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Theoretical aspects of electron transfer reactions of complex molecules
Electrochimica Acta 46 (2001) 3325– 3333 www.elsevier.com/locate/electacta
Theoretical aspects of electron transfer reactions of complex molecules A.M. Kuznetsov a,*, J. Ulstrup b a
A.N. Frumkin Institute of Electrochemistry, Russian Academy of Sciences, Leninskii prospect 31, Bldg. 5, 117071 Moscow, Russia b Institute of Chemistry, Bldg. 207, Technical Uni6ersity of Denmark, DK-2800 Lyngby, Denmark Received 9 November 2000; received in revised form 22 February 2001
Abstract Features of electron transfer involving complex molecules are discussed. This notion presently refers to molecular reactants where charge transfer is accompanied by large molecular reorganization, and commonly used displaced harmonic oscillator models do not apply. It is shown that comprehensive theory of charge transfer in polar media offers convenient tools for the treatment of experimental data for such systems, with due account of large-amplitude strongly anharmonic intramolecular reorganization. Equations for the activation barrier and free energy relationships are provided, incorporating vibrational frequency changes, local mode anharmonicity, and rotational reorganization, in both diabatic and adiabatic limits. Systems for which this formalism is appropriate are discussed. © 2001 Elsevier Science Ltd. All rights reserved. Keywords: Electron transfer; Complex molecules; Harmonic oscillator models
1. Introduction Theories of electron transfer (ET) reactions have reached high levels of sophistication, resting on different overarching concepts and formalisms. These are based on structureless and non-local dielectric continuum theory, stochastic chemical rate theory, or statistical mechanical molecular assemblies. ET theory frames broad classes of chemical, electrochemical, and biological processes [1–3], including new single-molecule and mesoscopic systems such as functional monolayers [4– 6] or scanning tunneling microscopy of molecular assemblies [7,8]. Methodologically ET theories, finally, cover important features beyond the simplest harmonic nuclear modes and linear reaction medium. These are, for example, low-temperature processes, anharmonic * Corresponding author. Fax: + 7-095-9520846. E-mail address: [email protected] (A.M. Kuznetsov).
nuclear modes, dissociative ET, proton and atom group transfer, and non-traditional systems such as superconducting electrodes and single-electron tunneling [2,3,8]. Simple analytical rate constants, and free energy relations such as the quadratic free energy relationship of chemical ET, were derived early [9 – 12], following notions in solid state physics [13]. The simplicity of the early formalism has made it the most common basis for experimental data analysis also at present. Use of these relations is often adequate for simple inorganic ET reactions [1,14] but a better account of molecular properties are required for increasing the number of processes involving complex molecules. These are, for example, organic and metallo-organic processes [15,16], hydrolytic and redox metalloenzyme processes [17– 19], processes involving ionic atmosphere and ion pair reorganization [20– 22], and new supramolecular and molecular scale interfacial processes [23– 25]. At the same time, more sophisticated ET formalism which
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accords better with such systems, can be given operational forms which, although requiring numerical analysis, are straightforward to use. The aim of this report is to discuss central features of ET reactions of complex molecules, with major bond reorganization that does not accord with the validity conditions of quadratic free energy relations. These properties are represented by more comprehensive but still transparent frames of charge transfer theory, and we provide a simple procedure for addressing these features in experimental data analysis.
2. Features of ET processes of complex molecules The quadratic free energy relationship originates from the theory of electronic relaxation in solids [13] and was introduced in chemistry by Marcus [9]. The explicit form is GA =
(Er +DG°)2 4Er
(1)
The activation Gibbs free energy, GA, is determined by the reaction free energy, DG°, and the molecular and tot solvent reorganization free energy E tot r . 4E r /4 is the ‘intrinsic’ activation free energy and −DG° is the driving force. Merits of Eq. (1) are obvious simplicity and framing rationale in a wide variety of contexts. Use of Eq. (1), however, prompts observations, regarding its validity conditions and mechanistically diagnostic value. The validity conditions are, particularly 1. All reactive nuclear modes [2,3] are purely classical. The modes are local nuclear modes, solvent inertial polarization modes, and some other kinds of collective modes. The electronic transition, i.e. the ET process, is via the minimum energy at the crossing of the initial and final state potential surfaces, Ui and Uf, respectively, with nuclear tunneling disregarded. 2. The potential surfaces are essentially diabatic surfaces with insignificant splitting at the crossing, and of parabolic shape. The latter reflects harmonic molecular motion with equilibrium nuclear coordinate displacement, and a linear environmental medium response. 3. The vibrational frequencies and the normal modes are the same in the initial and final states. The first condition confines the use of Eq. (1) to diabatic and weakly adiabatic processes. High-frequency internal modes such as CH, NH, OH, CC motion etc. are not represented by Eq. (1). The second and third conditions restrict the use of Eq. (1) to small changes in the molecular structures of the reacting molecules. Eq. (1) has frequently been taken as the sole experimental data frame also when the validity conditions are
far from adequate. Apparent accordance, for example in free energy relations, can here be misleading because Eq. (1) represents a relatively weak correlation over broad free energy ranges, and also in general, and does not disclose the physical nature of the linear response. Features of ET of complex molecules which do not accord with Eq. (1), are 1. ‘Large’ intramolecular reorganization, i.e. coordinate displacements in excess of, say :0.2 A, . 2. As a consequence of (1), significant vibrational frequency changes (potential surface distortion) and/or large anharmonicity. Notable cases for the latter are dissociative ET of alkylhalides [26 – 28], or ET function of new supramolecular systems such as the catenanes [29]. 3. Intramolecular rotational reorganization. The rotaxanes [23,25] are such examples. 4. Ion pair reorganization in weakly polar solvents [20,21]. 5. Multi-ET and long-range ET reactions [2,3]. All these features are inherent in comprehensive ET theory. We address specifically dissociative ET and rotational intramolecular reorganization. In Section 3 we discuss the effects of large-amplitude intramolecular reorganization on free energy relations based on multidimensional free energy surfaces, and provide a general procedure for calculation of the activation free energy. The procedure recasts the free energy relations in parametric form with different running parameters. Diabatic dissociative ET and rotational intramolecular reorganization in molecular ET are addressed in Section 4. In Section 5 we incorporate these effects in new expressions for the activation Gibbs free energy of adiabatic ET processes. In this section large-amplitude strongly anharmonic motion in both the diabatic and adiabatic limits is also illustrated numerically. Some concluding remarks are provided in Section 6.
