an... ”nun..." ,.,v "turn—u.9......"va gas/352W [TY lllll'llflllllllllllllfllllflljlllll 3 1293 00788 This is to certify that the dissertation entitled THE TRANSITION FROM NON-ADIABATIC TO SOLVENT CONTROLLED ADIABATIC ELECTRON TRANSFER: 4 SOLVENT DYNAMICAL EFFECTS IN THE INVERTED REGIME presented by DAH-Y EN YANG has been accepted towards fulfillment of the requirements for Ph. D. _ CHEMISTRY degree 1n EAST 5L Major professor Date JUNE 21, 1989 MS U is an Affirmative Action/Equal Opportunity Institution 0-12771 r \ LIBRARY Michigan State University K I PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. DATE DUE DATE DUE DATE DUE II I | TT—W MSU Is An Affirmative ActiorVEquel Opportunity Institution smut THE TRANSITION FROM NONADIABATIC TO SOLVENT CONTROLLED ADIABATIC ELECTRON TRANSFER: SOLVENT DYNAMICAL EFFECTS IN THE INVERTED REGIME By Dab-Yen Yang A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1989 ABSTRACT THE TRANSITION FROM NON-ADIABATIC TO SOLVENT CONTROLLED ADIABATIC ELECTRON TRANSFER: SOLVENT DYNAMICAL EFFECTS IN THE INVERTED REGIME By DAR-YEN YANG This study concerns the effect 'of solvent dynamics on non-adiabatic electron transfer reactions. A hamiltonian is designed to include: a reaction coordinate for two quadratic potential surfaces of donor and acceptor species and a heat bath which is characterized by a single Debye relaxation time 'CL. Solvent dynamical effects are described by an indirect coupling between the reaction coordinate and the heat bath. The time evolution of this system is obtained by use of the quantum Liouville equation. After averaging over the solvent fluctuations, the dynamics along the reaction coordinate are reduced to a classical Fokker-Planck operator, but the motion of the electron is still treated quantum mechanically. When the rate of nuclear motion in the potential well is comparable to the non-adiabatic transition rate, a consecutive reaction scheme leads to a rate constant expression k 12 = 1:131 1:2" / ( RI? + k3” ), the steps being the diffusion along the reaction coordinate with rate k3" followed by crossing at the intersection of the donor and acceptor potential surfaces with a rate k'l'S‘l' ns Arr .kd is dependent on the solvent dynamical effects through 1:1. and k1? is independent of tL. When the motion of the system in the transition region must be treated quantum mechanically, the transition region can be spread out over a length larger than the mean free path. Then RI? should be modified 'by solvent dynamic effects. When the separation into diffusive and crossing motion is no longer appropriate, we use an eigen—function method to expand the four coupled equations ' for the density matrix of the system, and solve for the reaction rate. A comparison of the numerical and analytical results is given. TO MY FAMILY ii ACKNOWLEDGMENTS I am very grateful to Prof. Robert I. Cukier for his guidance, patience, and countless time for help. I want to thank Prof. K.L.C. Hunt for her patience to read this manuscript and correct the errors. I also want to thank Prof. M. Morillo and Dr. P. Dardi for very useful discussions. I appreciate the Yates Memorial Summer Fellowship for support in 1987 and 1988. The encouragement and patience of my family enabled me to finish this work. iii TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF SYMBOLS CHAPTER I INTRODUCTION CHAPTER II DERIVATION OF EQUATIONS OF MOTION CHAPTER III ANALYTIC EXPRESSION FOR THE RATE CONSTANT CHAPTER IV NUMERICAL SOLUTION OF COUPLED EQUATIONS CHAPTER V RESULTS AND DISCUSSION APPENDIX A Formal derivation of reaction rate expression APPENDIX B Derivation of G 12 APPENDIX C Eigen-function L12 APPENDIX D Vector recurrence relation REFERENCES iv vii 10 40 48 54 59 64 68 74 LIST OF TABLES Table 1. Comparison of k” with 1:21, I: Table 2. Comparison of it12 with 1:3. irr rev 12 ’ 1‘12 51 52 LIST OF FIGURES Figure 1. Potential energy surfaces for the reaction coordinate. Figure Figure Figure Figure 2. 3. 4. 5. (a) Normal region (b) Activationless region (c) Inverted region Coordinate representation of the potential energy surfaces Branching ratio diagram for the forward rate constant Plot of I/(kfltl) versus fiEn. ( — ) corresponds to it“, ( ' ) corresponds to k3" Plot of k” x 10'3 versus i. ( x ) for G in ns 12 sq. (19.d), ( . ) for 61 1 in .‘q. (3.19s) vi 14 36 38 LIST OF SYMBOLS 4v12tL/n ZS/(foul) > u it : frequency factor H7 S-P mma’ u H II 1 / ksT cJ = L; Y (O) x = (l/xtt) (co/e,)3" D1, D2 : electric displacement of ion 1 and 2 A013 : overlap integral D : diffusion constant Ea : activation energy Er : reorganization energy AB = V 0 driving force so : optical dielectric constant 6,6 1 2 : electron energy of the fast sub-system E1, 32 : eigen-energy of reactant and product 0am) : wave function of slow (system or nuclei) variable 4", O2 : pure state wave function of the system in state 1 and 2 AF : difference of the force at the crossing point 1 = k0 x0 : slope at the crossing point . G“, Gzz’ G12 : Green function of L“, L22, L12 g : distribution matrix vii f1 : Planck constant hm : phonon energy H“ : Hermite function "i : rotation coordinate i = /_-_1 l : unit matrix kl3 : Boltzmann constant k1? : non-adiabatic rate constant from Transition State Theory kn. : non-adiabatic rate constant k d : adiabatic rate constant k : reaction rate constant irr irr k‘IZ ’ k21 I‘IV rev 1‘‘12 ’ k2: : irreversible rate constant : reversible rate constant k M, k” : reaction rate from RA to I and vice versa km, km : reaction rate from PB to I and vice versa x , i : rate kernel I r k 1 2, 1‘21 : forward and backward reaction rate K : reaction kernel k::, k: : forward and backward non-adiabatic reaction rate k d 1, k d 2 : adiabatic reaction rate in potential well RA and PB [(3 : non-adiabatic equilibrium constant Arr k d : Arrhenius adiabatic rate constant kw : force constant of the harmonic oscillator 118 z the size of the transition region If : mean free path I” : Landau-Zener length viii I = (1:2)“2 = ksT/kw : well motion length L : Fokker-Planck operator in potential surfaces V 11’ L22’ L12’ L21 11’ 1 1 V22’ '2' ( v11 + V:22 )’ § ( V“ + V22) In : left hand side eigen-function of L12 21' L3 : left hand eigen-vector of g and its conjugate Aj : eigen-value of M k : 1:le u : delocalization width In. : mass of electron u“ : eigen-value Na, Nb, Ni : population'in RA, PB, I 131 : Laplace transformation of N PB : potential well of product PL, PT : longitudinal and transverse part of polarization Y(r,R,qk) : wave function of the system va(r,qk) : wave function for the fast variable P : AE / th P ‘ J : Wigner representation of the probability distribution function g : projection operator P f : transition probability from RA to PB Pb : transition probability to stay at PB QatB : non-adiabaticity operator 9 : complement of projection operator g RA : potential well of reactant g : density matrix D : reduced density matrix g : Laplace transform of Q 31 : right hand side eigen-vector of g R 1. o : components of 30 r“ : right hand side eigen-function of L12 0 : position of crossing point - .1. 