MIT Department of Chemistry 5.74, Spring 2005: Introductory Quantum Mechanics II Instructor: Professor Andrei Tokmakoff

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1 MIT Departmet of Chemistry 5.74, Sprig 5: Itroductory Quatum Mechaics II Istructor: Professor Adrei Tomaoff p. 97 ABSORPTION SPECTRA OF MOLECULAR AGGREGATES The absorptio spectra of periodic arrays of iteractig molecular chromophores show uique spectral features that deped o the size of the system ad disorder of the eviromet. We ll ivestigate some of these features, focusig o the delocalized eigestates of these coupled chromophores: excito states. Much of this descriptio of has bee developed to describe molecular crystals, photosesitizers, ad chlorophyll light-harvestig arrays i photosythesis, although similar topics are appearig i the descriptio of properties of cojugated polymers ad i protei vibratioal spectroscopy. The absorptio spectrum for isolated molecule is: a ( ω )= f µ ˆ ε g δ ( E f E g ω) f I the case of expoetial dampig we write the lieshape as: a (ωω ) = µ Γ (ω ω eg ) +, eg eg Γ For a esemble of molecules with a distributio of slightly differet frequecies δω arisig from molecules i slightly differet eviromets, the lieshape is ω eg = ω +δω (frequecy shift of a molecule relative to the mea) ω = ω eg P (δω) = δω exp πσ σ a ( ω)= d (δω ) a (ω, ω + δω) P (δω ) Follows J. Koester, Optical Properties of Molecular Aggregates, i Proceedigs of the Iteratioal School of Physics "Erico Fermi" Course CXLIX (eds. Agraovich, M. & La Rocca, G. C.) (IOS Press, Amsterdam, ). See also Fayer, M. D. Elemets of Quatum Mechaics (Oxford Uiversity Press, New Yor, ), pp. 5-3.

2 p. 98 This is ow as the effect of static disorder or ihomogeeous broadeig of the lieshape. The lieshape is a covolutio of homogeeous lieshape a ad probability distributio P. This is ow as Voigt profile for Lorezia a ad Gaussia P. I limit σ >> Γ, a ( ω ) returs to δ fuctio expressio ad the lieshape reflects the distributio: a δ (ω + δω ω) a ( ) ω (ω ω exp σ ) Dimer of two molecules To describe the spectroscopy of a array of may coupled chromophores, it is first istructive to wor through a pair of coupled molecules. This is i essece the two-level problem from earlier. Two molecules i proximity: Each has trasitio dipole momet µ ad trasitio eergy ω. For o couplig betwee the sites: Dipoles ca have differet directios, ad the eergy ca have four states: g groud state,, with oe of the molecules excited, ad where both molecules are excited.

3 p. 99 More geerally, there will be couplig betwee the trasitio dipoles (a resoace iteractio) that couples the molecules: V = J ( + ) We ca tae the couplig to be a electrostatic iteractio betwee two poit dipoles: (µ µ J = r ) 3 (µ r )(µ r ) orietatio factor r 5 uit vectors K K = (µ µ ) 3 (µ r )(µ rˆ ) = µµ ˆ ˆ ˆ ˆ 3 ˆ r So we have H ω = J J ω, with two eigestates that result from mixig: ± = = ( ± ) with E ± = ω ± J (Here we eglect iteractio betwee g ad.) These symmetric ad atisymmetric states are delocalized across the two molecules. They are also referred to as oe-excito states. The dipole operator for the dimer is M =µ +µ

4 p. ad the trasitio dipole matrix elemets are: M ± = ± M g = (µ ±µ ) So the absorptio spectrum is two resoaces at (ω ± J ) which are polarized perpedicular to each other ad have amplitudes give by orietatio. (assumig upolarized light) Note for θ =, all amplitude i oe trasitio with magitude µ super-radiat.