3. Diabatic transition paths on multi-dimensional free energy surfaces We address first system motion on multi-dimensional potential surfaces spanned both by collective inertial polarization and local modes. The former are classical and actually represented by a single reaction coordinate the nature of which is treated elsewhere [30,31]. The local modes are deconvoluted from the solvent and represented according to their specific harmonic, anharmonic etc. character. Coupling of discrete modes to a continuum requires separate analysis [31 –33]. We address specifically: (1) the contribution of the molecular modes to the kinetic parameters; (2) their reflection in free energy relationships; and (3) procedures for calculation of the diabatic and adiabatic activation free energies.
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We shall base the discussion on a single molecular reactive mode, Q, in addition to the solvent continuum represented by the normal mode set {qk }. These can be recast as a single mode as discussed elsewhere [30,31]. The diabatic free energy surfaces Ui and Uf, take the form 1 Ui(qk,Q)= %' k (qk −qk0i)2 +ui(Q) 2 k 1 Uf(qk,Q)= %' k (qk −qk0f)2 +uf(Q)+DG° 2 k
the (free) energy scale being counted from the minimum of the initial state potential well. The local mode Q is regarded as purely classical. Nuclear tunneling is most conveniently represented by the appropriate nuclear Franck –Condon overlap integrals [2,3], leaving a set of reduced potential surfaces spanned by the low-frequency modes, with DG° modified by the vibrational zero point energy difference of the high-frequency modes. The electronic transition is effected by classical fluctuations of the low-frequency coordinates to the transition configuration. The fluctuations also bring the high-frequency vibrational levels in the initial and final states to resonance, followed by nuclear tunneling along Q between the resonating highfrequency levels. Classical reactive modes contribute to the activation free energy as Q GA = G sol A +G A
(3)
Q where G sol A represents the solvent polarization and G A the molecular degree(s) of freedom. Eq. (3) implies that the reaction Gibbs free energy can be decomposed into separate contributions from the solvent, Dgsol, and the local mode, DgQ, i.e. as
DG° =Dgsol +DgQ
(4)
Determination of Dgsol and DgQ is given below (Eq. (11)) and in Appendix A. The solvent contribution to GA is calculated by the partial symmetry factor hsol characterizing the symmetry of the transition state for the solvent polarization hsol =hsol(Dgsol)=
#U isol/#qk #U isol/#qk −#U fsol/#qk
(5)
U isol and U fsol are the first terms on the right-hand-side of Eq. (2), and the partial derivatives taken at the saddle point of the two potential surfaces spanned solely by the solvent coordinates. This point is uniquely determined once a value of DgQ is chosen. Then 2 sol G sol A =h solE r
(6)
sol r
where E is the solvent reorganization Gibbs free energy. hsol depends linearly on Dgs hsol =
1 Dgs 1+ sol 2 Er
n
The nature of Dgsol and E sol is most clearly appreciated r when the solvent is represented by a single stochastic coordinate [30,31], cf. Appendix A, but corresponding values along individual solvent coordinates can be identified in a multi-dimensional solvent representation [34]. A similar quantity can be introduced for the molecular degree of freedom hQ = hQ (DgQ ) =
(2)
(7)
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#ui(Q)/#Q #ui(Q)/#Q− #uf(Q)/#Q
(8)
taken at the crossing point between the two surfaces along Q. hQ (DgQ ) depends on the corresponding driving force DgQ. Q and DgQ, and the slopes of the potential profiles ui(Q) and uf(Q) are uniquely determined by Eqs. (2) and (4) once DgQ is chosen. G Q A can be calculated as GQ A = ui(Q*)
(9)
where the transition configuration Q* is determined by the condition of potential profile crossing ui(Q*)= uf(Q*) +DgQ
(10)
Dgsol and DgQ are uniquely determined by two conditions. One is Eq. (5). The obser6able symmetry factor h and partial symmetry factors at the lowest value of the activation free energy must, secondly, be identical (Appendix A), i.e. h=
n
1 Dgsol 1+ sol = hQ (DgQ ) 2 Er
(11)
These are the values where the activation free energy is minimum which is at the saddle point of the total potential surfaces spanned both by the solvent and local mode coordinates. Eqs. (5) and (11) thus determine the partial driving forces and their contributions to GA. We then proceed to the way the molecular degree(s) of freedom are reflected in the free energy relationships. As noted, this relationship is quadratic for simple charge transfer subject to equilibrium coordinate displacement only, corresponding to a linear dependence of the observable symmetry coefficient on DG° h=
1 DG° 1+ tot 2 Er
n
(12)