0y - O 2 10 s : Er/huL T : temperature 1: D, t : transverse and longitudinal relaxation time I. t = t/rl t ' kiZ + k21 : potential surface of reactant RA v11 V 2 z : potential surface of product PB V12 : electronic coupling constant V 0 : driving force ”I. = ' kw / ml. 1: : minimum point of PB x* (II/Bk” I. INTRODUCTION In oxidative phosphorylation, photosynthesis, and oxidation-reduction reactions, electron transfer (ET) is an important chemical process. Non-adiabatic transitions may occur in long-ranged ET or the spin change of reactants. Recently, Kosower and co—workers1 have found that ET rates in polar aliphatic alcohols are inversely proportional to the slowest dielectric relaxation time tL. This shows that the solvent dynamic effects can influence non-adiabatic ET, which has attracted considerable attention. The pioneering work of Marcus2 showed that, in the high-temperature regime, the ET rate has the thermally activated Arrhenius form k1? = AA exp (-Bs/kBT). The activation energy is the height of the nuclear barrier Ea : (AE-Er)Z/4 Er. This expression for the activation energy is valid in the classical limit when quantum effects of nuclear tunneling are omitted. In Marcus’ work, the ET process is adiabatic. Levich3 has treated ET as a non-adiabatic process using the Fermi-Golden rule rate expression. The medium is modeled by a set of harmonic oscillators. The frequency factor in the classical limit is AA = (Zn/fl) V1: (411 Er kBT)-“2. Zusman analysed‘ the situation when the dynamics of the reaction coordinate is diffusive rather than uniform; his approach rests upon the stochastic Liouville equation. In the extreme adiabatic limit, when the dynamics of the 1 2. reaction coordinate reduces to diffusion over the low adiabatic surface, Zusman’s expression reduces to Kramers’ results. Zusman’s result also connects the non-adiabatic limit for the frequency factor, which is determined by the electronic coupling, and the solvent-controlled adiabatic limit, which is determined by the reaction coordinate dynamics. In the non-adiabatic limit, the frequency factor is AA, while in the solvent-controlled adiabatic limit the frequency factor is proportional to tr. The most popular harmonic oscillator model is composed of two quadratic potential surfaces which cross, embedded in a polar mediated heat bath. When the system is at equilibrium, before the electron is transferred, the system is located at the minimum of potential well RA (see figure 1). When the solvent fluctuates (the system deviates from the equilibrium state), the system will move away from the minimum point. This deviation from the equilibrium state, which is activated by thermal energy, is described by a coordinate called the reaction coordinate. Since the dipole moment of the solvent always fluctuates due to thermal energy, and this may be described by diffusive motion, the system will diffuse along the potential surface. The splitting at the crossing point of the two surfaces is 2V12 (from the point of view of the adiabatic representation). When 2V12 > kBT, the system with thermal energy can not jump to the upper surface. The system will transfer from the left minimum to the right minimum within the same surface. This is called an adiabatic transition. When 2V1 < ksT’ the 2 system get enough thermal energy to jump to the upper surface several times before crossing. This is called a non-adiabatic transition. The 3 analysis of the relation between non-adiabatic and adiabatic ET is important for the understanding of solvent dynamic effects. We first summarize the outstanding problems6 as follows: 1) the relation between the ET dynamics along the reaction coordinate and the dissipation of the medium. Frauenfelder and Wolynes7 have considered the relationship between the mean free path and the Landau-Zensr length. They get two limiting physical situations: uniform dynamics and diffusive dynamics. 2) The relation between the ET dynamics and the strength of the electronic coupling. This depends on the distinction between the strong and weak electronic coupling in the Landau-Zener transition, which is determined by the L2 parameter I”. These two LZ limits are the non-adiabatic and adiabatic coupling limits. 3) The nature of the dissipative properties of the medium and of the strength of the electronic coupling. The interesting physical situtations are: non-adiabatic ET, solvent controlled adiabatic ET, and uniform-adiabatic ET. 4) The transition between non-adiabatic and the solvent controlled adiabatic ET. 5) The competition between ET and medium relaxation. The basic implicit assumption of general non-adiabatic ET theory includes the separation of the time scales for the fast medium dielectric relaxation and for the slow electronic processes. When the microscopic ET rates of a given state are comparable to the medium relaxation rates, the ET process is expected to be determined by the longitudinal dielectric relaxation time 1: 1. corresponding to the solvent controlled adiabatic limit. These two quadratic potential surfaces have three different combinations: normal, activationless, and inverted regimes. In the -__'T Er ENERGY 1--- Es. _’_-"’T’-' _ *. | AE L 6 k. Figure la. Potential energy surfaces for the reaction coordinate. Normal region ENERGY Figure lb. Potential energy surfaces for the reaction coordinate. Activationless region ENERGY V'T—_—"'_-—_—__———'.- Figure 1c. Potential energy surfaces for the reaction coordinate. Inverted region 7 normal regime, with barrier energy larger than ka’l‘, the non-adiabatic '1' L315 1: . . 1's _ _ 1/2 transition rate 18 km - 2 h ( B E .) exp(- B Ea). This is the Arrhenius law which describes localized transitions and kIfT is independent of tL. The diffusion motion inside the potential surface Arr gives kd = + ( g: )1/2 exp( - B Es ), which depends on IL. Both 1. km, and k3" include an exponential part and a prefactor. For kIfT, III the prefactor is proportional to the square of V 12. Usually this makes 131 k It. much smaller than k:". In the activationless or low barrier energy regime, it is still not clear how to describe the transition rate. The inverted regime which has been discussed by Marcus8 has strong quantum behavior. The region that nuclei can tunnel through is called the delocalization width :19 (see figure 1c). When u is spread out this means that the system exhibts strong quantum behavior. The diffusive motion inside the tunneling region is our main topic in this work. Rips and Jortner6 studied the reaction rate by using a path integral formalism that ultimately reduces to a consecutive mechanism. This method separates the reaction rate into diffusive and reactive 1. When part. The total rate is 1 / 1,12 = 1 / kg”- TST It. 131' it I'll Arr k d TS + l / k“. » TST k , k12 1: kn. ; the rate constant for the rate determining step is which is independent of ti. . . Arr 18 increased and k d . If we increase the friction of the TS‘I‘ « k ; solvent, 1: is decreased, to make kg" 1. Arr k d . The rate constant for the rate determining step is then k12 = Arr ks The solvent effect (through the longitudinal dielectric relaxation time IL) makes the transition rate slow down and the rate 8 constant k12 is changed from the non-adiabatic transition rate k1"? to the adiabatic transition rate k3". This shows how the solvent can controll the transition rate from non-adiabatic to adiabatic behavior (for an adiabatic transition, the reaction rate is independent of the TST electronic coupling V ). In the work of Rips and Jortner, km is 12 independent of IL, and the delocalization width 11 it If. Morillo and cinder1° showed, with I ) the delocalization width :1 » If, that it: = vi5(u/Er1ui'r)"zex (_LEr-j EILZ 1 1+x P 4ErkBT l+x ). Here it: depends on IL. The situation that we are going to study is when I 8 u » I r’ what is the reaction rate expression? And will the consecutive mechanism still be sensible? In this workzo, a Debye solvent has been used. The dielectric fluctuation has frequency lower than ksT / It. A classical diffusive motion of the system along the potential surfaces can simplify the calculation. The tunneling effect still needs to be kept in a quantum manner. The problem is formulated by using a density matrix in Liouville space. By use of the Wigner transformation, a set of semi-classical Zusman equations has been obtained, which were written down by Zusman