π21 2⁄– H Σ2

1 2⁄⟨ ⟩ ° ς≡ Π 2BLy Σ⟨ ⟩ J12---+

= =

Π23 2⁄ H Σ2

1 2⁄⟨ ⟩ Π23 2⁄– H Σ2

1 2⁄–⟨ ⟩ ° η≡ Π 2BLy Σ⟨ ⟩ J12---–

J32---+

= =

Π21 2⁄ H Σ2

1 2⁄⟨ ⟩ Π21 2⁄– H Σ2

1 2⁄–⟨ ⟩ ° θ≡ Π 2BLy ALy+ Σ⟨ ⟩= =

Π E hν Π Σ,( )= ΠΣ

α β γ2 J S⋅( )– 2ΣΩ–≡ δ ε

J xSx J ySy+ 2 J S⋅( )–

i Σ Sx Σ 1±⟨ ⟩± Σ Sy Σ 1±⟨ ⟩ 12--- S S 1+( ) Σ Σ 1±( )–= =

i Ω J x Ω 1±⟨ ⟩± ΣΩ J y Ω 1±⟨ ⟩–12--- J J 1+( ) Ω Ω 1±( )–= =

ζ η

i Λ Lx Λ 1±⟨ ⟩± Λ Ly Λ 1±⟨ ⟩=

Ly Lxθ B Lx

2Ly

2+( )

- 68 -

With thesematrixelementsthehamiltonianmatrixhastheform givenin thematrix.Thismatrixcan be reduced to two 3 x 3 matrices by using a symmetrized basis:

These functions have definite positive-negative (or Kronig’s even-odd) parity.Matrix elementsof thehamiltonianon thebasis caneasilybeobtainedfrom theex-pressions;for symmetric statestheequationsarevalid without anychanges,for antisym-metric statesthe matrix elementsindicatedby an asteriskget a negativesign.All matrixelementsbetweenstatesof differentparityarezero.This follows from invarianceof thehamil-tonian under the inversion operation.Theresultingmatricesyield two cubicsecularequations(onefor + andonefor the- sign)whichcannotbe solvedexactly.Dousmaniset al (1955)useda perturbationexpansionof the roots(similarto perturbationexpansionin quantummechanics)andobtainedratherlengthyandcom-plicatedexpressionsfrom which thedesired -splittingcanbeobtained.Theseformulaehaveto be used when high accuracy is desired. Below we shall consider three special cases.

|2Σ1/2>

α

δ

θ

ζ

η

0

|2Σ-1/2>

δ

α

ζ

θ

0

η

|2Π1/2>

θ

ζ

β

0

ε

0

|2Π-1/2>

ζ

θ

0

β

0

ε

|2Π3/2>

η

0

ε

0

γ

0

|2Π-3/2>

0

η

0

ε

0

γ

|2Σ1/2>

|2Σ-1/2>

|2Π1/2>

|2Π-1/2>

|2Π3/2>

|2Π-3/2>

1

2------- Λ2

Ω| ⟩ Λ2Ω–| ⟩±[ ] ΛΩ ±| ⟩≡

ΛΩ ± >Σ+

Σ-

Λ

|2Σ1/2±>

α ± δ η

|2Π >

θ ± ζ

|2Π >

|2Σ1/2±>

|2Π >

|2Π >

θ∗ ± ζ∗ β ε

η∗ ε∗ γ

- 69 -

Case (1): Effect of the 2Σ states negligible

In thiscasethematrixelements , and arezeroandthehamiltonianmatrix reducesto:

The resultingsecularequationcan be readily solved for the energiesof the two states, and perturbed wavefunctions. The results for the energies are:

This resultshowsthatin theextremecase(b) the -splitting originatesfrom thesmalldif-ference between the interactions of the positive and negative states.