is the total reorganization free energy incorwhere E tot r porating both the solvent and local modes. Local modes with features other than equilibrium coordinate displacement can modify this drastically. This applies for example to Morse or Rosen –Morse potentials [35,36], representative of strong chemical bond deformation along stretching and bending modes, respectively. As an immediate illustration of the need for modification of Eq. (12), harmonic motion subject to both equilibrium coordinate displacement and vibrational frequency changes is represented by the following set of equations [37]:
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Q GA = hDG°+ h(1−h)E sol r +wE ri
Q DG°=(2h−1)E sol r −wE ri
w=
i 2 ; f
h(1−h) wh+1−h
(13)
(1−h)2 −wh 2 [wh+1−h]2
(14)
1 2 2 EQ ri = m i (Q0i −Q0f) 2
(15)
Eq. (13) gives the activation free energy in terms of h while Eq. (14) is the equation for h in terms of the Q parameters E sol r , E r , and DG°. These equations show 1. They constitute a parametric relation between the activation free energy and the reaction Gibbs free energy. h is the running parameter. 2. They reduce to Eq. (1) when w= 1, corresponding to pure equilibrium coordinate displacement of the local mode. 3. The following form of h emerges when w 1 h=
1 DG°+wE Q r 1+ 2 E sol r
n
force. Eq. (18) is the correlation from which this value of Q is determined. The running parameter in Eqs. (17) and (18) is the value of Q= Q*, at the transition configuration between the molecular potentials ui(Q) and uf(Q). This parametric representation is entirely general. Practical implementation for the Morse potential, and illustrations of the difference from the harmonic model is given in the next section.
4.2. Rotational reorganization Rotational reorganization of larger molecular fragments, induced by molecular ET, has acquired new importance in ET-induced function of rotaxane [25,29], and other nanoscale supramolecular systems. The potential describing the rotational motion in the initial and final electronic states along the rotational angle q are suitably represented by
(16)
ui(Q) = V0i sin2 q
(19)
h therefore differs from 0.5 (h\ 0.5) for symmetric reactions (DG° =0) as in Eqs. (1) and (12). This can profoundly affect analysis of ligand substitution reactions, chemical and electrochemical dissociative ET (cf. below), and optical bandshape analysis of charge transfer transitions [38]. Other limiting analytical forms emerge when w is large or differs only slightly from unity.
uf(Q) = V0f cos2 q
(20)
with the equilibrium values q0i = 0 and q0f = p/2. Motion of this kind is representative of large-amplitude internal molecular motion, and obviously far from harmonic. This also becomes apparent by the calculation of the activation Gibbs free energy. We introduce first the reorganization energies along the rotational coordinate E qri = ui(q0f) −ui(q0i) =V0i
4. Dissociative ET and rotational reorganization
(21)
E qrf = uf(q0i) −uf(q0f) =V0f
4.1. Dissociati6e ET Theoretical ET frames were extended early to dissociative and associative mechanisms in ligand substitution processes of transition metal complexes [39] and organic nucleophilic substitution reactions [40]. Analogies to ET processes where chemical or electrochemical reduction is accompanied by chemical bond fission were also noted early [41]. Dissociative ET has attracted recent attention [26 –28,42] particularly in organic electrochemistry [16,43]. The molecular potentials here, however, do not accord with the harmonic approximation but arbitrary molecular potentials can be incorporated by the parametric procedure. The parametric relations analogous to Eqs. (13) and (14) are [37] GA = ui(Q)+E sol r
dui/dQ dui/dQ− duf/dQ
DG°=ui(Q)−uf(Q)+E sol r
n
2
dui/dQ+ duf/dQ dui/dQ− duf/dQ
(17)
n
(18)
Eq. (17) is analogous to Eq. (13) and provides the activation Gibbs free energy in terms of the transition configuration of the local coordinate Q at given driving
If the driving force is not large ( DG° B min{V0i, V0f}), the method above may be applied, giving the following equations equivalent to Eqs. (13) and (14), and (17) and (18) [44] GA = Er + h=
V0i V0i + V0f
2
+
V0i V0i + V0f
V0i DG° V0i + V0f
V0i V0i + V0f
V0f + Er
V0f − V0i V0i + V0f
(22) (23)
The symmetry factor is thus constant. Driving force variation results in a relative vertical shift of ui(Q) and uf(Q) while the position of the solvent polarization profiles remains unchanged. The activation Gibbs free energy for symmetric reactions (DG° = 0, V0i = V0f = V0) is 1 1 q GA = E sol r + Er 4 2 This is clearly different from Eq. (1)
(24)
A.M. Kuznetso6, J. Ulstrup / Electrochimica Acta 46 (2001) 3325–3333
1 1 q GA = E sol r + Er 4 4
(25)
Both the absolute value of the molecular reorganization energy and its appearance in the activation free energy thus differ from the harmonic approximation. Outside the limits DG° Bmin{V0i, V0f} the variation of DG° is reflected only in the potential profile shift for the solvent polarization. The activation barrier is GA = V0i +
(Er +DG°−V0i)2 4Er
for DG°\ V0i −Er GA =
V0f −V0i V0f +V0i
(Er +DG°+V0f)2 4Er
(26)
for DG°B Er
V0f −V0i −V0f V0f +V0i (27)
The molecular contribution to GA here either vanishes, Eq. (27), or is constant, V0i (Eq. (26)).