5 p. Dimer with disorder Now, let s cosider the case where the eergy of the molecular states of the dimer varies: ω H = J J ω E ± = (ω + ω ) ± (ω + ω ) + 4J For large variatio i the site (molecular) eergies (ω ω )/ >> J we have the wea couplig limit with two localized trasitios. For variatio i the site eergies (ω ω )/ << J we have the collective dimer states ± above. Disorder: Now we ca examie what happes whe we pic ω,ω from a Gaussia radom distributio: ω i = ω i +δω i ω i = ω i i =, a (ω,δω δω ) = δ (E E ω), M fg f g f =+ E ± = + (ω +δω ω +δω ) ± (ω + δω ω δω ) + (J / ) a ( ω )= d (δω ) d (δω ) a (ω, δω, δω ) P (δω, δω ) Here P (δω δω ) is the joit probability of havig a frequecy shift δω ad a shift δω. For, ucorrelated distributios with same width σ : P (δω ) = i δω i exp πσ σ P (δω δω ) = P (δω ) P (δω ) = exp, δω δω πσ σ σ

6 p. More geerally a joit probability distributio must accout for ay correlatio i the two frequecy distributios: where exp δω δω + ρ δωδω P (δω, δω ) = πσσ σ σ σσ ρ = δωδω σσ is the correlatio coefficiet. It varies from a value of + for correlated to for ati-correlated distributios. A value of reflects o statistical correlatio betwee distributios. Case : Large disorder σ>> J, most dimers are i ihomogeeous limit, ad spectrum is what you would expect for a distributio of oiteractig molecules. Set J ω ω = ω E + = ω + δω E = ω + δω a ( ω ) µ exp i =, σ ( ω ω i ) µ exp (ω ω o ) σ Gaussia lieshape with liewidth 8l σ (FWHM ) Case ) Wea disorder ( Γ<< σ << J ). If σ is fiite but σ << J, most dimers i wea ihomogeeity limit, with strog couplig ad delocalized oe-excito state. Set δω, δω

7 p.3 E ± = ω + ω ± J = ω ± J δω P(δω,δω ) becomes arrower exp σ πσ ( δω δω = small) Now we have two Gaussia peas split by J, ad the width of the Gaussias is arrower tha previously: (ω ω ) a ( ω ) i M i exp σ i=±. =.67σ This discussio was for diagoal disorder: variatio of the site eergies. You ca also imagie that the off-diagoal terms (couplig) could vary. Freel excitos Now let s cosider liear aggregate of N equidistat molecules. We will assume that each molecule is a two-level electroic system with a groud state ad a excited state. We will assume that electroic excitatio moves a electro from the groud state to a uoccupied orbital of the same molecule. That is, there is o charge trasfer to aother molecule, ad the groud state hole ad the excited state electro are the same molecule. Couplig betwee excitatios will lead to delocalized electroic states ow as Freel excitos. This behavior is usually applicable to describe molecular crystals ad molecular aggregates such as light harvestig complexes ad photosesitizers. Waier excitos refer to the case whe charge trasfer is cosidered.

8 p.4 The liear aggregate will be tae to have N molecules. We will label the molecules with umbers betwee ad N : The couplig betwee molecule ad molecule m is J iteracts oly with its eighbors, so that m. We will assume that a molecule J m = J δ,m ± Also, we use start with periodic boudary coditios, which for the couplig will mea that every molecule iteracts with two eighbors ad J,N = J N, = J. This leads to a Hamiltoia with high symmetry. If the molecules are separated alog the chai by a lattice spacig α, the the size of the chai is L = α N. The molecules each have a trasitio dipole momet µ, which maes a agle β with the directio of the chai. The case where β < 54.7 degrees leads to J <, ad is ow as a J- aggregate. We ca imagie specifyig the state of the system as product states i the eigestates of the N molecules. Whe all molecules are i the groud state, we have G = N = g,g,g,g If we excite oe of the molecules withi the aggregate, we have a sigly excited state i which the th molecule is excited. E = g,g,g,,e,,g It has a eergy ω. I the absece of couplig, you ca see that this is a N -fold degeerate state.

