Let s express the absorption of radiation by dipoles as a dipole correlation function.

Transcription

1 MIT Deparme of Chemisry 5.74, Sprig 004: Iroducory Quaum Mechaics II Isrucor: Prof. Adrei Tokmakoff p. 81 Time-Correlaio Fucio Descripio of Absorpio Lieshape Le s express he absorpio of radiaio by dipoles as a dipole correlaio fucio. Sar wih he rae of absorpio ad simulaed emissio bewee a iiial sae k iduced by moochromaic field: ad fial sae πe 0 w k = k ˆ µ ( ( [δ k )δ k +)] Le s cosider a wo-level sysem m ad wih E > E. m The rae of eergy absorpio is proporioal o he absorpio rae E m m ad he rasiio eergy: E rad = w. Bu more geerally we E m eed o cosider he differece bewee he raes of absorpio ad simulaed emissio, so he raes of rasiios bewee hese wo saes is w m w m E rad = p w,k= m, k k = π E 0 k p,k = m, k ˆ µ δ )+δ +) ( k ( k Here we have o sum over he raes of absorpio from m ad he raes of simulaed emissio from m. E rad = π E 0 p m ˆ m µ δ ( m ) absorpio + p ˆ µ m δ m m ( m + ) simulaed emissio

2 p. 8 Noe: δ( m +) = δ( m +)= δ ( m ) sice δ ( x) = δ ( x) Also: he marix elemes squared are he same. m = m E rad = π E 0 ( p p m ) m ˆ µ δ ( m ) m A equilibrium p = exp[ β E ]/ Z p p m = p (1 exp [ β m ]) Now, he eergy icide o he sample per ui ime is = Ei 8 c π E 0 ( ) E rad So we ca wrie he absorpio coefficie, α = E i ( )= 4π β α (1 e ) p m ˆ µ δ c ( m ) Now his is wrie for wo discree saes, bu i geeral we will wa o sum over all possible iiial ad fial saes. We ca ow separae α io a produc of facors ha represe he field, ad he maer, where he maer is described by σ () he absorpio lieshape. ( )= 4π (1 e β )σ α ( ) c ( )= p m σ m, ˆ µ δ ( m ) Now, sice he dela fucio ca be expressed as i δ d e π

3 p. 83 π m, + d e i( m ) σ ( )= 1 p ˆ µ m m ˆ µ 1 + d e i p ˆ µ m m = π m, e i m µ e i ˆ Sice m ad are saes of sysem wihou radiaio me i m = me ih / = m U () e i ih / = e = U () U µu = µ () ( ) 1 + e m, ˆ µ ( ) i σ = p 0 m m ˆ µ () π sum over m: uiy sum over : equilibrium esemble average σ d e i ˆ µ ( 0) µ ˆ () π If we average over a isoropic sysem: ( )= σ d e i µ 0 ( )µ () π 3 The absorpio lieshape is he Fourier rasform of he dipole correlaio fucio. The correlaio fucio describes he ime-depede behavior or spoaeous flucuaios i he dipole mome i absece of E field ad coais iformaio o saes of sysem ad broadeig due o relaxaio.

4 p. 84 Differe specroscopies are described by differe forms of he dipole operaor. Roaioal specroscopy The dipole operaor is he permae dipole mome: µ = µ = µ û 0 0 σ π d e i µ ˆ ˆ 0 ˆ ˆ 0 u ( ) u ( ) The lieshape is he Fourier rasform of he roaioal moio of he permae dipole vecor i he laboraory frame. The frequecy of he resoace would deped o he rae of roaio he agular momeum ad he mome of ieria. Collisios or oher dampig would lead o he broadeig of he lies. Quaum mechaically we expec a disribuio of resoaces for differe populaed roaioal saes which we would sum over whe performig he equilibrium esemble average. IR Vibraioal Specroscopy We ake he dipole operaor o be weakly depede o he displaceme of uclear coordiaes µ µ = µ 0 + q + q 0 Here he firs expasio erm is he permae dipole mome ad he secod erm is he rasiio dipole mome. If we are performig our esemble average over vibraioal saes, he lieshape ow becomes he Fourier rasform of a correlaio fucio i he vibraioal coordiae µ + σ d e i q ( 0 )q ( ) ( )= 1 π q The vecor aure of he rasiio dipole has bee dropped here. So he ime depede dyamics of he vibraioal coordiae dicae he IR lieshape.

5 p. 85 Rama Specroscopy We ca replace he dipole operaor wih a iduced dipole mome, he polarizabiliy esor o ge he correlaio fucio for Rama specroscopy. µ µ id = α ˆ µ ˆ α ˆ s i σ d e i ˆ α ( 0) ˆ ˆ α () ˆ i π s i s or d e i σ α ( 0 )α () π where we have wrie he polarizaio compoes of he icide (i) ad scaered (s) ligh. The polarizabiliy esor is a secod rak esor ha ells you how well a ligh field polarized alog i ca iduce a dipole mome (ligh-field-iduced charge displaceme) i he s direcio. For cylidrically symmeric sysems his usually akes he form α 1 α = α = αi + 3 β 1 α 1 where α is he isoropic polarizabiliy ad β is he aisoropic polarizabiliy. The polarizabiliy esor ca also be expaded i he uclear coordiaes α α = α 0 + q + q 0 where he leadig erm would refer o Raleigh scaerig ad he secod erm would give vibraioal Rama scaerig.

6 p. 86 ENSEMBLE AVERAGING ad LINE BROADENING A absorpio lieshape ca represe he dyamics of he dipole or eergy relaxaio (i.e., couplig o coiuum). I ca also reflec esemble averagig effecs. Homogeeous broadeig: dyamic Ihomogeeous broadeig: saic Homogeeous broadeig ( T ): 1 = + T + * T1 T τ or Populaio Relaxaio ( T 1 ): Ampliude decay due o fiie lifeime. This ca have coribuios from radiaive decay or o-radiaive processes (i.e., couplig o coiuum) (IVR) = + T 1 τ rad τ NR I his case, esemble averagig does chage he measureme. All members of esemble ideical measured decay is he microscopic lifeime. * Pure Dephasig ( T ): Radomizaio of phase by molecular ieracios esemble averagig effec. q1 q q3 q * T e Also, orieaioal dephasig (τ ) or

7 p. 87 Ihomogeeous Broadeig ( ) : Lie broadeig arisig from a saic disribuio of frequecies. No dyamics: esemble averagig effec. ime domai frequecy domai q1 q q3 3 1 q4 q 4 ~ e The oal broadeig reflecs he coribuio of all of hese effecs: ( 0) ( ) µ µ e e T * T 1 τ or g() e g(): lieshape fucio The lieshape for he broadeig of a give rasiio ca be wrie as he Fourier rasform over he oscillaig rasiio frequecy damped ad modulaed by he lieshape fucio: 1 σ ( ) = π + d e i i m g() e