without derivation. By averaging over all the solvent degrees of freedom excluding the reaction coordinate, a classical Fokker-Planck operator is obtained to describe the diffusive motion of the system. The quantum transition is kept in the off-diagonal terms. So, we have a set of semi-classical equations of motion. A phenomenological rate constant is defined“. To calculate the Laplace transformed rate kernel, an appropriate truncation of the Zusman 9 equations with reasonable assumptions gives us two reduced coupled equations. The well-known projection operator method12 is used to separate equilibrium and non-equilibrium parts of the motion. A consecutive mechanism can be sucessfully introduced to separate diffusive and reactive dynamics. The final rate expression interpolates between the adiabatic and the solvent-controlled non-adiabatic limits. The effects of the solvent on the adiabatic character are clarified. A careful study of the non-adiabatic transition rate shows the solvent dynamic effects in it. We show that, in the inverted regime, friction makes the quantum behavior more classical. Beyond the regime that the consecutive mechanism can be used, an exact numerical calculation has been done. An eigen-function expansion method” transforms our equations of motion into four coupled first order differential equations. The total reaction rate can be identified with the smallest non-zero eigenvalue". With large fluctuations included, a vector continued fraction method has been used to extract a relatively small eigen-value. The model hamiltonian for the rate process in condensed phases will be introduced in chapter II. In chapter 111, we define the rate expression and truncate the Zusman equations. The projection operator method is used to get the final rate constant expression. We also analyze the non-adiabatic rate constant in detail. The numerical calculations will be described in chapter IV. The results of numerical calculations and discussion of the validity of our consecutive mechanism are given in chapter V. II. DERIVATION OF EQUATION OF MOTION The electron transfer problem consists of a transfer of an electron from one localized state RA (D‘A) to another localized state PB (DA-) within the same molecule or between different molecules. We shall consider Levich's model3 in our following study to derive a quantum mechanical hamiltonian. The model consists of electron + donor 4- acceptor + solvent. Donor and acceptor carry charge M» and m+, and there is no first coordination layer of the ions in Levich's model. The solvent is a continuum polar medium which is characterized by the polarization P(r,t) of the optical vibrational branch of the solvent. A polaron model of this polarization has been used, so the solvent consists of a set of phonons. The total hamiltonian of the system is H:H+H+v +v (2.1) s e is es where 2 a) 2 5-1 3:991: (qz- )+—3-——I(nz.nz)dv (2.2) s 2h1 k a 2 enez i 2 . qk G 2 z - -9- 2 2.2 so “.-'2m.v.+.+1?:'§T (2'3) < ll ” -meI-Tl%—f(—:;i»dV-nel(i—iT-—Ei-El‘ldv (2.4) v" -eI‘1‘—i§—%;-}dv (2.5) cm is the optical dielectric constant. 10 11 H. describes the energy of the solvent: the first term is the energy of phonons which is described by the normal coordinates qk, and the second term is the energy of the solvent which is produced by the electric displacement D1 and Dz due to the ions. H. is the energy of the electron where the donor and acceptor are located at position 0 and R. V “ is the coupling between ions and solvent. V” is the coupling between electron and solvent. If we decompose P into P = PL + PT, from div PT 2 O, we can see that only the longitudinal part of P will contribute to VH and V". This is important in our choosing the relaxation time of the solvent. The Schrodinger equation is H ‘l’ (r,R,qk) = E V (r,R,qk). ( 2 . 6 ) In order to solve the Schrodinger equation, we can use the Born-Oppenheimer approximation to separate out the fast and slow motion of the system, which comes from the fact that both the fast motion of electron and the slow motion of heavy solvent particles are included in the total hamiltonian. Let s (r.R.qk) = (:1 243m) Wuhan) ( 2 . 7 ) and let va(r,q.) satisfy 2 8-]. fl 2 (I) 2 2 [-_E-ve+u1+v +v.+ 2J(D1+Dz)dV]\o1 81! so :ew (2-8) -iivzt. +v +V +£0-1(D2i’DzldV 2m e ‘12 ea is 8"62 1 2 W co 2 12 :sw (249‘) u -s_e_z, mez i'r Ir-RI 2 2 In 0 n 6 Substituting ‘1' into the Schr'odinger equation and multiplying with w‘, then integrating over the volume, we get [ H8 + Ci Ei ] ¢i v12 ¢2 [118».52-32](itzzvmtt1 (2.11) where vizzQiz-A12(Hs+€2'E) (2.14) V21:Q21-A12(H8+£1-E) (2.13) A = 0 01:3 (2.14) orB{ * , [wavfldv a“, not °° :6 a iaz'g 9.3“? if. [‘ji’ea‘q—R'devl‘a—‘akltl". aqzd" I t +IwaquBdV]. (2.15) This describes the Born-Oppenheimer states 4’1 and 412 of the system in initial and final state. The displacement of the heavy molecules of solvent caused by the electric field of the electron is small and a linear approximation of ca( qk ) relative to the equilibrium state can 13 be used. So do a s _, (o) a _ (a) Eaiqkl-ealqk l+hf1(5-a:-)qh_q:0’(qk 9k )f(2'16) where qim is the equilibrium position of qt. Now the total hamiltonian is o 2 _ gen _ (0) 2 _ a Hsiifia' 2 £{(th qt ) (a) z}+Ta’ hi 6 ( q. - q. l ( 2 . 17 ) where (0) hm Q (0) 2 'l‘aztzm('I )+-2—-Z(q. ); (2.18) hi ' i.e. co 2 - {131 (a) 2 a Hi-Zkfi{(qk-qki)_a( (0))2}+T1 (2'198) qt: q“ as 2 _ M (0) 2 a ‘ Hz'z z{(qt‘quz)' (0)2}+T2' (2.19b) "1 a ( it ' quiz ) If we set E _ _ (0) t‘qt qki (2.20) and E(0) _ (0) _ (0) it ‘ qt q“ (2 . 21 ) l4 Figure 2. Coordinate representation of the potential energy surface. 15 then on 2 hm 2 a H:—X(§- )+T 1 2 h, It aEma i (2.22s) so 2 n2=§§lz[(§k-g:°’)2- 62]+T2. (2.22b) III 65‘ Since qk is the normal mode of the vibration, Ek can be thought of as another N orthogonal coordinates in E-space. With a rotation of the coordinate Eh, a new coordinate set n‘ = 2 a” 8' is obtained. After the rotation of the coordinate Eh, the potential energy hypersurface of H is on axis 1‘)1 (see figure 2). The final hamiltonians are co 2 z 31=9§12(nk2- az)+§%1(12-az)+T1(2.23a) ksZ an.I an} a) 2 2 6 a H2:§§12(nk2- 2)+J§£2fl ("i-"0’2- 2 +T2 k-z an“ 6r)1 (2.23b) where n2-; (Em)2 (2 24) o -k-1 k C O The 771 axis can be thought of as the reaction coordinate; the other N-l dimensions are the same for H1 and Hz’ and can be thought of as solvent coordinates. Unfortunately, there is no coupling between reaction and solvent coordinates from this perspective. It is not clear how to 16 derive the coupling rigorously from a hamiltonian model. We will put the coupling term (between reaction and solvent coordinate) into the hamiltonian artificially (see eq. (2.25)). In the following, we will assume that the system moves on one dimensional potential surfaces (or reaction coordinate) which are Born- Oppenheimer surfaces. There are two well-separated minima which correspond to D-A and DA-. The heat bath will couple dynamically to the motion of the system. So the motion of the system on the reaction coordinate is described by the quantum Liouville equations. Using the Wigner transformation, four classical-like equations are obtained. We use classical Fokker-Planck operators to describe nuclear (solvent polarization) motion, and the electronic coupling is still kept in a quantum manner. The hamiltonian is H=H030+%(Vu-V22)oz+V12<;x, (2.25) where HO:[-§%; va2+v((qk})]+[-!2-‘;- vx2+%(v“+v22)] +v({qk}.x). (2.26) and OH Lo I.___J 17 are Pauli spin matrices. In Ho’ the first term is the heat bath hamiltonian, the second term is the motion of the system on reaction coordinate, V({q0),x) is the coupling between the reaction coordinate and the heat bath. We assume there are no off-diagonal terms in V ({qalm) i.e. V12({qal.X) V21({qa}.X)=0 (2.27) and V“({qk}.X) vzziiqkirx) viithoX). (2.28) The two-level pure state wave function is ¢ = ¢1({Qk},x) + Oziiqhiar) , so the density matrix p = 0 (it. The Schr3dinger equation of our system is . 