Case (2): A=0

This represents an extreme case (b). With A=0 the matrix elements are:

With thesematrixelementsthetwo cubicsecularequations(onewith and , theotherwith and bothfactorinto aproductof a linearandaquadraticequation.Thesecanbesolved to:

Herein is theenergyseparationbetweenthe and statesandthesubscript(+) and(-) indicatesa symmetricandanantisymmetricstate,respectively.However,when

as we haveassumed,the two statescoalescein energy.This implies that weshouldreplace and by N. The -splitting is thenobtainedby subtracting

η ζ θ

α δ+ 0 0

0 β ε0 ε° γ

Π21 2⁄ Π2

3 2⁄,( )

E±12--- α β+( ) BΠ 4 J

12---+

2 AΠBΠ-------

AΠBΠ------- 4–

+±=

Λ

κ + α δ+ E BΣ J 1 2⁄+( ) J 3 2⁄+( )+= =

κ– α δ– E BΣ J 1 2⁄–( ) J 1 2⁄+( )+= =

µ + Θ ς+ J 3 2⁄+( ) Π 2BLy Σ⟨ ⟩= =

µ– Θ ς– J 1 2⁄–( ) Π 2BLy Σ⟨ ⟩–= =

β BΠ J J 1+( ) 1 4⁄+[ ] BΠ J 1 2⁄+( )2= =

γ BΠ J J 1+( ) 7 4⁄–[ ] BΠ J 1 2⁄+( )22–[ ]= =

ε BΠ J 1 2⁄–( ) J 3 2⁄+( )=

η J 1 2⁄–( ) J 3 2⁄+( ) Π 2BLy Σ⟨ ⟩=

κ+ µ+κ- µ-

EΠ + BΠ J 1 2⁄+( ) J 3 2⁄+( ) 1–[ ] 8 J 1 2⁄+( ) J 3 2⁄+( )Π BLy Σ⟨ ⟩ +

2

∆ Π Σ,( )------------------------------------------–≅

EΠ – BΠ J 1 2⁄–( ) J 1 2⁄+( ) 1–[ ] 8 J 1 2⁄–( ) J 1 2⁄+( )Π BLy Σ⟨ ⟩ –

2

∆ Π Σ,( )-----------------------------------------–≅

∆ Π Σ,( ) E= Π Σ

A 0= ΠJ 1 2⁄+ J 1 2⁄– Λ

- 70 -

the difference between and .The result can be written in the form

where

This resultshowsthat in theextremecase(b) the -splitting originatesfrom thesmalldiffer-ence between the interactions of the positive and negative states.

Case (3):

Usuallytheenergydifference betweenthe and statesis muchlargerthanthe -splitting (in OH = 32 683cm-1), -splitting 0.05- 1 cm-1). In this case canbeassumedto bea constant . Thesecularequationreducesthento oneof secondorder:

which can be solved. If we disregard terms of order with the result is:

where again:

Thetwo signscorrespondto thetwo componentsof thespin-doublet:for (normaldou-blet) the (+) signcorrespondsto , the (-) sign to , for (inverteddoublet)thecorrespondence is reversed.The -splitting is obtainedby subtractingtheexpressionwith only from thatwith only :

After filling in some of the constants:

EΠ+ EΠ-

∆νΛ C1 C2–( )N N 1+( )=

C1 2Π BLy Σ⟨ ⟩+

2

∆ Π Σ,( )-----------------------------------= C2 2

Π BLy Σ⟨ ⟩–2

∆ Π Σ,( )-----------------------------------=

Λ

κ W– ∆ Π Σ,( )–=

∆ Π Σ,( ) Π Σ Λ∆ Π Σ,( ) Λ κ W–

∆ Π Σ,( )–≡

W2

W β γ 1∆--- µ± µ± ° ηη°+( )+ +– +

βγ εε°–1∆--- µ± µ± °γ ηη°β µ± εη°– µ± °ε°η–+( )++ 0=

Λ n–n 1>

2W β γ BΠX± 1∆--- µ± µ± ° ηη°+( ) ±+ +=

1BΠX----------- β γ–( ) µ± µ± ° ηη°–( ) 2µ± εη° 2µ± °ε°η+ +[ ]±

X 4 J 1 2⁄+( )2Y Y 4–( )+= Y AΠ BΠ⁄=

A 0>Π3 2⁄ Π1 2⁄ A 0<

Λ µ- µ+

∆νΛ1

2∆------- µ + µ + ° µ– µ– °–( ) 1

β γ–BΠX------------±

Φ+

=

Φ 2BΠX----------- µ + εη° µ + °ε°η µ + εη°– µ– °ε°η–+( )=

∆νΛ1∆--- J

12---+

1 2X----

YX----+−

± Π 2BLy ALy+ Σ⟨ ⟩ Π 2BLy Σ⟨ ⟩° cc+[ ] Φ2±

=

Φ24X---- J

12---–

J32---+

Π 2BLy Σ⟨ ⟩ 2=

- 71 -

Theuppersignsareappropriatefor theupperspin-component,thelower signsfor thelowercomponent. From the last expression only the real part should be taken.Evenin thepresentapproximationthetheoryneedssomeextension.All therelevantmatrixelementsshouldbereplacedby a sumoverthevibrational(v,v’) levelsof a givenelectroniclevel andoverelectroniclevelsof propersymmetry.In a morecompletetheoryalsointerac-tion between and levelsshouldbeincluded.Following Mulliken (1931)term-energiesand -splitting of levels are usually written in terms of the following (-)constants:

where:

( indicatesrovibrationalandelectronic stateswith positiveandnegativeparity).Withthese constants the expression for the term energy is:

where:

Thebracketedsigns referto thesymmetric(+) andantisymmetric(-) -levels.Theun-bracketed signs refer to the two components of the spin doublet.Althoughtheexpressionlooksrathercomplicated,theonederivedfrom it for the -splittingis simple:

( refers to components of the spin-doublet).

Π ∆Λ Π2 Λ

q C1 C2–=

p a1 a2–=

C1 8Σ v′ Σ2+,( )

Π 2BLy Σ⟨ ⟩ 2

∆ Π Σ,( )------------------------------------=

C1 8Σ v′ Σ2–,( )

Π 2BLy Σ⟨ ⟩ 2

∆ Π Σ,( )------------------------------------=

a1 8Σ v′ Σ2+,( )

Re Π ALy Σ+⟨ ⟩ Σ+BLy Π⟨ ⟩[ ]

∆ Π Σ+,( )------------------------------------------------------------------------=

a2 8Σ v′ Σ2–,( )

Re Π ALy Σ–⟨ ⟩ Σ–BLy Π⟨ ⟩[ ]

∆ Π Σ–,( )------------------------------------------------------------------------=

Σ± Σ

T J( ) T 0 Bv J12---+

21–

12---X± 1

2--- o

12--- p° q° J

12---+

2+ ++ + +=

12X------- 2 Y–( ) o

12--- p° q°+ +

p° 2q°+( ) J12---–

J32---+

+± +

±[ ]12--- J

12---+

1± 1X---- 2 Y–( )+

12--- p q+

12X-------q J

12---–

J32---+

+

+

p° a1 a2+= q° C1 C2+= o Σ ν′ Σ2,( ) <Π| ⟩ ALy Σ>2/Λ Π Σ,( )=

±[ ] Λ

Λ

∆νΛ J12---+

12--- p q+

1± 2X---- Y

X----–+

2qX------ J

12---–

J32---+

+=

±

- 72 -

Limiting Case (b)

In the extreme case (b) and we obtain:

with the(+) signfor andthe(-) signfor . In thesimplestcaseof a state the -doubling constant is:

Thematrix elementsappearingaredifficult to evaluate,usuallytheyareobtainedfrom a fit ofobservedspectra.A roughestimatecanbeobtainedfrom theso-calledPUREPRECESSIONapproximation. It is assumed (Van Vleck 1929) that:(1) vibrational excitation in the states is small so that is a constant,(2) momentof inertiais independentof (goodapproximationonly for electronsin Rydberg-

type orbits),(3)angularmomentum of constantvalueprecessessuchthatboththe andthe stateresult

from projection of this .In this approximation the matrix element the result for q is:

Often it is assumed that is that of a single electron .

