Dr Stuart Mackenzie - University of Oxford

1 atomic and molecular SAtomic and molecular SSpectroscopySpectroscopyDrStuartMackenz ieDr Stuart MackenzieAtomic StructureQuantum theoryatoms / moleculeskineticsthermodynamicsAtomic&Mo lecularStatisticalQuantum MechanicsRateAtomic & molecular SpectroscopyStatistical mechanicsValenceRate processesPhotochemistryReaction DynamicsLasers NMRS olids &surfaces ynamicsResourcesResources Handouts (colour online) Tutorialsuto a s Books: Modern spectroscopy (4thed. 2004) JM Hollas High Resolution spectroscopy (2nded., 1998) JM Hollas molecular spectroscopy (OUP Primer) JM Brown SpectraofAtomsandMolecules(2nded2005)Ber nath Spectra of Atoms and Molecules(2nded. 2005) Bernath fundamentals of molecular spectroscopy (4thed. 1994) Banwell & McCash atomic Spectra(OUP Primer) TP Softley molecular Quantum Mechanics(4thed.) Atkins and Friedman Electronic and Photoelectron spectroscopy , Ellis, Feher and WrightPil Practicals: II 03 HCl, DCl spectraII 04 Fluorescence and quenching II 05 I2visible spectrum II 08 Flame atomic absorption2pp II 10 Na/Na+ atomic spec II 17 Computational Raman II 18 N2+spectrumLecture 1: General Aspects of SpectroscopyLecture 1: General Aspects of spectroscopy Electromagnetic Electromagnetic radiation Transverse wave of perpendicular, sinusoidallyoscillating electric and magnetic fields ()EEik t with wavevector, k= 2 / andangularfrequency 2 ()0 EEsinkxt = +Band angular frequency, = 2 Characterised by.

2 Wavelength, (in m) or A plane electromagnetic frequency, (in Hz) Speed in vacuodefinedas cvac= 299 792 458 ms 1 c= = /kwave propagating in the z direction c= = /k cvacis related to the permittivity (electric constant) and permeability (magnetic constant) of free space:21=c(proof comes from Maxwell s Equations)00 Itillllbittidli Quantised Light: Quantised Light: PhotonsIt will usually be convenient to consider light as a stream of zero rest mass particles or packages of radiation called photons with the following properties: Energy, E= h in which his Planck s constant, h= x 10 34 JsMax Planck (1855 1947) Linearmomentump=E/c=h /c=h/ (deBroglie) Linear momentum,p =E/c = h /c = h/ (de Broglie)Louis de Broglie(1892 1987) (spin) Angularmomentumequivalent to a quantum number of 1:jph1 ,===j(18921987) , 1) photons are Bosons ( , obey Bose Einstein statistics)2)photonshavehelicity(project ionofangularmomentumonthej phph,j2) photons have helicity(projection of angular momentum on the direction of travel) of 1 only ( , not 0) Quantities and Units Quantities and Units Wavelength, : SI unit = m[or m, nm or Angstr m, 1 = 10 10m] is dependenton the (refractive index of the) medium in which the wave travelsFrequency, :SI unit = Hz ( , cycles s 1) [or MHz = 106Hz , GHz = 109Hz]frequency is independent of the mediumEnergy, E:SI unit = J,BUT : It is hard to measure energy directly.

3 Spectra are recorded as line intensities ftifflthas a function of frequencyor conversion to energy appearssimple: E = h = hc/ But h is only known to 8 significant figures. Hence, it is convenient to introduce1 Wavenumber, a propertydefined as reciprocal of the vacuum wavelength: 1vac =and whose units are universally quoted as cm 1( not m 1)Wavenumberis directly proportional to energy, and thus we commonly quote energies in units of cm 1. =EhcThe total energy of a molecular system Energy levels: The Born Oppenheimer Energy levels: The Born Oppenheimer Approximation The translation of the whole molecule, Ttrans Kinetic energy ,TeandTnof electrons and nuclei, respectively( we ll neglect this as trivial)e r 2 Potential energy, Veeand Vnnof electrons and nuclei, respectively Potential energy between nuclei and electrons, Vne Valences HT yeanneeeenetotnn=++++=+The Born Oppenheimer Approximation (Annal.)

4 Phys., 84, 457 (1927))See noteTheornOppenheimerApproximation( ,84,457(97))Due to the difference in mass between the electron and nuclei, the motion of the two may be separated and the total molecular wavefunction, tot,may, to a good approximationbewrittenapproximation, be written()()tot elnq,Q Q =nuclear coordinateselectron coordinatesEtot= Eel+ Enucand the resulting total energy is a simple sumIt will be convenient, though less rigorous, to further factorise nfurther into vibrational and rotational parts so tot= el vib rotandEtot= Eel + Evib+ ErotMolecular lMolecular lEnergy LevelsEnergy , typically Eel>> Evib>> ErotDifferent electronic states (electronicarrangements(electronic arrangements,configurations or terms) E2 x 104 105cm 1500 100 nm102 5 x 103cm 1100 m 2 m3 300 GHz ( 10 cm 1)Transitions at Vis UV 00 infrared10 cm 1 The Population of Energy The Population of Energy levelsTheBoltzmannLaw EAt thermal equilibrium, the population of the ithenergy levelis given by: EniEiThe Boltzmann Law = iiiENngexpqkT En0E=0 Where: = iilevels ,iEqgexpkTWhere:qis the molecular partition function (see HT Stat.)