6 _ Ifi'a—EP-[H,P]. (2.29) Its components' forms are .9. , t_ x 1 t lfiat¢1¢i '[ HO’¢1¢1]+[-2-(v11-v22)'¢i.¢1] t 3 +[v12¢2¢1-¢1¢2v12] (2.30s) pa {3‘ 9 9 II 3 1 3 atiz [Ho'¢1¢z]+{§(vii'vzz)’¢i¢z} t s +[V12¢2¢2-¢1¢1V12] (2'30b) 18 ifi_a_.¢ @‘ :[ Ho,¢ OIJ-{%(V11-szli¢z¢:} atzi 21 *[V,2°i¢:'¢2¢;v12] (2'30c) . a s_ s 1 t 11g‘é—Eiiizitz"I:Ho'4.24.2]"['2'(“’11'V22)’4’2¢2] +[v12¢1¢:-¢2¢:v12]° (2'30d) Two kinds of terms appear: commutator and anti-commutator. A non-adiabatic transition is a quantum transition, so we must start from a quantum description. But we want to treat the solvent classically. For a quantum system, especially a quantum statistical mechanical system, the Wigner transformationis gives an appropriate procedure to transform from quantum to classical motion, because a Poisson bracket-like term which describes a Liouville equation is obtained and this describes classical motion. The Wigner transformation is defined by P :(uh)-(N+1) U Idydlyalexpigfiipy+payaH "-ya')¢:(X+y.q"+y'). (2.31) ¢(X'Y9q t! a i a We simplify the expression to one degree of freedom only. The commutator part is 2 fi 2 :1 ['25" ”WM” 21 _ --py 2 2(nh)‘ Jaye“ [-g-DVZ+V.¢‘(X—y)¢:(x+y)] (2.32) 19 -- -21 .. 2 -1h{ maxJanka.+21sm[2vxvp]v1)”. (2.33) The Poisson bracket term is similiar to a Liouville equation classically. The sine term can be expanded by using a Taylor expansion and the lowest order term can be included in the Poisson bracket term. The higher order terms which include h“ (n 2 2) describe the quantum correction, and are not of interest here. The anti-commutator is _ 4 27:“ Vi¢i¢J ~(nfi) e V.¢i(x-y)¢J(X+y) =2003[§VxVP]VPU. (2.34) We want to treat the solvent classically, so higher order terms of the Taylor expansion of the cosine term are omitted too. The Liouville equation that we obtained yields the Fokker-Planck operator, following the well-known procedure (see ref. 16). Let pijzidppij’ (2.35) then we have 1p :L p +Xl£(p -p ) (2 368) at 11 11 11 in 21 12 ' a _ 12 atpzz‘L22922+ih (912 921) (2'36b) V - V ) V a __ ( 11 22 12 _atP.z-L,ZP,2+ if, 912+;g-(922-pn) (2.36s) (V - V ) V- a _ 11 22 12 57921-151ng- i‘h 921+i-fi—(p11_pzz). (2.36d)‘ The potential surfaces in Li J are, from equation (1 - 4): ( V 11 +v22)/2+(vi -v22)/2=v11’(v11+v22)/2,(vii+v22) 1 /2,(Vn'tvzz)/2-(VII-V22)/2=V22inL“,L12,L21,L22 where L i j are the Fokker-Planck operators. The motion of the system in the initial and final potentials (V and V 2 2) is determined explicitly, and the off-diagonal terms 11 are described by the motion on the averaged potential surfaces. These semi-classical equations describe the solvent dielectric fluctuation as a low frequency fluctuation with energy smaller than kBT. So the solvent is treated classically. The relaxation time in the Fokker-Planck operator is ti. (longitudinal relaxation time) instead of to (transverse relaxation time), because only the ‘longitudinal part of the solvent polarization contributes to the interaction with electrons (see eq. (2.1c,d)). Recently, many investigators" have claimed the relaxation time should have an order of magnitude between 1: and to (usually ID > tL). In our model, a continuum medium has I. been used, so the relaxation time is fixed at IL. The diagonal elements p and 922 are the probabilities of finding the system in the initial 1 1 and final potential well. The diffusion motion (now it is the fluctuation of solvent polarization) is slow since it is activated by thermal energy. The off-diagonal terms include not only the diffusion 21 motion but the high frequency quantum transition behavior (1 ( x - o ) / f1 term). These properties are important in our future assumptions. III. ANALYTICAL EXPRESSION OF RATE CONSTANT We derived four coupled equations for electron transfer in a polar medium in the last chapter. In this chapter, we are going to derive an approximate expression for the rate constant by using a projection operator method. A consecutive mechanism will be used to separate the diffusive and reactive dynamics (see figure 3). Electron transfer can be expressed as motion from RA (D-A) to PB (DA_) through an intermediate state I. The boundaries of RA-I and PB-I are located in the region where the tunneling probability is negligible. Such a division has been used in chemical kinetics. An interesting example is given by Northrup and Hynes in ref. 11. A double-well potential surface has been used in a chemical reaction system. The potential surface has been separated into three different regions RA, PB and 1. RA and PB correspond to wells and I corresponds to the intermediate region. During the course of the reaction, spatial equilibrium in each well has been obtained. But the intermediate state has been perturbed by passage across the barrier. Thus the intermediate state will deviate from the equilibrium distribution, and this produces a net change of the populations which makes the meaning of the rate constant sensible. Northrup and Hynes use the internal rate constants (which are equal to our kd1 and k d2) to describe the rate of approach to equilibrium inside wells RA and PB. The motion within the region of the intermediate state 22 23 I is characterized by the barrier rate constants (which are equal to our k1: and. k::) which describe the rate from RA-I to PB-I and vice versa. The final overall rate constants then are geometric sums of the internal rate constant and barrier rate constants. We shall use a similiar point of view in the following discussion. The kinetic scheme for the occupation probabilities Ne, Nb, Ni of the states RA, PB and I is a - fiNs--kAle+klANi (3.18) -a—Ni"k N-k Ni-k Ni+k Nb (3 1b) at ' AI ' IA 13 BI ’ isz'k Ni-k Nb (3 1c) at [B B! ‘ ‘ To maintain these expressions, the macroscopic quantities N s, Nb, and Ni should exist; the barrier height should be high enough to obtain an equilibrium state at each potential well and the rate constant should be small enough to be measurable. The macroscopic quantities can exist only when they can be explicitly identified. Distinguishability between N. and Nb exists when RA and PB are well seperated by a high barrier. Passage across the high barrier can perturb the equilibrium distribution in the intermediate state, and make it non-equilibrium. A net change of the populations per unit time in region I produces the reaction rate. In other words, an oscillation of the change of the population from RA to PB which is fast forward and backward can not h 24 di 1” 8 kdlpf (l-Pa) it 9 di 1’ 1,1,8”, kMPr(l-P8)PrPB Figure 3. Branching ratio diagram for the forward rate constant. 