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Eq. (29) determines Q* at given h. Substitution of Q*=Q*(h) into Eq. (30) gives an equation for h. The activation barrier is [37]
n
h 1 GA = h 2E sol r + ui(Q*) − DE 1−h 2
1/2
(31)
Eqs. (29) – (31) can be viewed as a parametric dependence of the activation barrier on the reaction free energy, with Q* as running variable. h can be calculated by Eq. (29) at any value of Q*. Insertion of h(Q*) into Eqs. (30) and (31) then gives GA. Eq. (29) is equivalent to Eq. (13) or (17), Eq. (30) to Eq. (14) or (18). Eqs. (29) –(31) also apply to arbitrary potentials ui(Q) and uf(Q). We now illustrate this general result by specific forms for ui(Q) and uf(Q). The most widely used anharmonic potential is the Morse potential. Intramolecular reorganization along Q in a Morse potential resulting in a displacement of the equilibrium chemical bond length of the Q-mode, Q0i Q0f gives ui(Q) = Di[1− e − k(Q − Q0i)]2;
5. Adiabatic transitions
uf(Q) = Df[1− e − k(Q − Q0f)]2
The activation Gibbs free energy relations above apply when the resonance splitting of the electronic levels in the crossing region of the diabatic potential surfaces, DE, is small [2,3]. Presently this implies that the diabatic potential surfaces are not significantly distorted, i.e. DE5kBT. If DE is large, the process belongs to the adiabatic limit of strong donor –acceptor interaction, and the character of the process different in two respects. Charge transfer is now effected by system motion along the lower adiabatic potential surface (electronic ground state) instead of crossing between the diabatic electronic states representing the reactants and products. The lower adiabatic surface is, secondly, significantly distorted (lowered by the resonance splitting) in the crossing region of the diabatic potential surfaces. In the two-state approximation the adiabatic potential surface is constructed from the diabatic free energy surfaces as 1 U = [Ui +Uf − (Ui −Uf)2 +(DE)2] 2
(28)
The activation barrier is determined by the saddle point on this surface. If we neglect the dependence of DE on the reactive mode coordinates, the equations for the value of Q at the saddle point, Q*, are [37] (1 −h)
dui duf dui/dQ +h =0 or h= dui/dQ−duf/dQ dQ dQ
DG°=(2h−1)E sol r +ui(Q)−uf(Q)−
(2h−1) 2 h(1−h)
(29) DE (30)
(32)
Introducing the new variables X= e − k(Q − Q0i);
X0 = e − k(Q0f − Q0i); and d=
Df DiX 20 (33)
we transform Eq. (29) (1 − h)(1 − X) +hd(X0 − X) =0
(34)
Calculating X and substituting into Eqs. (30) and (31) we obtain
GA = h 2 E sol r + −
d 2E Q DE r − [(1 −h) + hd]2 2 h(1− h)
DE
h(1− h) 2
DG° = (2h− 1) E sol r + +
n
(35) d 2E Q DE r − 2 [(1 −h) + hd] 2 h(1− h)
(1−h)2(d− 1)2dE Q r [(1 −h) + hd]2
n
(36)
where the reorganization energy along Q in the initial state has been introduced 2 EQ r = ui(Q0f) =Di(1− X0)
(37)
Eqs. (35) and (36) give the dependence of GA on DG° with h as running variable. The diabatic limit is reached if terms involving DE are omitted. If the repulsive branches of the initial and final potentials are also identical (i.e. if d= 1), the result of Saveant [26] is recovered. In the limit of large intramolecular reorganization and for identical intramolecular potentials (Di = Df),
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displaced along Q, X0 is small, and Eqs. (35) and (36) reduce to
2 GA = E Q E sol r +h r −
DE 2 h(1−h)
n
−
n
DE
h(1−h) 2 (38)
DE +E Q (39) r 2 h(1−h) As h(1 −h) varies slowly in broad range of h, we can also write Eqs. (38) and (39) in the form
DG°=(2h−1) E sol r −
eff Q 2 eff GA = E Q r +[E r (h) +DG°− E r ] /4E r (h)
−
DE
h(1 −h) 2
(40)
Q DG°=(2h−1)E eff r (h) +E r ;
sol E eff r (h)= E r −
DE
n
(41) 2 h(1−h) Eqs. (40) and (41) show that large intramolecular reorganization gives a constant activation barrier contribution. The latter is decreased by the quantity DE h(1− h)/2, while the solvent part of the reorganization energy is decreased by DE/2 h(1−h). It is worth noting that the model of two identical (but shifted) Morse-like potentials is clearly different from the model of Morse potentials used in Ref. [26] where only the final state repulsive branch was used. Figs. 1 and 2 illustrate numerically the parametric procedure for displaced Morse potentials (Eqs. (35) – (37)) in both the diabatic and adiabatic limits. The
dissociation energy ratio, Di/DfX 20 was in the range 0.5–2, the adiabatic resonance splitting (4 –8)kBT or 0.1– 0.2 eV, and both reorganization (free) energies E sol r and E Q r 20kBT or 0.5 eV. The auxiliary parameter p is zero or unity, and represents the diabatic and adiabatic limit, respectively. The results of the displaced harmonic oscillator model are shown for comparison. Figs. 1 and 2 show 1. All correlations are smooth and appear to resemble one another. However, the absolute activation free energy values depend sensitively on the local mode potential. 2. The activation free energy increases with increasing d. This parameter characterizes the repulsive branches of the Morse potentials in the initial and final states. Small d represents a shallower potential in the final than in the initial state, and vice versa. Smaller values of d therefore give a smaller activation Gibbs free energy (Fig. 1). 3. Harmonic and anharmonic potentials are compared in Fig. 2. The harmonic local mode corresponds to d= 1. The activation free energy for a shallow final state anharmonic mode (d=0.5) is lower than for the harmonic mode (d=1) but higher for a steep repulsive final state branch (d=2). In addition the correlation is notably asymmetric around zero driving force for the anharmonic mode, with larger curvature for negative than for positive DG°. 4. Resonance splitting (DE= 0.1 and 0.2 eV) lowers the activation free energy by approximately DE/2 over most of the driving force range.
