The Trasitio Dipole Momet Iteractio of Light with Matter The probability that a molecule absorbs or emits light ad udergoes a trasitio from a iitial to a fial state is give by the Eistei coefficiet, B fi B if = B fi = µ if 6ε! 2 Equatio 1 o More details coectig a absorbace measuremet to the Eistei coefficiet are give i the OPTIONAL additioal otes. the oly compoet ot a costat i Equatio 1 is the trasitio dipole momet µ fi o the trasitio dipole momet is a quatum mechaical quatity o it couples the total molecular wavefuctio of the iitial Ψ i ad fial Ψ f states of the molecule via the molecular dipole momet µ. 2 IMPORTANT µ fi = Ψ f µψ i = f µ i Equatio 2 o the trasitio dipole momet must be o-zero for a trasitio to occur (or for a pea to be preset i a experimetal spectrum) o the trasitio dipole momet ca be regarded as a measure of the size of the electromagetic jolt give to a system IR Spectroscopy ad the Dipole Momet What does the dipole momet have to do with light iteractig with a molecule? o the static dipole momet (µ ) is a vector with compoets i the x, y ad z directios. We ca thi of the dipole momet as "summarisig" the charge distributio withi a molecule, Equatio 3, o where r α is the distace from the CoM of the molecule to the charge q α ad has two compoets, a electroic (e) ad uclear (Z ) compoet. µ = µ x µ y µ z = e Z R + e i r i = q α r α i α Equatio 3 o light is electromagetic radiatio ad is represeted by a oscillatig electric field ( ˆε ) o this electric field iteracts with the charges distributed throughout a molecule, the oscillatig electric field of the photo iduces oscillatios i the electro desity (egative) ad uclei (positive charges). o this oscillatio of charges withi the molecule ca couple fial ad iitial states, that is cause the system to chage from a iitial to a fial state o electromagetic radiatio also has a oscillatig magetic field, however the magitude of the iteractio is much smaller ad a trasitio is 1 4 times less liely to occur. o More details o the form of the CoM motio ad the trasitio dipole momet are give i the OPTIONAL additioal otes. Hut / Lecture 4 1
Rama Spectroscopy ad the Polarizability Rama spectroscopy is a idirect techique. The molecule is irradiated with very high itesity laser radiatio, most of which is scattered elastically (ie with o eergy trasfer betwee photos ad molecule) bac towards the source, producig the Rayleigh lie i spectra, Figure 1. Figure 1 Rama spectrum of CCl4 (488. m excitatio) 1 a very small umber of ielastic iteractios occur where eergy is trasferred betwee a photo ad the molecule. These very rare evets, approximately 1 i every 1 7 photos, ad are very difficult to observe i the strog bacgroud of the elastically scattered photos. whe the light iteracts with the molecule it eters a virtual eergy state (light + molecule) for a short period after which the light is re-emitted. Durig this process the photo ca trasfer a (vibratioal) quata of eergy ito the molecule, loosig this quata itself, ad producig the Stoes lies i a Rama spectrum. The ati-stoes lies are due to a quata of eergy beig removed from the molecule ad beig gaied by the photo. because there are more molecules i the groud vibratioal state (Boltzma s distributio) the Stoes lies are much stroger tha the ati- Stoes lies ad it is customary to measure ad report the Stoes lies. Because of the depedece o the populatio of vibratioal states, Rama spectroscopy is sesitive to temperature. hν " h(ν "-ν 1 ) hν ' h(ν '-ν 1 ) hν h(ν -ν 1 ) scattered light scattered light mius 1quata of vibratioal eergy v=1 v= Figure 2 Stoes trasitios for Rama spectroscopy 1 from Naamoto Ifrared ad Rama Spectra of Iorgaic ad Coordiatio Compouds, 5 th Editio (1997), Joh Wiley & Sos, New Yor, PartA, p1, Fig I-6. Hut / Lecture 4 2
i additio totally symmetric Rama active modes leave plae polarised light polarised, otherwise the light will be depolarised. o polarised light is light that oscillates i oly oe plae (ormal light oscillates i all plaes) o whe plae polarised light iitially iteracts with the molecule it is with compoets of the vibratio that are aliged to plae of oscillatio. o however the molecules i a liquid sample are movig ad radomly reorietatig, the emitted photo will be i a differet orietatio. If the active vibratio is ot-totally symmetric light will be depolarised. o if the vibratio totally symmetric, the orietatio of the molecule does ot matter ad the scattered light remais polarised. o this is a very simplified justificatio of a much more complex pheomea beyod the scope of this course to discuss further. light wave directio of propagatio arrows represet