25 have a reaction rate. The initial conditions are: N. ( t = o ) = 1, Ni ( t = o ) = o, andNb(t=0)=0. Using the well-known steady state approximation for the intermediate state 52? Ni : 0, we obtain Z s It —Idt§r(r)Ne(t-t)+Jdt§r(t)Nb(tft) (3.2a) BETszIdti-t’f(t)Ns(t-t)-Idti_tr(t)Nb(t-t) (3.2b) where the rate kernels i r and it are defined as ir:k“6(t)-k“kuexp{-(ku+k”)t} (3.3a) irzkmkmexp{-(kui-km)t}. (3.3b) Here 5(t) is the delta function. The memory effect in (eq. (3.2)) means that the rate of change of the stable state population at time t is dependent on the intermediate state population at time t - t. The intermediate state population depends on the previous gain from and loss to the stable state. Since we are interested in a long time rate t > I" (transient time which is the time period required for the maintaining of a 26 non-equilibrium steady state condition in the intermediate state), the rate expression can be simplified to a .. ath—-k12Ns(t)+k21Nb(t) (3.4a) -a-Nb-k Na(t)-k Nb(t) (3 4b) at " 12 21 ' where k12=Jdtxr(t) (3.5a) k21=Jthr(t). (3.5b) Then the population equations can be written as Nb 4:12 1:21 Nb "“‘l"'] N" (3.6) andwedefine Q Nezjdxpn(x,t) .0 a) szjdxpzz(x.t) .m in our case. G After Laplace transformation, f(s) = I e- at F(t) (it, we have 0 27 [si+1_;]1§(s)=§(t=0). (3.7) The reason to do this Laplace transformation is: There are an infinite number of relaxation modes which are distributed from low to high frequency. The slow ET rate is assumed to have a comparable rate to the slowest relaxation frequency (of course not in the low barrier and activationless regimes). If a long time rate exists, we want to extract this slowest process from the fast relaxation modes. Letting s 9 0 (i.e. t -+ 00), we can get a long-time rate expression. Fortunately, our four coupled equations of eq. (2.36) can be reduced to two coupled equations similar to the reduction to eq. (3.16) and the rate expression can be identified by using a projection operator method. We now carry out this reduction. The formal solution of eq. (2.36c) is v12 p12:'-K_Idxiidti 612(x,t| xi' t'1) [p22(xi'ti)-pii(xi'ti)] ‘ (3'8) where - 1 - a - Giz'[8_t-L12+l(vii-v22)/fi] = 1 1 exp{% 2/21t/D'cl(l-exp(-2t)) 2 7t 1 L -t D1:L(1-exp(-2t))[2D h (l’e ) 28 2 71: 2 . -t 1 -t i. -1[x-xie -§xo(1-e )]]-D( h)ti.t yrI. 1 +iT(x-xi-2x0t+at)} (3.9) V +V _ a a a 11 22 Liz-Dn[fi+53—i‘_‘z——’] for the potential of eq. (4.1). This propagator includes both the motion induced by the , and the surface splitting, (V 22 - V 11)' for stochastic process, L 1 2 the tunneling effect. As we discussed in the last chapter, the diffusion along the reaction coordinate is treated classically because its characteristic frequency to a 1 cm.1 (i.e. smaller than kBT / h) is lower than the quantum transition frequency a) a 1000 cm.‘1 (i.e. 1 ( x - o ) / 11). Assuming the time variation of G 12 is faster than that of the local population difference, [922 ( x, t ) - t)11 ( x, t ) ], then 912:"1deifiizlxl xi)[922(xi.t)-P“(Xi.t)] (3. 10) here - V12 K12(x|xi) : "fi- I dt G12( x,t|x‘,0) (3 . 11) i1 15L 1 (in G12, the _—_h ( a - 2 x0 ) t term shows that the oscillation frequency will increase as t increases). 29 The formal solution of eq. (2.36s) with eq. (3.10) yields v . p“(x,t):-%—2-IdxiIdxildtGfl(x,t|x1.0) Rahal",’[Pzz‘xi-H“Pn‘xr“]' (3.12) 11 . a -1 where G11 18 the propagator corresponding to [ g-t- - L ] . 61 ‘ describes the motion in well A. If we assume its spatial variation (with frequency smaller than ksT / ft) is slow relative to that of _ 7 1’ _ K12 (with frequency 1 ( x - a ) / it) (see i f) L x term in K12), then v12 m - pnz—g—deiIdth(x,tl x1 ) [Idxi K12‘31l xi) 0 .[pzztxi,t)-p“(x‘,t)]]. (3.13) This is the key assumption and leads ultimately to the consecutive rate scheme. Note that we separated the diffusion (G 11) motion and the transition (R12) behavior in this equation. That is the basis of a consecutive mechanism. We define V e * .- _12 - K12(x,x)-fi—Idx K12(x1|x) 30 v12 2 o =(.._fi) d". dtG12(x1,IIX.0)- (3.14) 0 Applying G1? to eq. (3.13), we obtain a _ . x [fi-L11]911-2Rex‘2(x,x) [932(x.t)-p,,(x.t)] (3.15) A similiar treatment of 022 gives us the final two coupled equations EQEE=[E+§]E (3-16) with p:[pii] - p22 't‘ N f'_""! C L" i H L" o N N h—————J 'W II N R sub N f——"1 I he I H t-o t—u—u—l the if term can also be thought of as a sink term as discussed in ref. 18. The projection operator method will now be used to separate the diffusive and reactive dynamics. The reactive dynamics arise from the equilibrium part of the rate kernels, see eq. (3.16). 31 We define the projection operator 1: = g I dx and its complement Q l - E! where 2,0 °° g: . 81=GXP(-anl/ [dxexm-BV“). 0 s 2 4» By applying 2 and 9 to the Laplace transformation of eq. (3.16), we get =Bg(t=0). (3.17) In obtaining this expression, we have used 2 L, = 0 (this projects out the diffusive motion), the boundary condition 8 9 0 as x + x a; and 1!! = 0 for the equilibrium distribution conditions in each well. Comparing with the reaction kernel in eq. (3.7), we can identify the rate kernel as at: i=lim c"{i§-§§[al+9"9<9+§)1}§. s90- ( 3 . 18 ) The first term describes the instantaneous rate constant which is obtained by assuming equilibrium in the donor and acceptor wells. The second term accounts for the non-equilibrium effects caused by diffusion and reaction dynamics. After simplification of the formal expression (see Appendix A), 32 weget 12 kiZ: 12 kn. 21 (3.198) l+klie/kdili'kns/kdz and 21 k21= 12 km 21 ' (3'19b) 1{'knslkdi‘.,kns/kt:|2 where (D h"- dt c°("t)"(‘)-1 (319s) di " ii x,x, Si 1: ' o and k‘J-dx(-‘) () (319d) M— xU x,x gl x. . The k c“ are the internal rate constants which describe the approach to equilibrium in each well. The non-adiabatic transition rate kit: is a surface crossing rate which depends on the initial equilibrium distribution g i and the transition term in the intermediate region. Our KU is derived from (it12 (see eq. (3.14)). For large friction it is convenient to use G 11 (see ref. 10), because the difference between G1 1and 612 is small. For moderate friction, their difference must be treated explicitly. The formula that we get for the rate constant can be explained by using a branching diagram (see figure 3). In this diagram, k M represents the diffusion rate in surface RA, starting from an equilibrium distribution. P f = k:: / ( k d 1 + k3: ) is the probability 33 of reaching the PB-I boundary rather than staying at RA. Pb = k d2 / ( k d 2 + kii ) is the probability of staying at PB rather than crossing back to RA-I. The total foward rate constant should include the one way flux and all of the recrossing trajectories, so ‘9 i k12=kd1Pr{J§O[(I'Pb)Pr] }Pb k Pth/[l-Pr(1-Pb)] d1 12 kns de 12 ‘ 12 _ ( kd1 + kns ) ( de + kna ) k12 kZI us as 12 21 ( kdi + kns ) ( de + kns ) " k: ( 3 20 ) " 12 21 ’ ° 1 + kna I kd1 + kna / de and similiarly for k21° If the passage to PB is irreversible (i.e. k3: = 0), then _ 12 12 kiz-km/(l+kM/k“). (3.21) . . . . _ 12 21 _ This is the one way reactlon rate. The ratio k13 / k21 - km / k"a — Kr: , and satisfies detailed balance. If it: « k d 1, the rate determining step win be the non-adiabatic transition rate kit. If kl: )9 k d 1, the rate determining - 34 step will be the diffusion rate k d 1. Thus, the solvent controlled transition from a non-adiabatic to adiabatic rate constant is involved in our rate expression. Sometimes conditions can satisfy k: / li¢n « 1, k: / k d2 >> 1, i.e. the system reaches an equilibrium state in RA very 12 as fast and is slowly stabilized in PB. The forward rate then is k12 = k k d 2 / k:: and the rate is controlled by the rate of stabilization in PB. The equilibrium constant [(22 = k: / k3, so the forward rate is independent of ET rate. This can happen for an endothermic reaction. The k (H rate constants describe the rate of approaching equilibrium in each well in the absence of the reaction. The propagator s G‘ i ( x ,t I x*, 0 ) describes the probability density for being at the . . t . . . . i . crossing pomt x at time t, given that the particle is at x at t1me t = 0. The average time for the particle’s evolution until it reaches 3 - equilibrium, starting from x , is k d:. For barrier heights B Ea > 5, we can use the Arrhenius reaction rate (k:" = % ( “E. ) t. ”2 exp (- B E-)) instead of using k; = 3:1 I ( Gii - gi ) dt (see figure 4). As we can see, for B Ea > 5, they match very well. The rate constant kl: describes, through G12, the stochastic process and surface crossing transition in the neighbourhood of the crossing point starting from the equilibrium initial well RA (see eq. (3.19d)). This off-diagonal propagator G12 describes the transition process (see Appendix B) as 1: K12(x;x*):(%)2%Reldtexp{-ix hL(1-e't) 35 71:1. 