Fig. 1. Activation Gibbs free energy dependence on reaction free energy for charge transfer with coupling to a local Morse potential and a harmonic solvent continuum, calculated by Eqs. (32), (35) – (40). Different values of the parameter d = Df/DiX 20 (Eq. (33)). Q Energies in units of kBT, T= 298 K. E sol r = E r = 20. DE= 4. The following parametric variations: 1, d = 2, diabatic limit; 2, d = 2, adiabatic limit; 3, d =1, diabatic limit; 4, d= 1, adiabatic limit; 5, d =0.5, diabatic limit; 6, d =0.5, adiabatic limit. Large d corresponds to strong repulsion in the repulsive branch of the Morse potential, small d to weak repulsion. Large d therefore gives the higher activation free energy.
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Fig. 2. Activation Gibbs free energy dependence on reaction free energy for charge transfer with coupling to a local mode potential and a solvent continuum, calculated as in Fig. 1. Comparison between a strongly anharmonic shallow local Morse potential (d = 0.5) Q and a virtually harmonic potential (d= 1). Energies in units of kBT. E sol r =E r =20. DE=8. 1, harmonic potential (d = 1), diabatic limit; 2, harmonic potential, adiabatic limit; 3, anharmonic Morse potential (d =0.5), diabatic limit; 4, anharmonic Morse potential (d =0.5), adiabatic limit.
6. Concluding remarks Large-amplitude nuclear dynamics and coordinate displacement along strongly anharmonic local nuclear modes is important in a range of known, and non-traditional molecular and supramolecular systems. Rotational reorientation in rotaxanes [25,29], and chemical and electrochemical dissociative ET [26 – 28,41– 43] were noted. Dissociative ET is mostly associated with alkylhalide reduction, but is also encountered in biological macromolecular function. An example is the reduction of the Type II/III trinuclear copper cluster in the blue oxidase ascorbate oxidase [45] by long-range intramolecular ET from the Type I centre [46]. Other cases are chemical and biological proton and atom group transfers [2,3] where bond breaking rests on stretching and bending which by far exceed the amplitudes of zero-point energy vibrations. These cases are natural targets for molecular charge transfer theory. In quantitative approaches none of them, however, accord with the limiting rate constants based on displaced harmonic oscillator models, but the present report shows that contemporary charge transfer theory including new achievements [2,3,8] offers appropriate other means. The contribution of large-amplitude intramolecular reorganization along arbitrary local mode potentials to the observable activation free energy was addressed. A general procedure was first provided where the dependence of the activation free energy on the reaction free energy is recast in parametric form. The running parameter is either the symmetry factor, or the local mode coordinate at the transition configura-
tion. Corresponding values of the activation and reaction Gibbs free energies can next be calculated for all values of the running parameters. The formalism extends, moreover, both to the diabatic and adiabatic limits of molecular charge transfer. Incorporation of large-amplitude local mode reorganization in general has to resort to a numerical procedure, but the parametric form is straightforward, and reduces to analytical form in representative cases. In addition to the harmonic form this applies, particularly, to rotational reorganization (Eqs. (22) – (27)) and specific cases of harmonic vibrational frequency changes and anharmonic potentials. The numerical analysis (Figs. 1 and 2) shows that free energy relationships retain qualitatively similar form for rather different local mode potentials. This is one reason for the apparent successful use of the simplest harmonic model also in cases when the validity conditions of this model are significantly violated. The apparent accordance is, however, misleading. The character of the free energy relationships is thus dominated by local mode reorganization when these modes dominate over the harmonic environmental modes. This modifies the curvature of the correlations, symmetry coefficients etc. The parameters and values of the activation free energy are also sensitively correlated in spite of the smooth character of all the correlations. The sensitivity of the activation Gibbs free energy to the local mode potentials subject to large-amplitude reorganization, and the attractive simplicity of the parametric procedure, therefore warrant broader use of this formalism in new experimental data analysis.
