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Date Created: 10/15/15
Introduction to Polarizability Motivation Raman scattering KramersHeisenbergDirac Timedependent formulation Energy Raman scattering Scattered Incident l 4 2 O 2 Nuclear Displacement Raman scattering is an inelastic light scattering process In the resonant picture it involves evolution in the excited state so it also depends on the FC factor and the transition dipole moment On the left a sumover states picture is shown KramerHeisenbergDirac Raman scattering depends on the molecular polarizability The electric field drives the system into excited state ngt and then from ngt it returns to the final state fgt a 1 ltilengtltnlrtclfgt ltiuclngtltnluplfgt p6 if hn1 wO wnf Irn Two electric field interactions are required In the above expression p and G are any two of the Cartesian coordinates x y or z The energy differences mnf include both electronic and vibrational contributions eg an Deg 39 r10vib39 We can consider both nonresonant and resonant cases for KHD Nonresonant KHD Far from resonance we can ignore the vibrational energy difference and the KHD expression becomes 1 0 0 ltOngtltnfgt ltOngtltnfgt apG0f Eev Mgepugeo 00 Deg DO Deg The quantities ltOngt and ltnfgt are vibrational overlaps The square of a vibrational overlap is a FranckCondon factor so the Raman excitation profile bears a defined relationship to the absorption spectrum Here the n quantum numbers refer to the intermediate state vibrational energy levels Since excitation is off resonance there are in principle many vibrational and electronic states that can contribute We can use the closure relation 2Vngtltn 1 to simplify the expressions Nonresonant KHD 2coe 5 OLPG0f H2epuge0 2 93f02 eg 200eg 2 2 eg DO The bottom equation describes Rayleigh scattering The initial and final vibrational states are the same in Rayleigh scattering There are no selection rules All molecules are active Rayleigh scatterers The Kroenecker delta in the top equation is 50f ltOfgt where 50f 0 if 0 7 f Thus the first term and the second term are the same here This means that nonresonant Raman scattering will not occur within the Condon approximation Nonresonant KHD In order to explain nonresonant Raman scattering we must consider the coordinate dependence to the transition dipole moment eg expand the transition dipole moment in a power series We a 96Q 26 21 Q I o and keep only the first term we find that the coordinate dependence of the transition moment can play a role The reason for this is that even though ltOfgt O for O 7 f in general ltOQfgt does not need to be zero Thus for Raman scattering to be allowed we use the linear term above and make the substitution 5 Mgelt0fgt gta e ltOQfgt 0 Nonresonant KHD The transition polarizability is 6 0 5 Se M394 86lt0Qfgt u386plt0ofgt l apcov39 hie 030 0 eg CO0 Deg The selection rules arise from the requirement that ltOQfgt does not vanish We can define a polarizability derivative such that the transition polarizability is does upclwrr a cilt39IQIfgt 0cm ltIIQIIfgt where oc39rs is the polarizability derivative also called the derived polarizability The terms oc39rs and doors6Q are equivalent For a harmonic oscillator ltiQfgt vanishes except when f i i 1 Thus the selection rule of Av 1r 1 applies to nonresonant Raman scattering as well as infrared spectroscopy within the harmonic approximation Resonant KHD If the incident frequency 030 is in resonance with an electronic transition of the molecule the antiresonant term with 030 meg in the denominator can be neglected and only the resonant term contributes to Raman scattering If we keep terms up to linear in Q we may express the transition polarizability as a sum of two terms polf T Aif Bif These terms are called the Albrecht A and B terms The first of these terms arises from the Condon approximation The Condon approximation states that there is no nuclear coordinate dependence to the wave function The second term arises from vibronic coupling This term depends on the derivative of the transition moment with respect to nuclear coordinate The Albrecht Aterm For resonant Raman scattering within the Condon approximation the following expresssion applies A ugepu26 m ltquot gt lt 39fgt 00 iFeV In this expression the energy of an incident photon is equal to that of the energy difference between a ground state vibrational energy level gv39 and an excited state level ev The term ifeV is a phenomenological damping term This term arises from dephasing and lifetime broadening in the excited state levels One can envision the contribution of F as an energy width to each of the excited state energy levels Note that this is a sumover states expression and requires explicit calculation of all possible excited state contributions to the transition polarizability engi Quantum Mechanical