Limiting case (a)

For this case we use the approximation

By substitutingthis expressionwe obtainin orderfor thesplitting of the state(withthe lower signs in the equation for ).:

with:

hereinthe(+) signcorrespondsto andthe(-) signto . If thegeneralequationis usedthe result for is:

Theuppersignsin yield zeroin order.If we neglecttheJ-independenttermsin theexpan-sion we obtain for the splitting of the state in order:

A 0= Y 0=

∆νΛ qN N 1+( )±=

N J 1 2⁄+= N J 1 2⁄–=Σ+ Λ

q 8Π BLy Σ⟨ ⟩ 2

∆ Π Σ,( )--------------------------------=

Σ B Bv≡Ω

L Π ΣL

q2Bv

2L L 1+( )

∆ Π Σ,( )-------------------------------=

L L 1→( )

Y ∞±→( )1X---- 1

Y------ 1 2

Y---

2

Y2

------ J12---+

2–+≅

Yo Π2

1 2⁄∆vΛ

∆νΛ1 2⁄

a J12---+

±=

a2

∆ Π Σ,( )-------------------- Π 2BLy ALy+ Σ⟨ ⟩ Π 2BLy Σ⟨ ⟩° cc+ =

A 0> A 0<∆vΛ

∆νΛ1 2⁄

p 2q+( ) J12---+

±=

Yo

Π23 2⁄ Y

1–

∆νΛ

32---

b J12---–

J12---+

J32---+

=

- 73 -

where

(with the approximation ). The expression yields to order :

with the (+) sign for corresponds to and the (-) sign to .Theseresultsagreewith theexpressionsproposedin the introductionandwe havetheex-pectedB dependence( for for ). However,theaccuracyis limited bytheapproximation = constant,neglectof couplingbetween and states,neglectof the spin-rotation, etc. However, the J-dependence derived is confirmed by experiments.Fromtheresultsfor thelimiting casesit is possibleto drawa correlationdiagrambetweenthe -split levelsof a multiplet for case(a)and(b) coupling.Suchadiagramis shownin thefigurebelow.Thebasisrule for thisdiagramis thatlevelsof agivensymmetry“com-bine” only with levelsof thesamesymmetry;thespin-splitting,of course,disappearsin caseA = 0 andthetwo spin-doubletcomponentscorrelatewith asingleN-level.Weseethatthe

-levelsfor eachJ-valueexcept of the multiplets(bothnormalandinvert-ed) cross. So there are values ofY such that the -splitting disappears for these levels.Transitionsareallowedbetween -sublevelsof different parity both of different andthesamero-vibrationallevels.The latter typeof transitions,alsoknownas -transitions,arewell known from interstellar OH and CH radicals and from laboratory spectra of NO.

b4

Y∆ Π Σ,( )----------------------- Π 2BLy Σ⟨ ⟩ 2 16B 4( )

A∆ Π Σ,( )------------------------ Π BLy Σ⟨ ⟩ 2

= =

16B 4( )∆ Π Σ,( )∆ Π1 2⁄ Π3 2⁄,( )-------------------------------------------------------- Π BLy Σ⟨ ⟩ 2

=

E Π21 2⁄( ) E Π2

3 2⁄( )– A= Y2–

∆νΛ

32--- p

Y2

------ 2qY------+

J12---–

J12---+

J32---+

±=

A 0> A 0<

~B3 Π2

3 2⁄ ~B Π21 2⁄

κ W– Π ∆

Λ Π2

Λ J 1 2⁄= Π1 2⁄Λ

ΛΛ

1 -+

2 +-

3 -+

4 +-

A=0-+

N

1/2

J

+-

3/2

-+

5/2

+-

7/2

2Π1/2

+-

3/2

-+

5/2

+-

7/2

2Π3/2

A>0

-+

3/2

+-

5/2

-+

7/2

-+

3/2

+-

5/2

-+

7/2

+-

1/2

A<0 J

- 74 -