Helsinki Winterschool in Theoretical Chemistry 2013

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1 Helsinki Winterschool in Theoretical Chemistry 2013 Prof. Dr. Christel M. Marian Institute of Theoretical and Computational Chemistry Heinrich-Heine-University Düsseldorf Helsinki, December 2013 C. M. Marian (HHU Düsseldorf) Lecture I 1

2 Molecular Transitions Radiative Intramolecular Transitions Non-Radiative Intramolecular Transitions Intermolecular Transitions Lifetime and Rates Time-Dependent Perturbation Theory Molecular Transitions C. M. Marian (HHU Düsseldorf) Lecture I 2

3 Radiative Intramolecular Transitions Energy S 1 T 1 ISC IC VR VR ISC P F A S 0 Jablonski Diagram 1. Vibronic transitions (a) Absorption (A) (b) Fluorescence (F) (c) Phosphorescence (P) 2. Vibrational transitions (a) Absorption (b) Emission Within molecular system: No energy conservation C. M. Marian (HHU Düsseldorf) Lecture I 3

4 Non-Radiative Intramolecular Transitions Energy S 1 T 1 ISC IC VR VR ISC P F A S 0 Jablonski Diagram 1. Vibronic transitions (a) Internal conversion (IC) (b) Intersystem crossing (ISC) 2. Vibrational transitions (a) Intramolecular vibrational redistribution (IVR) (b) Vibrational relaxation (VR) Energy conservation in IC, ISC, and IVR System-bath energy transfer in VR C. M. Marian (HHU Düsseldorf) Lecture I 4

5 Intermolecular Transitions Radiative transitions 1. Förster resonance energy transfer (FRET) Non-radiative transitions 1. Dexter energy transfer 2. Reactive and non-reactive collisions Förster and Dexter energy transfer concept ( c chemwiki.ucdavis.edu) C. M. Marian (HHU Düsseldorf) Lecture I 5

6 Lifetimes and Transition Rates Eigenfunctions of the time-independent Schrödinger equation are stationary, i.e., they have an infinite lifetime. Excited states exhibit limited lifetimes. Sooner or later the molecular system reverses to the ground state. The computation of lifetimes and transitions rates requires a time-dependent approach. Cases that can be treated by time-dependent perturabtion theory Interaction of molecules with a weak electromagnetic field Weak non-adiabatic interactions (vibronic coupling) Spin-orbit coupling in light-element compounds Excitation energy transfer... C. M. Marian (HHU Düsseldorf) Lecture I 6

7 Molecular Transitions Time-Dependent Perturbation Theory Time-Dependent PT Constant Perturabtion: Nonradiative Transitions δ distribution Fermi s Golden Rule Resume Condon Approximation Cosine-like Perturbation: Radiative Transitions Resume FC Approximation Time-Dependent Perturbation Theory C. M. Marian (HHU Düsseldorf) Lecture I 7

8 Time-Dependent Perturbation Theory Goal: Approximate solution of the time-dependent Schrödinger equation Ansatz: Partition Hamiltonian into Ĥ = Ĥ(0) +λĥ(1) (1) where the unperturbed Hamiltonian Ĥ(0) is time-independent the perturbation Ĥ(1) is a time-dependent potential ˆV( r,t) The interaction potential causing the perturbation can originate from external fields internal coupling terms C. M. Marian (HHU Düsseldorf) Lecture I 8

9 Time-Dependent Perturbation Theory Solve time-dependent Schrödinger equation i Ψ( r,t) t = (Ĥ(0) + ˆV( r,t)) Ψ( r,t) (2) in terms of an orthonormal set of functions Ψ( r,t) = k c k (t)φ k ( r)e ie kt/ (3) where the c k (t) are time-dependent coefficients, Φ k ( r) solutions of the time-independent Schrödinger equation Ĥ (0) Φ k ( r) = E k Φ k ( r) (4) and e ie kt/ phase factors. C. M. Marian (HHU Düsseldorf) Lecture I 9