Definition of the Polarizability We return to the definition of the polarizability far from resonance a l2 ltiluplngtltnlualigt ltiluplngtltnlualigt PGif nn D Dni 00an If we use the timedependent wave functions that we have already derived we can obtain a first order correction to the transition moment lt wtlulvft gt e mrufperm ufind Timedependent Treatment The term uifperm is the transition moment evaluated with the zeroorder states uifperm ltiufgt The permanent transition moment is responsible for absorption and emission The induced transition dipole is the molecular polarizability The dipole and polarizability must be Hermitian uh u and or oc This means that the induced dipole will be of the form l ioot ioat umd 2oche oc e E0 The derivation will proceed as follows We substitute in the firstorder wave functions expand and compare to the above results The comparison will give us an expression for or Timedependent Treatment The first order wave function for state f is watgt C itiequotEn Ingt with a similar expression for state i The coefficients are given by t j L I iconzquot I ct 5 h dt e 1 t The Kroenecker delta 5 1 if n j and 5 0 if n 7 The matrix element is V t ltnLl E0jgteit e ioat E nceioat e ioat In this representation ano represents the magnitude of the pertubation 0 1 an Hnj 39 E0 and pm is the transition Timedependent Treatment The coefficients are found by direct integration 39 0 IVt In dtle Iwnjtelot e th n o 033V 5n 0 8 I Viv ewmnfy 1 em wnfy 1 n h 3 03m 0 pm A similar expression holds for coefficient cnit The coefficients are now substituted into the expression for the perturbed wavefunction 0 0 t 0 03 t 39Et39 6 quotf 1 e quotf 1 IWftgt e I fn fgt Z D 0m ngt n f 03an Timedependent Treatment Again a similar expression holds for wit Physically we are interested in the induced dipole moment on a molecule at the frequency of an applied electric field The wavefunction is perturbed at the field frequency eii03t The part of the transition frequency that depends on expioanft does not contribute to the induced transition moment This is called the rotating wave approximation We can rewrite the wavefunction as V0 iwt 4m gt Efth gt n e e gt Mt e If h 3 W 03 can Iquot and the complex onjugate is Mg e ioat eioat h CD con 0 COni Note that the ind x changes from fto i ltwt lti ltnI Timedependent Treatment The zero order wavefunctions lti and fgt combine to give the ordinary transition moment Hifperm lt iIHIf gt The induced dipole moment is 1 eimt e imt inr70fm an Q an i03t foot ani2 e e h D 03m D Dni Substituting in the form of the matrix elements ano we have 1 eioat e ioat Hifmd Hianf mnf 03 03nf Eo 1 e ioat eiwt ivlnrivlm m mm D Dm Eo Timedependent Treatment Comparing this expression to our original hypothesis regarding the form of the induced dipole moment ufind ocfe ageI39m E0 gives the following expression for the transition polarizability 1 mun Mn in a 0 032 03 f 0 It is crucial to keep in mind that each of the terms um and pm is a transition moment connecting two states in the molecule Thus um and um are vectors and have components It is customary to represent these by p and o where these refer to the x y and 2 directions in the molecular frame Lifetime and dephasing contribution The derivation above assumes that states are infinitely narrow in energy As we discussed for the Fermi Golden Rule in reality states have a finite width in energy due to their lifetime In fact they also have a finite energy width due to pure dephasing Pure dephasing can be thought of as the loss of phase information terms such as eXpi0iJt for states i and j due to fluctuations in the electric field in the environment We can define a rate F at which population decays or dephasing occurs Thus the probability of stationary state ngt decays exponentially in time Pnt ct2 0C e To separate out contributions from population relaxation with time constant T1 and pure dephasing with time constant T2 we define the overall T2 time for a state as the inverse of the rate F Lifetime and dephasing contribution As discussed previously the damping has contributions from lifetime and pure dephasing ri 1 1 T22T1T The coefficientris Imm This term is incorporated into the expression for the coefficient N0 t nj dtre Kmnj IT2t eioaz e ioat h 0 021 5m 8 eio0nf IT2t 1 ei nf IT2t 1 nj h D OM irlz D 03nf iF2 Lifetime and dephasing contribution When propagated through the steps above the expression for the polarizability becomes 1 ltirtclngtltnluplfgt DJr h co com ITZ This expression suggests that polarizability has real and imaginary parts These represent contributions to dispersion and absorption respectively This formalism is also related to higher order coupling in nonradiative transitions socalled superexchange coupling by replacing the dipole operator with the appropriate perturbation that connects two states