1 -t +1 h[§-xo(l-t-e )+ta] -AiZ[-3+2t+4e“-e‘2‘]} (3.22) where 1 = k" x0 = AF and AF is the difference of slope (or force) at the crossing point. In the normal region, x o -) large (i.e. the separation of the wells is large, see figure 1c), this implies 118' I -) 0, and we get 1.2 more localized behavior. In the inverted region, x0 ., 0, then 118' I” -> large, and the quantum behavior becomes more obvious. The interesting region that we discussed in the Introduction is 1 1O 1, (overdamped region), and inverted region (A is small). If we assume A 12 x» 1, then only the short time behavior in eq. (3. 22) is important. Then 11: x 1 . 1- K12(x;x)=(%)iReJdtexp{-1 h[(x--o)t 1 2 2 2 1 '17ti +§(-2-xo-x)t 1-[5A1 +3x—fi— 1 -iytt To accurately describe the delocalization width, )1, is very difficult. We define it qualitatively as the ratio of the 1/3 power of the coefficient of t3 and t, and we have u=lx-0| 36 l 0.25L A L. b—J x 0.15L U .1 \ " 0.05- 4 8 12 16 20 B x Eo Figure 4. Plot of l/(kdtl) versus BEs. (——-) corresponds to k“, ( - ) corresponds to k3". 37 1 3 T 1/ = ‘/[§AV] -1/3 =(xotL) . (3.24) As the friction increases (i.e. IL + large), the delocalization width )1 will shrink to 0 (i.e. localized transition). The small x0 (inverted region) will increase 11. After the integration over x in k: (see eq. (3.19d)), we have on 12_ (321 -t _ kna-(Z) intCOS[A(S(l-e ) Pt] 0 2 -t exp{-Ak [e -1+t]}. (3.25) When A 12 >> 1, the short time expansion is appropriate, and we obtain 12 azl A12 kn.=(z) iIdtCOB[ABt]exp{- 2 t} 1 21: B2 :_ exp{___-}, (3.26) 2 A12 2A As we can see, in the last equation there are no friction effects in the exponential term. This means that in the extreme overdsmped regime, the transition rate depends on the barrier height only. The width of splitting is zero (relative to the diffusion length), and it is a localized transition. This localized transition gives us the Arrhenius 38 12- n l O 8r- r- X x NO .- c .X x 4L. 0 X x x x o x e e x 1: ° 0 Figure 5. Plot of k: x 10-3 versus 1. ( x ) for G12 in eq. (3.19d), ( ° ) for G11 in eq. (3.19d). A = 0.06, S = 0.05, P = -4., i 1 ho: 300 om" , V12=O.3cm- ,1“: 300 K. 39 result. When A 12 is not too large, it should influence the activation part (i.e. exponential part). As we can see from figure 4, (x means k12 as is calculated with the use of G12 and o is calculated with the use of G 11), which is in the inverted region, our kit will decrease. as the friction increases. If we increase the friction to have large A 12, r which is independent of friction. Also we than k12 will approach k" I'll “I can see that the difference between k12 (from G ) and k12 (from. G ) ns 12 ns 11 is large when k (friction) is moderate. To conclude this chapter, the key step in getting the consecutive rate expression is the assumption that the spatial variation of G 11 is slower than that of E12. This separates the diffusion and transition parts of eq. (3.10). So we have two different rates in the wells and the intermediate region. The behavior of the transition depends on the relative size of the splitting width )1 and diffusion length It. If the diffusion length is smaller than u, than within the length )1 the system is still undergoing diffusion motion and the friction effect is obvious. If the diffusion length is larger than u, than within the transition region the system will not feel the friction effects. Therefore, the quantum behavior of the transition can be reduced to a classical (i.e. localized) transition by increasing the friction. Friction effects (i.e. solvent dynamic effects) will contribute to our k;:. The solvent controlled transition fron non-adiabatic to adiabatic behavior is obtained in our result. IV. NUMERICAL SOLUTION OF COUPLED NUATIONS We have solved the reduced Zusman equations eq. (3.10) by using a projection operator method’ and obtained approximate analytic expressions for the rate constants. The consecutive mechanism relies on the separation of diffusive and reactive dynamics. To go beyond the region of I > u, when the transition is no longer confined to the crossing point, a unified method should be used. In this chapter we solve the Zusman equations exactly for the long-time reaction rate. The Fokker-Planck operator describing the Brownian motion in the harmonic potential surfaces can be cast into a tri-diagonal vector recurrence relation by basis expansion of the probability density. The Zusman equations with four coupled (second order in space, first order in time) differential equations can be transformed into four coupled first order (in time) differential equations. A vector form of first order differential equations is obtained. The Laplace transformation of its solution can change this first order vector differential equation into an eigen-value problem. We can identify the reversible forward and backward reaction rates with the smallest non-zero eigen-value. The efficient way to solve for this small eigen-value is to use tri-diagonal vector recurrence relations, from which a matrix continued fraction equation is obtained. This method has the advantage that without the detailed balance condition we still can get accurate and fast results. For computer programming, it is a good method especially 40 41 to save space. We use the harmonic potential surfaces -1 2 vii-kax v --1-ir(x-x)+v (4 1) 22'2 w o o' ' The stochastic processes corresponding to overdsmped Brownian motionsre — .59.. .9. _§_ - L -Dax[ax+saxv“],i-1,2 (4.2) Ii also (4.3) t" I C" I NIH I" + I." 12" 21' To simplify the calculation, we need to rearrange the definition of the density matrix and have the same differential operator for its elements. Also we can eliminate the complex components. This can simplify the computer calculation. We define our new density matrix elements as t 93(91‘1922) (4.48) 2Rep122'p12-tt)21 (4.4b) 21mp12=912-921 (4.40) With these definitions, the Zusman equations can be written as a + a2 a + 1 a - 57p-[ax2+(x--xo)5—;+l]p+§xo-a—;p (4.58) a - a2 a 1 a 6—3;" =[ax2+(x--2-xo)5-;+1]p +2x061;p -aImpn (4.5b) -a—Rep = a2+(x--1-x )—a-+1 Rep at, 12 a 2 2 0 air 12 x +b(x-O)Imp12 (4.5c) .3... - a?” -- ,6... at mpiz" a 2 x 2110 ax mpiz x 8- -b(x-a)Rep12+zp. (4.5d) Now we introduce a basis set expansion of our new density matrix: :1: a) :t P=Ean(t)rn(y//—2) (4.68) nIO . m + Rep12=2bn(t)rn(y/f2) (4.6b) n-O m - Imp12=2bn(t)rn(y//—2). (4.6c) 11-0 The argument y = x - -;- x0. The set { rn } are the right eigen-functions of the Fokker-Planck operator L12: 2 6 1 a - [ax2+(x--2-xo)5-;+1]rn—unrn (4.7) 43 With -1,2 _ n -1/2 __l__1/4 2 __ rn-(Zn!) (2*) e Hn( 2y) and Hz-n,n=O,1,2,... (see Appendix C for details)o The left-hand eigen-function { In ) also has been introduced and the integration over x leads to a + + - 5min":uflan«é-xoy/Tiana1 (4.38) a -_ 1 'l— a-Tan-uan-é-xo na_1-ab (4.8b) 5%b:=unb:+b/—fib;_1+b/n+lb;”-bayb; (4.8c) a -_ - + + + 1 - fibn—unbn-bmbnd—by’nribani-baybni-zaan. (4.8d) We assemble all of the coefficients into a matrix form d .. 572(t)-1§y(t) (4.9) where 44 is a column vector of the time-dependent coefficients. Rate expression: The formal solution of eq. (4.9) is I t y(t)=e 11(0). This formal expression can be rewritten in eigen-functions and eigen-values of 13]. Let I : Al I 1:. ‘2‘ J E. U = A . non-degeneracy implies L and I}, are orthogonal, i.e. J E3 _|‘ : 61k 1:; g 31.: A) -3 But: A): 1" -|t‘ Thus, (4.10) terms of the (4.11s) (4.11b) (4.12s) (4.12b) (4.13) 45 This shows a multi-exponential relaxation process or modes of fluctuation in the solvent. The eigen-values are the relaxation frequencies. We can show, from the conservation of probability, that AD =0andcoRoozl. D Recalling the rate expression from Chapter III, our scheme is - t Ns(t)=e-tt [k erxdx+N(0) ,N(0)=1 o k 21 12-tt (4.14) Now from the definition (a Ns(t)=Ip11(x,t)dx m 46 Q %I[p++p-]dx .