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Acknowledgements Financial support from the Danish Technical Science Council (grant No. 9801629, Russian Foundation for Basic Research (grant No. 00-03-32239) and the EU programme INTAS (grant No. 99-1093) is acknowledged.
Appendix A. Decomposition of the reaction free energy into solvent and intramolecular terms [3] Decomposition of DG° according to Eq. (5) is at first arbitrary and corresponds to all different trajectories across the dividing surface between the potential surfaces in the reactants’ and products’ states. The solvent part of the activation Gibbs free energy is 2 sol G sol A =h solE r =
hsol =
2 2 (E sol (E sol r +Dgsol) r +DG°−DgQ ) = ; sol sol 4E r 4E r
n
1 Dgsol 1 DG°−DgQ 1+ sol = 1+ 2 2 Er E sol r
n
(A1)
and corresponds to crossing at an arbitrary point of the dividing surface or equivalently, crossing at the saddle point of the potential surfaces spanned solely by the solvent coordinates at arbitrarily chosen Dgsol. The total activation Gibbs free energy is GA =
(E sol r +DG°−DgQ ) +ui[Q*(DgQ )] 4E sol r
(A2)
The value of Dgsol or DgQ for which GA is minimum is given by
n
#GA 1 DG°−DgQ #ui[Q*(DgQ )] =0 or 1+ =
hQ #DgQ 2 #DgQ E sol r (A3) The obser6able symmetry factor is, from Eq. (A2) h=
dGA 1 DG° = 1+ sol Er dDG° 2
n
(A4)
Eqs. (A1), (A3) and (A4) thus show that the three symmetry factors are identical. Together with Eqs. (5), (9), (10) and (A2) this provides the general procedure for the free energy relations and for separate determination of Dgsol and DgQ.
References [1] R.A. Marcus, N. Sutin, Biochim. Biophys. Acta 811 (1985) 265. [2] A.M. Kuznetsov, J. Ulstrup, Electron Transfer in Chemistry and Biology, Wiley, Chichester, 1999. [3] A.M. Kuznetsov, Charge Transfer in Physics, Chemistry and Biology, Gordon & Breach, Reading, UK, 1995.
[4] H.A. Finklea, in: A.J. Bard (Ed.), Electroanalytical Chemistry, vol. 19, Marcel Dekker, New York, 1996, pp. 109 – 335. [5] J.F. Smalley, S.W. Feldberg, C.E.D. Chidsey, M.R. Lindford, M.D. Newton, Y.-P. Liu, J. Phys. Chem. 99 (1995) 13141. [6] K. Weber, L. Hockett, S.E. Creager, J. Phys. Chem. B 101 (1997) 8286. [7] A.M. Kuznetsov, J. Ulstrup, J. Phys. Chem. A 104 (2000) 11531. [8] A.M. Kuznetsov, J. Ulstrup, Electrochim. Acta 45 (2000) 2339. [9] R.A. Marcus, J. Phys. Chem. 24 (1956) 966 (see also p. 979). [10] (a) N.S. Hush, J. Chem. Phys. 28 (1958) 962; (b) N.S. Hush, Trans. Faraday Soc. 57 (1961) 557. [11] R.R. Dogonadze, in: N.S. Hush (Ed.), Reactions of Molecules at Electrodes, Wiley – Interscience, New York, 1971, p. 135. [12] V.G. Levich, in: P. Delahay (Ed.), Advances in Electrochemistry and Electrochemical Engineering, vol. 4, Interscience, New York, 1966. [13] R. Kubo, Y. Toyozawa, Prog. Theor. Phys. (Kyoto) 13 (1955) 160. [14] For a review see: E.D. German, Rev. Inorg. Chem. 5 (1983) 123. [15] G.W. Parshall, S.D. Itel, Homogeneous Catalysis, 2nd ed., Wiley, New York, 1992. [16] L. Eberson, Electron Transfer in Organic Chemistry, Springer, Berlin, 1987. [17] R.H. Holm, E.I. Solomon (Eds.), Bioinorganic Enzymology (special issue), Chem. Rev. 96 (1996) 2237. [18] A. Kohen, J.P. Klinman, Acc. Chem. Res. 31 (1998) 397. [19] A.M. Kuznetsov, J. Ulstrup, Can. J. Chem. 77 (1999) 1985. [20] R.A. Marcus, J. Phys. Chem. B 102 (1998) 10071. [21] R.R. Dogonadze, A.M. Kuznetsov, J. Electroanal. Chem. 65 (1975) 545. [22] T.J. Jensen, H.B. Gray, J.R. Winkler, A.M. Kuznetsov, J. Ulstrup, J. Phys. Chem. B 104 (2000) 11556. [23] J. Jortner, M. Ratner (Eds.), Molecular Electronics, Blackwell, Oxford, 1997. [24] G. Mahler, V. May, M. Schreiber (Eds.), Molecular Electronics, Marcel Dekker, New York, 1996. [25] V. Balzani, A. Credi, F.M. Raymo, J.F. Stoddart, Angew. Chem. Int. Ed. Engl. 39 (2000) 3349. [26] J.M. Saveant, J. Am. Chem. Soc. 109 (1987) 6788. [27] R.A. Marcus, J. Phys. Chem. 101 (1997) 4072. [28] E.D. German, A.M. Kuznetsov, J. Phys. Chem. 98 (1994) 6120. [29] S.J. Langford, F.M. Raymo, J.F. Stoddart, in: J. Jortner, M. Ratner (Eds.), Molecular Electronics, Blackwell, Oxford, 1997, pp. 325 – 342. [30] L.D. Zusman, Chem. Phys. 49 (1980) 295. [31] A.M. Kuznetsov, Stochastic and dynamic views of chemical reaction kinetics in solutions, Pressses polytechniques et universitaires romandes, Lausanne, 1999. [32] A.O. Caldeira, A.O. Leggett, Ann. Phys. 149 (1983) 374. [33] B.J. Gertner, K.R. Wilson, J.T. Hynes, J. Chem. Phys. 90 (1989) 3537.