5 Mech. notes)giis the degeneracyof the ith level (the no. states with same energy)EistheenergyoftheithlevelEiis the energyof the ithlevelkis the Boltzmann constant( = R/NA= x 10 23 J K 1) Tis the Kelvin temperatureHencerelativeton: iingEexpLudwig Boltzmann 1844 1906 Hence, relative to n0:= iiexpng The Interaction of Light and Matter I: A simple classical The Interaction of Light and Matter I: A simple classical pictureConsider the ways in which a single photon might interact with a system of two energy levels E1and E2, with populations n1and n2, respectively:ildbih*E2n2A. Stimulated absorption, M + h M* The photon is lostE1n1 The system absorbs energy E= h = E2 E111rateofabsorption dndnEnB En E1n11121 112 21 1rate of absorption EnB Endtdt = = In which B12is the Einstein Coefficient of Absorption and (E21) is the radiation energy density (energy of radiation field m 3) at energy E21, which, for a black body at temperature T, is given by Planck s Law yp,gy()3381radiation density, hEcE = 1cEexpkT EnB.

6 Stimulated emission M* + h M + 2h Additionalphotoncreatedwithsamefrequency E2n2 Additional photon created with same frequency, polarization, direction and phase as the original The system relaxes, ,emits energyE1n12221 221 21 2rate of stimulated emission dndnEnB Endd = = in which B21is the Einstein coefficient of stimulated 221 21 2 dtdt Einstein showed that for a system to reach equilibrium a 3rdprocess must occur:E2n2C. Spontaneous emission M* M + h A photon is created with E= E2 E1= h Thtliit= = dndnnAn2222rateofspontaneousemissionE1n1 The system relaxes, ,emits energyand Ais the Einstein coefficient of spontaneous emission (or Einstein A coefficient ) nAndtdt22rate of spontaneous emission The Einstein Coefficients [A. Einstein, Z. Phys.,18, 121 (1917)] The Einstein Coefficients [A. Einstein, Z. Phys.,18, 121 (1917)] equilibrium:()()112 21 1 21 2 21 21 2 0, , B E n A n B E ndt ==+ ()(){}21 2212112112 1 21 2gEAnAEBn BnBexp B == Rearranging, (){}12 1 21 212212gkTBexp B()321381 hE = Planck s Law()213211cEexpkT f31 12 2 21212138and AhgBgBBc ==Yielding:There is only one independent Einstein coefficientWhat are the implications of the fact that the A coefficient, A 3?

7 Interactions of Light and Matter II: A time dependent Interactions of Light and Matter II: A time dependent treatmentE2n2E2n2We will often use pictures liketo consider our approach will bei) to determine the eigenstates (stationary states) of a system and then ii) consider allowed transitions between these , the photon doesn t expicitly figureThe total wavefunction, tot, satisfies the time dependentSchr dinger equation:00 where and H iH H Vt Vt Ecostt ==+= =Eigenstates are the solutions of the t independent Schr dinger eqn: 00nnn HE =and the full (t dep) wavefunction is{}onnexp iE t / = totis a linear combination of stationary states{} onn nnct expiEt/ = =Time-dependent coefficientsAftilti(MQMCh6)itthtftitittt After some manipulation (see MQM, Ch 6), we arrive at the rate of transition to state mfrom a well defined, , pure, initial state, j, to be:()()0000 ()()00000 2mjmjmmjiE E t iE E tEdc t expexpddt i + =+ ==== = 123 Thus, for non zero transition probability ( , allowed transitions): 1 0 theremustbenon zeroradiationintensityandE 0001.

8 0 there must be non zero radiation intensity, 2. , energy must be conserved, 30 The""mustbenon ztransitiondiperolaemoomentndandmjEEE d = =3. 0 The must be non ztransition diperole moo mentmjd The Transition Dipole Moment, The Transition Dipole Moment, R21 The transition dipole moment, TDM, is defined as212 12 1* Rd == P f hlwhere the dipole moment operator, iii qr = Charge on ithparticlePosition vector of ithparticleCharge on ithparticle operates upon our initial wavefunction 1producing a new state1 = TDM, R21,thus represents the transition amplitudeof ending up in our particular state, 2, determined by the overlap integral of 2with : , 2,ypg 2 221 = The rate of transition (or intensity) is the square of this amplitude:()222titiitit* iRd ()22212 121transition intensity ,Rd == The TDM is, unsurprisingly, closely related to the Einstein B coefficient (after all they both describe the same thing).

9 32281 ()22212121220081643 BRRh === The Transition Dipole Moment and spectroscopic selection The Transition Dipole Moment and spectroscopic selection rules()222212 121* == TheTDMis thus the ultimate source ofspectroscopic selection rulesfor dipoleallowedtransitions , of all the conceivable energetically allowed transitions it determines whichactually occur and encompasses symmetry and angular momentum have R21= 0 AllowedtransitionshaveR21 0 Allowedtransitions have R21 0