‘Q O 1‘” + - 1 --2-n}.30[an(t)+an(t)]Jrn(x—-2-xo)dx —O 1‘” w :§n§o[a:(t)+a;(t)]Irn(x)dx -@ =%(2u)"‘(a:(t)+a;(t)), (4.15) Also from probability conservation 0 I(p,,(x.t)+pzz(x.t))dx -® 0 =Ip+(x,t)dx —(D 0° + 1 =It.§.a.r.1dx —co G -n§oan(t) rn(y)dy.y-x-2xo 85—H8 .0. 1/4 (21¢) aO (t) -1. (4.16) So we have 47 - 1 1/4 - . . -§[1+(2n) ao(t)], (4.17) i.e. 2 k 2 k 114 - _ 21 _ 12 - r 1. (2x) ao(t)—( 1: 1‘)+ 1: e . (4.18) Comparing eq. (4.13) and eq. (4.18) k --(El'-9+1)A/2 (4 19a) 21 ' R 1 ‘ 0,0 kizz-A1_k21° (4.19b) The quantities that we are going to calculate now are A1 and R1 0 / R0 0 of the matrices 1:1. The method of solution relies on the useful feature that 1;) is a block tri-diagonal matrix, where the blocks are 4 x 4 matrices, corresponding to the four coefficients at, bf for a given i. This tri—diagonal matrix actually satisfies a vector recurrence relationship. The details of the method are presented in Appendix D. From the structure of the zeroth 4 x 4 block, there is at least one zero eigen-value corresponding to the equilibrium state. Of the three other eigen-values, one is real and corresponds to 1 / ( k 12 + k21 ), and the other two are complex conjugates. For the existence of the reaction rate, A1 4! A2 - A1 (where 0 = A < A1 < A2 <...) and 0 R1 0 < R1 1 must be satisfied. Otherwise, ,an initial condition dependent rate constant expression should include all of the relaxation modes. It will be observed in the ativationless region. V. RESULTS AND DISCUSSION The numerical results of our algorithm can be checked with the Arr _ 2 kne-(v12/2fi) ( BnEa )1]2 e-BEa) at high barrier. Table 1 shows Be 2 B2 / 2A = 2.69 crossing rate of Arrhenius’ result (i.e. and the friction changes from k = 10 to 500. As we can see when we . . . . TS‘I’ irr rev increase the friction, the ratio k12 / kn. < 1‘12 / k12 < 1:12 / It12 , but all of the ratio are close to one. This means, for the normal region, with small splitting of the delocalization width, we can get the Arrhenius result from our numerical method. To check the validity of the consecutive mechanism, the parameters were chosen to have approximately equal diffusive (k d 1 , k d 2) and crossing (k3: , kil) contributions in the approximate rate expression eq. (3.19a,b). In this situation, the effect of non-separability of the dynamics will be explicit (i.e. the assumption of eq. (3.10) will not be valid). The friction values ranging from ti. = 10/3 to 200/3 represent liquids with moderate to large 1". values. The range ti. = 1000/3 to 2000/3 may represent a protein solvent. The electronic matrix element . . _ 2 1/2 V12 is chosen to satisfy td - (V12 / f1) [Id (v11 - V22) / dxl D 1-1/3 for non-adiabatic transitions (see ref. 9). This is the condition for the validity of perturbation theory. If this condition is violated, 48 49 for example, for large friction, the transition which is non-adiabatic in the ballistic region will change into an adiabatic transition. The factors in eq. (3.19 a,b) must be evaluated. The diffusion rates km and k d2 are obtained by direct integration of eq. (3.19c). The non-adiabatic rate constants kg: and k: are defined in eq. (3.19d) They would follow directly from a Golden-rule calculation, with the propagator corresponding to evolution on the averaged surface. The results presented in Table 2 show that for sufficiently inverted behavior the overall rate cannot be broken up into the consecutive steps of diffusion in the donor well and surface crossing. The width of the surface crossing region becomes so large, due to nuclear tunneling, that this separation can no longer be made. For not too inverted behavior, Table 2 shows that the approximate expression is good, when kii is evaluated by incorporating the nuclear tunneling effect. That is, if we were to use kifr instead of k;: , the result would be completely incorrect. For future work, we can expand our present work to include activationless ET with a localized initial condition (a delta function) which is different from, our gaussian equilibrium distribution. In the activationless case, there is no barrier, and the reaction rate is very fast. Before the equilibrium is reached, the reaction may be finished. Therefore, we should consider all of the following factors: 1) a delta function initial distribution. 2) For the activationless case, the 50 reaction rate is governed by the shorter time dynamics rather than by the longer time dynamics. Thus the high frequency relaxation modes (i.e. high frequency friction) are most important. Rips and Jortner use Zusman’s method to find the activationless solvent-controlled ET rate with thermal equilibrium and a localized initial state. Qualitatively, they found when the initial state is in thermal equilibrium, within a relevant time t / 1: < 4, an asymptotic rate expression is in good I. agreement with activationless ET kinetics. In the localized initial state, the system will move from its initial position to the bottom of the initial well. The time it takes is called the delay time t‘ which is smaller than the thermalization time. For t > t*, a single exponential decay description is appropriate. Our numerical algorithm gives a quantitative understanding of the contribution of the fast relaxation modes to the transition rate (i.e. the modes in eq. (4.13). with j 2 2 should be considered, because the splitting of the eigen-values between Ii1 and A 2 is small). In the two crossing point case, Kakitani and Mataga et. al.19 have argued that the force constant of the potential well should increase significantly when the solute charge is increased (for example, the reaction A+B- 9 AB) and the ET rate will be dependent on the large change of the activation energy. Using our method, it is possible to solve numerically for the ET rate when the potential wells have different curvatures. 51 . . TST lrr rev Table 1. Comparison of k,12 with kn. , k12 and k12 1 1 s = 40, p = -10, T = 300 K, v12 = 10 om" , not = 100 cm' -1 TST irr rev A k12(sec ) kiZ / kn. k12 / R12 12 / kiZ 10 2.3411109 0.95 0.97 0.98 50 9.76x107 0.99 0.994 0.997 100 2.45::107 0.9957 0.9978 0.999 200 6.13x106 0.9958 0.9969 0.9975 500 1.00x106 1.018 1.018 1.018 irr _ TST Arr TST Arr I kiz ' kna kd1 / ( kns + kd1 )’ rev _ TST TST Arr TST Arr k12 ' na,12 / ( 1 + kna,12 / kd1 + ns,21 / kd2 ) 52 Table 2. Comparison of k12 with k:: 1 T=300K,‘huL=10cm-,P=-4,7L=10 s V12(cm-1) tL(psec) ammo“) tin/kg 0.00948 0.1 100/3 136 2.99 200/3 68 2.99 0.0248 1.5 100/3 2.72::10'I 2.82 200/3 1.41::107 2.74 0.174 2.5 100/3 3.55x109 1.19 200/3 1.82x109 1.17 1 T=300K,huL=100cm-.P=-4,k=10 -1 -1 Ip S V12(cm ) tL(psec) k12(sec ) k‘Z/k12 0.0938 0.1 mm 2.49x103 2.362 20/3 1.247x103 2.362 0.304 20 10/3 1.74x109 1.97 20/3 8.89x10° 1.92 0.68 23.75 10/3 1.98x10‘° 1.09 20/3 1.01x10‘° 1.06 53 Table 2 (cont’d) T: 300 Last = 10 on", p: -4,).: 100 s V12(cm-1) tL(psec) k12(sec-1) tin/k3 0.00948 0.01 1000/3 13.64 2.99 2000/3 6.83 2.99 0.0166 0.375 1000/3 7.4iir10‘| 2.97 2000/3 3.88x10‘ 2.88 0.039 0.788 1000/3 1.50::107 1.06 2000/3 7.67x10‘ 1.04 T=300K,th2100cm-,P=-4,k:100 s V12(cm-1) rL(pseo) hams“) trig/2:; 0.0753 0.01 100/3 1.85 2.31 200/3 0.93 2.31 0.126 1.0 100/3 4.27x10‘ 2.58 200/3 2.27x10‘ 2.69 0.313 7.5 100/3 1.22x10° 1.16 200/3 6.25x107 1.14 as 1! R12 see eq. (3.19) APPENDICES APPENDIX A Formal derivation of reaction rate expression We use a projection operator method to derive the formal reaction kernel and the reversible forward and backward reaction expression. First, Laplace transformation (LT) of eq. (3.16) gives us (A.l) 1'0) 4. '9! re) 1...: 88-8<0)=[y where g; is the LT of [ p", 922 ]T and g (0) is the initial distribution. In our model, at t = 0, the particle is distributed in potential well A, with an equilibrium distribution. For a harmonic potential this is a gaussian distribution. So 2 (0) = [gf 0 Jr, where g‘ = exp ( - B V“ ) / I exp ( - BV11 ) dx. Applying g, gto eq. (A1), we obtain aié-§e<0>=-[2499+9999+r§£§+299§1 (14.2) and 899-9p<0>=-[9929+999§+999§+9§9é1 (A.3) 54 55 Rearranging them, and using the following conditions, 9 ( o ) I d p“ ( 0 ) 1 8‘ ( ) p =g x =g = A.4a " " 922(0) ' O O _ __ gi g‘l _ 0 where 911(0)=81.Pzz(0)=822 Ee+s<9+§>199=-<9+9+9§>98 (1.5, then 899-22<0) ='(EE+§§)E§-E(E+§)Q§ A -‘I =-‘E£*§§’28+(§E+§§>[8+9(E+§)l <99+9§>ié (1..) Finally we have 56 -1 {8+[9<9+i>-§<9+§)[8+9<9+§>1 9(i_.