the electric field oscillatig up ad dow looig dow the ceter lie the electric field oscillates up ad dow vertical polarizer horizotal polarizer upolarized light upolarized light ONLY vertically polarized light Figure 3 Stoes trasitios for Rama spectroscopy as the electric field of light is oscillatig the uclei ad electros get pulled first oe way ad the the other. o a trasiet dipole momet depeds o the electromagetic wave ca be iduced i the sample, Figure 4. µ = αε where ε is the magitude of the electric field ad α is the polarizability. o α measures how easy it is to deform the electroic structure, typically "soft" elemets ad ligads are said to be highly polarizable. Examples iclude iodie atoms or sulphur cotaiig ligads o Rama spectroscopy depeds o the iduced trasiet dipole momet ad a chage i the electric polarizability with respect to the vibratioal mode is required for a mode to be Rama active. o light! δ- δ+ hν δ+ δ- oscillatig electric field Figure 4 Cartoo represetig the formatio of trasiet dipoles uder the ifluece of a oscillatig electric field, the dipole momet vector ad polarizability tesor Hut / Lecture 4 3
This ca be expressed more formally such that if we tae the molecular dipole ad expad it i a Taylor series with respect to the electric field ( ˆε ) we obtai the followig: # µ = µ + µ & % ( ε + 1 # 2 µ & % ( ε 2! $ ε ' 2 $ ε 2 ' # α = µ & % ( $ ε ' # β = 2 µ & % ( $ ε 2 ' µ = µ +αε + 1 2 βε 2 µ x µ y µ z µ = αε Equatio 4 the polarizability α xx α xy α xz = α yx α yy α yz α zx α zy α zz o µ is the permaet electric dipole idepedet of the electric field, α is the polarizability ad β is the first hyperpolarizability. Both α ad β are depedet o the electric field. the dipole momet is a vector, as is the electric field, this meas that the polarizability must be a matrix, i this case a special type of matrix called a tesor. o otice the subscripts (they are similar to the biary fuctios!) o it is assumed that there is o permaet dipole momet, ad that we are taig the expasio i the electric field oly to first order. Where does the trasitio dipole momet come from? to fid the origis of the trasitio dipole momet we eed to go right bac to quatum mechaics ad probabilities we ow that the probability of beig i a particular state: ρ = ψ 2 ψ = a ϕ = a ϕ a m ϕ m = a a m ϕ ϕ m = 2 a m,m,m ε x ε y ε z Equatio 5 o here = is the groud state ad > are excited states, thus to fid the probability of the molecule beig i a excited state we eed to determie the coefficiets a o we eed to determie how has the light has affected the iitial state to light produce the fial state, Ψ i """ Ψ f settig up the problem o i the basic Schrödiger equatio we use the molecular Hamiltoia (H mol =T e +T +V ee +V e +V ) which is idepedet of time, HΨ = EΨ o ow we eed to itroduce a perturbatio due to the lightwave, "pert" below stads for perturbatio (the molecule is perturbed by the electromagetic wave of the light) H = H mol + H pert Equatio 6 o the light iteractig with the molecule is mathematically represeted by the dipole momet of the molecule iteractig with the electric field (ε) of the lightwave H pert = µε Equatio 7 Hut / Lecture 4 4
o lightwaves are oscillatig i time so we should use a perturbatio expressio that represets this oscillatig ature (time depedet perturbatio theory), it is traditioal to use a cos fuctio for this (see below, the extra 2 is to mae some of the mathematics easier) ad we should use the time depedet Schrödiger equatio! V = µε(t) H (t) pert = 2V cos(wt) = 2V e iwt + e iwt HΨ = i! Ψ t Equatio 8 o teachig you time depedet perturbatio theory is beyod the scope of this course. What I will do is itroduce you to some time-idepedet (o-degeerate) perturbatio theory so you ca obtai a uderstadig of where the trasitio dipole momet comes from (ad so that you ca move oto time-depedet perturbatio theory more easily if you wat to tae this further). o time depedet perturbatio is a core part of quatum mechaics ad if you are iterested i quatum mechaics I suggest you tae a loo at this theory i Atis ad Friedma "Molecular Quatum Mechaics". (No-degeerate) Time Idepedet Perturbatio Theory Start with a geeral expasio of the Hamiltoia, H is our uperturbed Hamiltoia, H () are perturbatios of decreasig magitude ad of differig types, the superscript gives the "order" of the term, ad λ is just a mathematical tool for a eat tric (ad represets a coefficiet for cotrollig the stregth of a perturbatio) H = H + λh (1) + λ 2 H (2) +! Equatio 9 ow if the Hamiltoia is chagig, so too must the wavefuctios (solutios) ad the eergy, but they will both be perturbed i a similar maer to the Hamiltoia, so we use a similar expasio ψ = ψ + λψ (1) + λ 2 ψ (2) +! E = E + λe (1) + λ 2 E (2) +! Equatio 1 if we pac it all bac ito the Schrödiger equatio Hψ = Eψ ( H + λh (1) + λ 2 H (2) ) ψ + λψ (1) + λ 2 ψ (2) +! ( ψ + λψ (1) + λ 2 ψ (2) +! ) = E = E + λe (1) + λ 2 E (2) +! Equatio 11 collect all the terms with the same λ coefficiets ( H ψ ) = E ψ +λ H ψ (1) + H (1) ψ +! +λ 2 H ψ (2) + H (1) ψ (1) + H (2) ψ +! Equatio 12 +λ E ψ (1) + E (1) ψ +! +λ 2 E ψ (2) + E (1) ψ (1) + E (2) ψ +! Hut / Lecture 4 5