10 Time-Dependent Perturbation Theory The left hand side (l.h.s) of Eq. (2) yields i Ψ( r,t) t = k E k c k (t)φ k ( r)e ie kt/ +i k c k (t) φ k ( r)e ie kt/ t The right hand side (r.h.s) of Eq. (2) yields (Ĥ(0) ˆV( r,t)) + Ψ( r,t) = Ĥ(0) Ψ( r,t)+ k ˆV( r,t)c k (t)φ k ( r)e ie kt/ Comparison shows that i k c k (t) φ k ( r)e ie kt/ t = k ˆV( r,t)c k (t)φ k ( r)e ie kt/ (5) C. M. Marian (HHU Düsseldorf) Lecture I 10

11 Time-Dependent Perturbation Theory Multiplication from the left by φ l ( r) and integration over r i c l(t) e ie lt/ t = k φ l ˆV φ k c k (t)e ie kt/ yields a system of coupled differential equations that determine the time-dependent coefficients c. i c l(t) t = k φ l ˆV φ k c k (t)e i(e l E k )t/ (6) c l (t) 2 is a measure for the probability to find the system in state φ l at time t. C. M. Marian (HHU Düsseldorf) Lecture I 11

12 Time-Dependent Perturbation Theory Suppose that the perturbation operator ˆV( r,t) is of the form ˆV( r,t) = Â( r)ŷ(t) (7) Consider, for simplicity, only two states Φ a and Φ b and assume that initially c a (0) = 1 and c b (0) = 0. i c b(t) t = φ b Â φ a Ŷ(t)c a(t)e i(e b E a )t/ (8) Introduce abbreviation ω ba E b E a (9) C. M. Marian (HHU Düsseldorf) Lecture I 12

13 Time-Dependent Perturbation Theory In the course of time, c b (t) adopts values different from zero. c b (t) t = i φ b Â φ a Ŷ(t)c a(t)e iω bat (10) The probability of finding the system in state φ b at time t is then c ba (t) 2 = 1 2 φ b Â φ a 2 t 0 Ŷ(t )e iω bat dt 2 (11) Eq. (11) is the starting point for the evaluation of transition rates between molecular states. C. M. Marian (HHU Düsseldorf) Lecture I 13

14 Time-Dependent Perturbation Theory In the case of nonradiative transitions, the perturbation is assumed to be time-independent Ŷ(t) = ˆ1 (12) For radiative transitions, we assume the time-dependent perturbation to behave like a cosine wave Ŷ(t) = 2cosωt = e iωt +e iωt (13) This cosine wave can also be perceived as superposition of an incoming and an outgoing wave. C. M. Marian (HHU Düsseldorf) Lecture I 14

15 Constant Perturabtion: Nonradiative Transitions Assume that the perturbation is constant in time. Integration of Eq. (10) then yields c ba (t) = i φ b Â φ a = i φ b Â φ a t 0 e iω bat dt [ 1 iωbat e iω ba = 1 φ b Â φ a (eiω bat 1) ω ba (14) The probability of finding the system in state φ b at time t is given by ] t 0 c ba (t) 2 = 1 2 φ b Â φ a 2(eiω bat 1) 2 ω 2 ba (15) C. M. Marian (HHU Düsseldorf) Lecture I 15

16 Constant Perturabtion: Nonradiative Transitions Evaluation of the numerator of the frequency-dependent term ( e iω ba t 1 )( e iωbat 1 ) = 1+1 e iωbat e iω bat = 2 2cos(ω ba t) ( ) 1 = 4sin 2 2 ω bat and switching back to energy units finally yields the expression for the probability of finding the system in a specific state φ b at time t c ba (t) 2 = 4 φ b Â φ a 2 1 (E b E a ) 2 sin2 ( ) (Eb E a )t Due to the highly oscillatory behavior of the sine function, it adopts substantial values only if E b E a. 2 (16) C. M. Marian (HHU Düsseldorf) Lecture I 16