9 p.5 The optical properties of the aggregate deped o the eigestates of the Hamiltoia, which we ca write H = H + H m N H = ω + J N =,m = m m Rather tha diagoalizig this Hamiltoia (for which the aggregate has a complete basis of N states), we ca tae advatage of symmetry to solve for the eigestates. The symmetry of our Hamiltoia (with periodic boudary coditios) is such that it is uchaged by ay umber of lattice traslatios (a itegral umber of lattice spacigs). This also dictates that the eigestates must also be uchaged by lattice traslatio. Maig use of our spatial displacemet operator (problem set ), this suggests that the N eigestates will have the property: iκ α ψ ( x +α) = e where the wave vector ψ ( x) π κ = =,, N ( idexes the eigestate) L Sice ay lattice traslatio α is possible, the eigestates ca be writte N e iκ α = N = Note that this is just expressig oe of the eigestates as a superpositio of states where the coefficiets are just a phase factor c = exp i κα). Each ( has equal amplitude for each, but with a differet, positio-depedet phase factor. The eigestates have the same umber of odes as. For =, all dipoles oscillate i phase. For = N there is a ode betwee each dipole.

10 p.6 For example, if N =, we recover the dimer: = = = + = ) (κ = ) = = ( iπ = = e = ( = = ) (κ = π / α) = Schematically for N = 4 we see: Now we ca solve for the eergy eigevalues: H =Ω H = H + H m N N N e i κ ( ) α H = ω + J m = m,= N = H =ω

11 p.7 With J m, = J δ,m ± we write: N = J e iκ α iκ α H + + e m = N = N κα i i+ ( ) κ α +κα i i ( ) κ α J e e + + e e N = = + = Jcos κ α κα i +κα i J e e With P.B.C., sums over these terms are just Ω =ω + Jcos κ α You predict that the bad of states varies i eergy from ω J to ω + J. This is the excito bad. κ is tae from π /α to +π /α. If we tae J as egative, = is at the bottom of the bad. A more careful approach that does t use periodic boudary coditios has solutios that are remiiscet of particle i a box states. Here you use the coditio that ψ = at sites ad N+. Here the eigestates are boud at the ed of the chai. The chage i boudary coditio gives sie solutios: = N si π N + = N + Here =, N, ad the labels o the molecules also rus from =, N. The eergy eigevalues are π Ω =ω + Jcos N +

12 p.8 You ca cofirm that these also give the dimer solutio. If you calculate the oscillator stregth: ( ) µ π µ g = M g = N + cot ( N +) This result shows that most of the oscillator stregth lies i the (ow) = state for which all oscillators are i phase. For large N, µ =.8µ. Much of the rest is i the = 3 state with = odes µ =.9µ. = 3 Absorptio spectra for N = 3,7,. ω = ad J =. The shift i the pea of the absorptio relative to the moomer gives the couplig J. Icludig log-rage iteractios has the effect of shiftig the excito bad asymmetrically about ω. Ω =ω +4. J (=, bottom of the bad with J egative) Ω =ω 8. J (Top of bad) N

13 p.9 Exchage Narrowig If the chai is ot homogeeous, i.e., all molecules have same eergy ω, the we ca model this as Gaussia radom disorder: ω = ω + δω H = H + H dis + H m H dis = δω The effect is to shift ad mix the homogeeous excito states. π δ Ω = H dis = si N + δω N + We also fid that these shifts are also Gaussia radom variables, with a stadard deviatio: σ =σ 3 ( N +) where σ is the stadard deviatio for site eergies. So, the delocalizatio of the eigestate averages the disorder over N sites, which reduces the distributio of eergies by a factor scalig as N. So, delocalizatio has the effect of arrowig the absorptio lieshape exchage arrowig. This depeds o the distributio of site eergies beig relatively small. Specifically: 3π J σ<< N 3/ which is strogly size-depedet. Absorptio spectra for N =,6,3 ormalized to the umber of oscillators. σ =.3, ω = ad J =.