these must be equal ad so we obtai separate equatios ad we throw away the λ's as they have doe their job the first equatio is the equatio for the uperturbed system, the secod equatio is called the first order correctio ad so o H ψ = E ψ H ψ (1) + H (1) ψ = E ψ (1) + E (1) ψ H ψ (2) + H (1) ψ (1) + H (2) ψ = E ψ (2) + E (1) ψ (1) + E (2) ψ Equatio 13 we ow focus o the first order correctio, we mae the assumptio that the (1) slightly perturbed wavefuctios ψ ca be expaded i terms of the uperturbed wavefuctios ψ { H E }ψ (1) = { E (1) H (1) }ψ ψ (1) = a ψ Equatio 14 i the uperturbed system there will be a umber of solutios, each with a differet eergy, the lowest eergy state E will have the wavefuctioψ, the ext highest state will have E 1 ad ψ 1 Hψ = E ψ Hψ 1 = E 1 ψ 1 ad so o Hψ 2 = E 2 ψ 2 Equatio 15 o The uperturbed groud state is described by E ad ψ. This state is perturbed slightly to become ψ (1). Oe way to thi of this is as a mixig of small amouts of the origial excited states ito the groud state wavefuctio, ie addig a small amout ψ 1 ad ψ 2 etc ito the origial uperturbed groud state ψ. Now we substitute for ψ (1) ito our perturbatio expressio, I have ow added subscripts related to the differet explicitly H { E }ψ (1) = { E (1) H (1) }ψ o startig with the left had side (LHS) Equatio 16 { } a LHS = H E = a E E ψ { }ψ o puttig the equatio bac together agai: sice H ψ = E ψ Equatio 17 a E E { }ψ = { E (1) H (1) }ψ Equatio 18 ow we pre-multiply ad itegrate with a sigle uperturbed wavefuctio ψ % ' & ( { }ψ * = ψ % E (1) H (1) { }ψ ( & ) ) a E E Equatio 19 Hut / Lecture 4 6
we pull costats out of the itegratio, ad loo at the itegrals, we ow that the uperturbed solutios are orthogoal ad ormalised, so we ca simplify the equatios, a E { E } ψ ψ = E (1) ψ ψ ψ H (1) ψ = if =1if = Equatio 2 = if =1if = however o the RHS we fid that we must have = if we wat to fid the first order correctio to the eergy E (1), so settig = for the whole equatio a E { E } ψ ψ = E (1) ψ ψ ψ H (1) ψ = if =1if = Equatio 21 o we ca see that the itegral o the LHS will oly be o-zero if = so settig ==, the LHS eergy term becomes zero! { } a E E!#" # $ ψ ψ = E = =1 = E (1) ψ H (1) ψ (1) Equatio 22 ψ ψ ψ =1 H (1) ψ o we obtai the result that the first order correctio to the eergy depeds oly o the uperturbed wavefuctios ad o the perturbatio i the Hamiltoia. E (1) = ψ H (1) ψ Equatio 23 o however we are also after the correctio to the wavefuctio, so if we focus o the LHS ad set = { } ψ a E E ψ!#" # $ = E = if =1if = a E E (1) ψ ψ!#" # $ ψ = if =1if = { } = ψ a = ψ E ( E ) H (1) ψ H (1) ψ = ψ H (1) ψ E E = H (1) ω Equatio 24 o we obtai the first order correctio to the wavefuctio H (1) ψ Hut / Lecture 4 7
ψ (1) = a ψ (1) H ω ψ (1) = a ψ + ψ Equatio 25, what does this mea? o after the system has bee perturbed there is some probability of fidig the system i a higher eergy state, this probability is related to the size of the electric field, the trasitio dipole momet, ad to the eergy gap betwee the states (the larger the gap the smaller the chace of a trasitio) a 2 = H (1) 2 = εµ 2 ω ω Equatio 26 o hece it is possible to see why Β fi is depedet o the square modulus of the trasitio dipole momet. B fi = 8π 3 µ fi 2 3h 2 Equatio 27 the above derivatio is soud for a time idepedet perturbatio betwee o-degeerate states, ad provides liig justificatio for the form of the trasitio dipole momet a full treatmet of time depedet perturbatio theory ad subsequet derivatio of the Eistei coefficiet ad Fermi's Golde Rule ca be foud i Atis ad Friedma's Molecular Quatum Mechaics. Key Poits be able to write the equatio for the Eistei coefficiet B fi be able to defie ad discuss the importace of the trasitio dipole momet be able to briefly describe ey features of Rama spectroscopy be able to defie the dipole momet ad polarizability ad describe how they relate to light iteractig with matter ad IR ad Rama spectroscopy be able to preset the derivatios for time idepedet o-degeerate perturbatio theory be able to derive the first order correctio to the eergy ad the first order correctio to the wavefuctio Hut / Lecture 4 8