17 Constant Perturabtion: Nonradiative Transitions The transition rate W ba (t) is defined as the transition probability per unit time W ba (t) = c ba(t) 2 = 4 φ b t Â φ a 2sin2( 1 (E 2 b E a )t ) (E b E a ) 2 t = 4 φ b Â φ a 2 π ( sin 2 1 (E 2 b E a )t ) 4 2 π( 1 (E (17) 2 b E a )) 2 t The probability of finding the system in state φ b after a long time is W ba = lim t W ba (t) = π 2 φ b Â φ a 2 lim t [ sin 2 ( 1 2 (E b E a )t ) π( 1 2 (E b E a )) 2 t ] (18) C. M. Marian (HHU Düsseldorf) Lecture I 17

18 The Dirac δ Distribution One of the limiting representations of the Dirac δ distribution is given by ( 1 sin 2 ) tx δ(x) = lim (19) t π tx h(x) k(x) Plot of sin 2 tx/(tx 2 ) vs. x: solid line t = 1, dashed line t = In the limit of inifinite t, sin 2 tx/(tx 2 ) is only different from 0 at x = C. M. Marian (HHU Düsseldorf) Lecture I 18

19 Constant Perturabtion: Nonradiative Transitions Making use of the δ distribution we obtain for the transition rate W ba = π 2 φ b Â φ a 2 δ( 1 2 (E b E a )) (20) A scalar constant c multiplying the argument of the δ distribution can be taken outside, i.e., δ(cx) = δ(x) c (21) This yields the final expression for the transition rate between the states φ a and φ b W ba = 2π φ b Â φ a 2 δ(e b E a ) (22) C. M. Marian (HHU Düsseldorf) Lecture I 19

20 Fermi s Golden Rule The rate for a transition to any state other than the initial state φ i is given by integrating over all possible final state energies E f and weighting with the density of state ρ(e f ) W i = 2π 0 φ f Â φ i 2 δ(e f E i )ρ(e f )de f (23) This is the Fermi Golden Rule expression for the transition rate from an initial state to a set of final states when the perturbation is time-independent. C. M. Marian (HHU Düsseldorf) Lecture I 20

21 Fermi s Golden Rule Molecular Transitions Time-Dependent Perturbation Theory Time-Dependent PT Constant Perturabtion: Nonradiative Transitions δ distribution Fermi s Golden Rule Resume Condon Approximation Cosine-like Perturbation: Radiative Transitions Resume FC Approximation For a system that experiences a time-independent perturbation the rate for a transition from an initial state φ i to a set of final states φ f is W i = 2π 0 φ f Â φ i 2 δ(e f E i )ρ(e f )de f (24) C. M. Marian (HHU Düsseldorf) Lecture I 21

22 Condon Approximation Let the initial state be a vibronic state represented by a product of an electronic wavefunction φ a and a vibrational state χ ia. Likewise the final set of vibronic states are supposed to be product states consisting of an electronic state φ b and corresponding vibrational states χ bf. If the perturbation does not depend strongly on the nuclear arrangement, Eq. (23) can be simplified. In the Condon approximation, the electronic and vibrational contributions to the transition rates are separated. W i = 2π 0 χ bf φ b Â φ aχ ai 2 δ(e bf E ai )ρ(e bf )de bf 2π φ b Â φ a 2 0 χ bf χ ai 2 δ(e bf E ai )ρ(e bf )de bf (25) C. M. Marian (HHU Düsseldorf) Lecture I 22

23 Cosine-like Perturbation: Radiative Transitions Assuming a cosine-like perturbation, integration of Eq. (10) yields c ba (t) = i φ b Â φ a t 0 t ( e iωt +e iωt ) e iω bat dt = i φb Â φ a e i(ω ba+ω)t +e i(ω ba ω)t dt 0 = 1 [ e i(ω φb Â φ ba +ω)t 1 a ω ba +ω ] + ei(ωba ω)t 1 ω ba ω (26) Starting from an initial population of state φ a at time t = 0, we observe a substantial probability of finding the system in state φ b after time t only if one of the terms in brackets is large. Since t was assumed to be small, e i(ω ba±ω)t 1. Hence, c ba (t) can only become large, if one of the denominators in Eq. (26) approaches 0. C. M. Marian (HHU Düsseldorf) Lecture I 23