A.M. Kuznetso6, J. Ulstrup / Electrochimica Acta 46 (2001) 3325–3333 [34] R.R. Dogonadze, A.M. Kuznetsov, M.A. Vorotyntsev, Phys. Stat. Sol. (b) 54 (1972) 125 (see also p. 425). [35] N. Bru¨ niche-Olsen, J. Ulstrup, J. Chem. Soc. Faraday Trans. II 74 (1978) 1690. [36] E.D. German, A.M. Kuznetsov, J. Chem. Soc. Faraday Trans. I 77 (1981) 397 (see also p. 2203). [37] A.M. Kuznetsov, J. Ulstrup, Phys. Chem. Chem. Phys. 1 (1999) 5587. [38] E.M. Itskovitch, J. Ulstrup, M.A. Vorotyntsev, in: R.R. Dogonadze, E. Ka´ lma´ n, A.A. Kornyshev, J. Ulstrup (Eds.), The Chemical Physics of Solvation. Part B. Spectrocopy of Solvation, Elsevier, Amsterdam, 1986, pp. 223 – 310. [39] E.D. German, R.R. Dogonadze, J. Res. Inst. Hokkaido Univ. 20 (1972) 34.
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[40] E.D. German, R.R. Dogonadze, Int. J. Chem. Kinet. 6 (1974) 457 (see also p. 467). [41] J. Ulstrup, in: H. Lund, (Ed.), Proceedings of the Sandbjerg Meeting on Organic Electrochemistry, Aarhus University, Aarhus, 1982, p. 1. [42] M.T.M. Koper, G.A. Voth, Chem. Phys. Lett. 282 (1998) 100. [43] H. Lund, K. Daasbjerg, T. Lund, S.U. Pedersen, Macromol. Symp. 134 (1998) 73. [44] E.D. German, A.M. Kuznetsov, Izv. Akad. Nauk SSSR, Ser. Khim. 1 (1989) 77. [45] A. Messerschmidt, H. Luecke, R. Huber, J. Mol. Biol. 230 (1993) 997. [46] O. Farver, I. Pecht, J. Biol. Chem. 269 (1994) 22933.
.
Theoretical aspects of electron transfer reactions of complex molecules A.M. Kuznetsov a,*, J. Ulstrup b a
A.N. Frumkin Institute of Electrochemistry, Russian Academy of Sciences, Leninskii prospect 31, Bldg. 5, 117071 Moscow, Russia b Institute of Chemistry, Bldg. 207, Technical Uni6ersity of Denmark, DK-2800 Lyngby, Denmark Received 9 November 2000; received in revised form 22 February 2001
Abstract Features of electron transfer involving complex molecules are discussed. This notion presently refers to molecular reactants where charge transfer is accompanied by large molecular reorganization, and commonly used displaced harmonic oscillator models do not apply. It is shown that comprehensive theory of charge transfer in polar media offers convenient tools for the treatment of experimental data for such systems, with due account of large-amplitude strongly anharmonic intramolecular reorganization. Equations for the activation barrier and free energy relationships are provided, incorporating vibrational frequency changes, local mode anharmonicity, and rotational reorganization, in both diabatic and adiabatic limits. Systems for which this formalism is appropriate are discussed. © 2001 Elsevier Science Ltd. All rights reserved. Keywords: Electron transfer; Complex molecules; Harmonic oscillator models
1. Introduction Theories of electron transfer (ET) reactions have reached high levels of sophistication, resting on different overarching concepts and formalisms. These are based on structureless and non-local dielectric continuum theory, stochastic chemical rate theory, or statistical mechanical molecular assemblies. ET theory frames broad classes of chemical, electrochemical, and biological processes [1–3], including new single-molecule and mesoscopic systems such as functional monolayers [4– 6] or scanning tunneling microscopy of molecular assemblies [7,8]. Methodologically ET theories, finally, cover important features beyond the simplest harmonic nuclear modes and linear reaction medium. These are, for example, low-temperature processes, anharmonic * Corresponding author. Fax: + 7-095-9520846. E-mail address: [email protected] (A.M. Kuznetsov).