+§)]}gg=gg(0). (74.7) Comparing this with eq. (3.7), we have -1 §=“m§{3‘b+§>'§‘9*§’[8+9‘itill s90 Q(L+K)}Eo (A.8) Note that P L. = 0, as a boundary condition, and L g = 0,since g is the equilibrium distribution. Our final formal result is '9: -1 i=1ms”{9§-iifs+9ii+ )1 9§}i “ .. -. .. -. kG111+kG11 '31 82 Finally, k12=(141:6;‘1’141.5.22)"'1 :k::/(1+k::/kd1+k::/kdz) (A.12a) k21=(1+kG:1+kG:2)-‘-2 :k::/(1+k::/kd1+k::/kd2). (A.12b) APPENDIX B Derivation of G 1 2 The potentials that we need are given in eq. (4.1). The crossing point can be calculated at V11 = V 2 2, so -_1_ _ a..2x0 Vo/kuxo (3.1) and VH-szzkwxo(x-o) (8.2) =7(x-0) where yzkwxo. V Assume the coupling term 'i-‘hl-Z- ( p22 - p ) is a source term 11 then 612 should satisfy the Fokker-Planck like equation a a2 1 a . [<9—t-D'a—;2-50(kwx-2kwxo)-a_;—Bbkw+1%(x-a)] 612:6(x-x0)5(_t-t0)' (8.3) 59 60 Use the space Fourier transformation, -Iex A [Gas dx.-G12 yields A a 2 a 1 . .1._§__ [57+Ds -3ka(-Ba—§-2xols)+lfi(las 0)]612 e""‘06(t-to). (3.4) Now we want to calculate G12 from [-2E+(BkQDs-%)-駗+(Dsz+i%Bkquos—i%0)]G, a s 2 :0 (8.5) by solving the partial differential equations. Weuse dt_ ds ._ dGiz ' T' 1' 2 .1 .1 (8'6) BkuDs- -D8-l'2-Bkax08+lfiO and l. Q. 5)) dt 12 G12=G12(t)eXP{‘D[° 7 1 +2———-c Bkah tfika Att=t =0. 12(0):e 02612 0) From 61 2 1 1 2BkmD (eBkun 1 ( I t 2 - 1 ) + ( B. kw'D fl ) t'] eBkth _1) 62 7 7 8____:(8-____)eBRth Bkafi o Bkah 1 7 B): Dt s:(s-——-—)e (0 +-———-—— 0 Bkah Bkqul 1 c: Bkat; ( s - ————— ) e- 1 B kw D it finally we have G "ex -ix [(s-——Z——)-Bkunt+ Y 1 12' p o Bka‘h e BkuDE 1 7 2 -2Bth —D[m(S-W)(l-e In) ) I 7 -Bk 01. +2 (s-—-—--)(l-e (i) ) (Bka)zfi Bkqui I 2 +(BkmDfi) ‘1 -i[%xo(s-F-E;I—D-g)(l-e-Bkwni) +%xo%t-%ot]}. (B-7) Inverting the Fourier transformation, we get 812:217518'128‘ .xds : 1 1 exp{-;-[2D1—:;:(1-e-i) {—252/DtL(l-exp(-2t)) 63 - 2 -%x(l-eflt)]]/[DIL(1-e-2t)] +i-—i£(x-xi-%xot+ot)} (3.8) 63 - 2 (l-e‘t)]] /[DIL(1-e-Zt)] -D( fi)tLt+i-—fi£(x-x‘-%xot+ot)} (B.8) APPENDIX C Eigen-functions of L12 The L12 operator with simplified notation is 32 1 a PHI-55’5“.” ¢.=".°. ”3'“ X with The boundary condition at x 9 :1: 00 is (in = 0. Then the eigenvalue problem is a -1 2 1 _ a x Z + (‘—Z y“ + §'- A“ ) ‘Wn - 0 ( C . 2 ) where _ -'y I4 4)“ - '9“- The solution is 2 -1/2 1/4 6- y I! wn=(z“nz) Hun/f2). (0.3) 64 65 So it -1/2 1 1/4 -2/2 ¢n=(2n!) (fl) 6" Hn(Y//—§) =rn (0.4) An3-n,n=0,1,2,.... 1 1/4 -( - /2 )Z The ground state, 11 = 0, is ( n) e x no . Adjoint operator For an operator L which satisfies 2 _ 8 a Lu-Po(x)a 2u+P1(x)3—-;‘-u+P2u, (0.5) x then ~ 62 d Lu==a (Pou)--a-;(P1u)+qu x 62 I a II I :Poax2u+(2P0-P1)3-_xu+(P0-P1+P2)u° (0.6) For our diffusion equation 66 2 a a a 12 6x2 81: 5t ~ -__9_2_ _a_._a_ 12.6 2 "6x at" (0.7) (0.8) the left-hand eigen-function of L 12 should be the right eigen-function of L12. Eigen-functions and eigen-values of :12 2/4 Let ¢n 3 By (P 1 N then L 0 = A' 41 is 12 n n 62 '1 I 1 _ a 2*‘(73’ +An+2) 'Pn-Or where _ n —1/2 _1_1/4 -y/4 (pm-(2 :) (2,) Hun/F2 _ n 1/2 1 1/4 _ (tn-(2 ') (77,-) Hn‘Y/fil-I. A! = n ' n = 0’ l. 2’ 0.. O The identities that we use to derive the basis expansion are: ) (0.9) (0.10) (0.11) (0.12) 67 l. 3 3 InLizrn-(L121n) r“ (C.l3a) II I! 2. 00 3 Idxlrzfi (C.l3c) In an -o 3. 3-4-- +14 () (0 13d) ayn- n n+1 y ° 4. a) 36 __ 11".5-;;rndy.--y’n+16m."H (C.l3e) -00 5. Q * - Jlnyrndy-y’ni-l6mnnttf—E m,n-l (0.13f) -® 6. a) __ 0 for n21 I¢n(y)dy-{(2n)1/4, “:0. (0.13g) (D APPmDIX D Vector recurrence relation Eq. (4.8) can be written in the form a _ .. fish gash-1 +971 Sn+9n Emil where __ + —1' _n-[a,8,bn,b] ro gin/T. 0 _ -%xof71 0 0 9,: 0 0 0 of; _0 0 -b/'Ti -n 0 0 0 -n 0 -a Q : 'n 0 0 -n -b0 1 v 0 :8 b0 -n .. y J i0 0 0 0 00 0 9+ - "‘ 00 0 b/nTi 0 —by’n+1 0 J 68 (0.1) (D.2a) (D.2b) (D.20) 69 Let 9“ ( t ) = c_ne- A t; we have A 9" +(9n+Al)§n+-Q:§n+1:o. (D.3) —n En-i We define then 9'2 +[(9+AI)+9*§+]2:0 (D.4a) -n —n-1 —n -n -n -n or [Q;§;+(Q+Al)]0n+9:§n”=0. (D.4b) From eq. (D.4b) [Q;§;+(gn+u)]§n=-9:§m (D.5a) [92§2+(9.+“)1§2..sn..=--an.n (D-Sb) wehave "+ + “+ '1 - §n:-[(9n+1+Al)+9n+1§n+1] 9n+1 (D.6&) ._ _ ._ -1 + §n:-[9n-1§n-1 +(9n-I+Al)] git-I. (D'Sb) We can calculate s: and 8; by expanding them into 70 -n+1 15+ + _1 s:-{(Q +Al)+9n.1(-l)[(9n.z+Al)+“°] 9n+2 } gnoi (D.7) and §;=o ._ -1 §.=-[9.+HJ 9. “- - -1 +1 '1 §2:-[91+Al+91”(-1)(QO+AI) 90] 91 (D.8) Our equation can be expressed in terms of S:, S; as [(9;§.+2:§:>+<9.+M>1§.=o 7..., i.e. [Qn+Al+§n(A)]§n=0 (0.10) where ) §n(A)=9;s’+9 3". 71 The eigen-value A can be determined from Det[gn+Al+§n(A)]:0. (0.11) For the zeroth block, n = 0, 0et[go+hl+f_(o()\)]:o. (D.12) is be» The structure of r0 0 k31 32 33 k WWOO ”3'00 41 42 43 “j. b 80 A Det[go+Al+§o(A)] A[A(A+k33)(A+k“)-ak32(boy+k‘3) +a(a+k‘2)(A+k33)-A(k34-boy)(bay‘tknfl Z :0. (0.13) This shows there is at least one A = 0 root. How are k12 and 1:21 determined? 72 First note that y(t) H —‘1 m ')(t)=i:4y(0) OIO-O' II 0 From probability conservation a3=<2n)"”. i.e. -A t_ -1/4 coRo,o+J§10Ro,j° J -(2n) Ast-HD, _ -1/4 CORO,O-(2n) , so c :(2u)""‘/R . (D.14) (D.15) (D.16) 73 -1/4 - We need 00R -(2n) Ri'o/Ro'otocalculateao(t). 1,0 From [ 90 +. A l + 20 ( 0 ) ] so : 0, for the first eigen-value A : 0, 0 0 0 0 R0.0 0 0 0 0 R1,0 =0'(D 17) km kaz k3: k34 - b 0y R2,0 a R _k“ 152+; k43+b0y k“) _ 3,0 Solving these algebraic equations yields a ksz -1 R1.0/120,0 : [4+k42 -_ ( boy+ k43)] 33 k31 [E—(boy+kw)-h“]. (v.18) 33 So k12+k21=A1 (D.19) 2k2‘-1~R /R (0 20) A1 - 1'0 0.0. 0 Our final results are R1 0 k21(§-—+1)A1/2 (D.21a) 0,0 LIST OF REFERENCES LIST OF REFERENECS 1.(a) E. M. Kosower and D. Huppert, Chem. Phys. Lett. 9_6_, 433 (1988). (b) D. Huppert, H. Kanetz and E. M. Kosower, Faraday Discuss. Chem. Soc. 74, 161 (1982). 2.(a) R. A. Marcus, J. Chem. Phys. E, 966, 979 (1956). (b) R. A. Marcus, Annu. Rev. Phys. Chem. _l_5_, 155 (1964). (c) R. A. Marcus, Faraday Disc. Chem. Soc.(London) 14, 7 (1982). 3. V. G. Levich, Adv. Electrochem. Eng. 4, 249 (1965). 4. L. D. Zusman, Chem. Phys. 19, 295 (1980). 5. H. A. Kramer, Physica(Utrecht) 1, 284 (1940). 6. I. Rips and J. Jortner, J. Chem. Phys. fl, 2090 (1987). 7. H. Frauenfelder and P. G. Wolynes, Science 299, 337 (1985). 8. R. A. Marcus, Faraday Disc. Chem. Soc. (London) 14, 7 (1982). 9. R. I. Cukier, J. Chem. Phys. .8_8, 5594 (1988). 10. M. Morillo and R. I. Cukier, J. Chem. Phys. _8_9, 6736 (1988). 11. S. H. Northrup and J. H. Hynes, J. Chem. Phys. 62, 5246 (1978). 12. S. H. Northrup and J. H. Hynes, Chem. Phys. Letter. 54, 244 (1978). 13. H. Risken, The Fokker-Planck Equation (Springer-Verlag, Berlin, 1984). 14. N. S. Snider, J. Chem. Phys. _4_2_, 548 (1965). 15. E. Wigner, Phys. Rev. 42, 749 (1932). 16. J. M. Deutch and I. Oppenheim, J. Chem. Phys. 5_4, 3547 (1971). 17. (a) M. Maroncelli, J. MacInnis and G. R. Fleming, Science 243, 1674 74 75 (1989). (b) R. F. Loring and S. Mukamel, J. Chem. Phys. _81, 1272 (1987). 18. J. D. Morgan and P Wolynes, J. Phys. Chem. 9_1, 874 (1987). 19. T. Kakitani and N. Mataga, Chem. Phys. 93, 381 (1985). 20. D.Y. Yang and R.I. Cukier, J. Chem. Phys. 91, 0000 (1989). HICHI RIES (1)1111111111111