24 Cosine-like Perturbation: Radiative Transitions Case E b > E a : Then ω ba > 0 and only the second term of Eq. (26) may become large. The transition rate is the probability per unit time for a transition from φ a to φ b. It is given by W ba (t) = c ba(t) 2 t = 1 2 φ b Â φ a 21 t [ e i(ω ba ] ω)t 2 1 (27) ω ba ω Squaring the numerator of the term in brackets requires complex conjugation ( e i(ω ba ω)t 1 )( e i(ω ba ω)t 1 ) = 1+1 e i(ω ba ω)t e i(ω ba ω)t = 2 2cos((ω ba ω)t) ( ) 1 = 4sin 2 2 (ω ba ω)t (28) C. M. Marian (HHU Düsseldorf) Lecture I 24

25 Cosine-like Perturbation: Radiative Transitions The transition rate W ba (t) can then be written as W ba (t) = 4 2 φb Â φ a 2sin2( 1 (ω 2 ba ω)t ) (ω ba ω) 2 t = 4 2 φb Â φ a 2π sin 2( 1 (ω 2 ba ω)t ) 4 π( 1(ω 2 ba ω)) 2 t (29) (30) The probability of finding the system in state φ b after a long time is then given by W ba = lim t W ba (t) = π 2 φ b Â φ a 2 lim t [ sin 2 ( 1 2 (ω ba ω)t ) π( 1 2 (ω ba ω)) 2 t ] (31) C. M. Marian (HHU Düsseldorf) Lecture I 25

26 Cosine-like Perturbation: Radiative Transitions Making use of the δ distribution and its properties we obtain for the transition rate W ba = π 2 φ b Â φ a 2 δ( 1 2 (ω ba ω)) (32) This yields the final expression for the transition rate in terms of the energy difference between the states φ a and φ b and the frequency of the perturbation W ba = 2π φ b Â φ a 2 δ(e b E a ω) (33) C. M. Marian (HHU Düsseldorf) Lecture I 26

27 Generalization and Resume Molecular Transitions Time-Dependent Perturbation Theory Time-Dependent PT Constant Perturabtion: Nonradiative Transitions δ distribution Fermi s Golden Rule Resume Condon Approximation Cosine-like Perturbation: Radiative Transitions Resume FC Approximation For a system that experiences a time-dependent perturbation of the form Â( r)eiωt +Â( r)e iωt the rate for a transition from an initial stationary state φ a to a final stationary state φ b is W ba = 2π φ b Â( r) φ a 2 δ(e b E a ω) for E b > E a (34) W ba = 2π φ b Â( r) φ a 2 δ(e b E a + ω) for E b < E a (35) C. M. Marian (HHU Düsseldorf) Lecture I 27

28 Franck-Condon (FC) Approximation Also in radiative transitions, a Condon approximation is possible, if the transition moment does not depend strongly on the nuclear arrangement. C. M. Marian (HHU Düsseldorf) Lecture I 28

29 Franck-Condon (FC) Approximation In the Franck-Condon approximation, the probability for a radiative transtion is W i 2π φ b Â φ a 2 χ bf χ ai 2 δ(e bf E ai ± ω) (36) where the plus sign applies to emission, the minus sign to absorption. C. M. Marian (HHU Düsseldorf) Lecture I 29

11.1. FÖRSTER RESONANCE ENERGY TRANSFER

11-1 11.1. FÖRSTER RESONANCE ENERGY TRANSFER Förster resonance energy transfer (FRET) refers to the nonradiative transfer of an electronic excitation from a donor molecule to an acceptor molecule: D *