nuclear modes, dissociative ET, proton and atom group transfer, and non-traditional systems such as superconducting electrodes and single-electron tunneling [2,3,8]. Simple analytical rate constants, and free energy relations such as the quadratic free energy relationship of chemical ET, were derived early [9 – 12], following notions in solid state physics [13]. The simplicity of the early formalism has made it the most common basis for experimental data analysis also at present. Use of these relations is often adequate for simple inorganic ET reactions [1,14] but a better account of molecular properties are required for increasing the number of processes involving complex molecules. These are, for example, organic and metallo-organic processes [15,16], hydrolytic and redox metalloenzyme processes [17– 19], processes involving ionic atmosphere and ion pair reorganization [20– 22], and new supramolecular and molecular scale interfacial processes [23– 25]. At the same time, more sophisticated ET formalism which
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accords better with such systems, can be given operational forms which, although requiring numerical analysis, are straightforward to use. The aim of this report is to discuss central features of ET reactions of complex molecules, with major bond reorganization that does not accord with the validity conditions of quadratic free energy relations. These properties are represented by more comprehensive but still transparent frames of charge transfer theory, and we provide a simple procedure for addressing these features in experimental data analysis.
2. Features of ET processes of complex molecules The quadratic free energy relationship originates from the theory of electronic relaxation in solids [13] and was introduced in chemistry by Marcus [9]. The explicit form is GA =
(Er +DG°)2 4Er
(1)
The activation Gibbs free energy, GA, is determined by the reaction free energy, DG°, and the molecular and tot solvent reorganization free energy E tot r . 4E r /4 is the ‘intrinsic’ activation free energy and −DG° is the driving force. Merits of Eq. (1) are obvious simplicity and framing rationale in a wide variety of contexts. Use of Eq. (1), however, prompts observations, regarding its validity conditions and mechanistically diagnostic value. The validity conditions are, particularly 1. All reactive nuclear modes [2,3] are purely classical. The modes are local nuclear modes, solvent inertial polarization modes, and some other kinds of collective modes. The electronic transition, i.e. the ET process, is via the minimum energy at the crossing of the initial and final state potential surfaces, Ui and Uf, respectively, with nuclear tunneling disregarded. 2. The potential surfaces are essentially diabatic surfaces with insignificant splitting at the crossing, and of parabolic shape. The latter reflects harmonic molecular motion with equilibrium nuclear coordinate displacement, and a linear environmental medium response. 3. The vibrational frequencies and the normal modes are the same in the initial and final states. The first condition confines the use of Eq. (1) to diabatic and weakly adiabatic processes. High-frequency internal modes such as CH, NH, OH, CC motion etc. are not represented by Eq. (1). The second and third conditions restrict the use of Eq. (1) to small changes in the molecular structures of the reacting molecules. Eq. (1) has frequently been taken as the sole experimental data frame also when the validity conditions are
far from adequate. Apparent accordance, for example in free energy relations, can here be misleading because Eq. (1) represents a relatively weak correlation over broad free energy ranges, and also in general, and does not disclose the physical nature of the linear response. Features of ET of complex molecules which do not accord with Eq. (1), are 1. ‘Large’ intramolecular reorganization, i.e. coordinate displacements in excess of, say :0.2 A, . 2. As a consequence of (1), significant vibrational frequency changes (potential surface distortion) and/or large anharmonicity. Notable cases for the latter are dissociative ET of alkylhalides [26 – 28], or ET function of new supramolecular systems such as the catenanes [29]. 3. Intramolecular rotational reorganization. The rotaxanes [23,25] are such examples. 4. Ion pair reorganization in weakly polar solvents [20,21]. 5. Multi-ET and long-range ET reactions [2,3]. All these features are inherent in comprehensive ET theory. We address specifically dissociative ET and rotational intramolecular reorganization. In Section 3 we discuss the effects of large-amplitude intramolecular reorganization on free energy relations based on multidimensional free energy surfaces, and provide a general procedure for calculation of the activation free energy. The procedure recasts the free energy relations in parametric form with different running parameters. Diabatic dissociative ET and rotational intramolecular reorganization in molecular ET are addressed in Section 4. In Section 5 we incorporate these effects in new expressions for the activation Gibbs free energy of adiabatic ET processes. In this section large-amplitude strongly anharmonic motion in both the diabatic and adiabatic limits is also illustrated numerically. Some concluding remarks are provided in Section 6.
3. Diabatic transition paths on multi-dimensional free energy surfaces We address first system motion on multi-dimensional potential surfaces spanned both by collective inertial polarization and local modes. The former are classical and actually represented by a single reaction coordinate the nature of which is treated elsewhere [30,31]. The local modes are deconvoluted from the solvent and represented according to their specific harmonic, anharmonic etc. character. Coupling of discrete modes to a continuum requires separate analysis [31 –33]. We address specifically: (1) the contribution of the molecular modes to the kinetic parameters; (2) their reflection in free energy relationships; and (3) procedures for calculation of the diabatic and adiabatic activation free energies.
A.M. Kuznetso6, J. Ulstrup / Electrochimica Acta 46 (2001) 3325–3333
We shall base the discussion on a single molecular reactive mode, Q, in addition to the solvent continuum represented by the normal mode set {qk }. These can be recast as a single mode as discussed elsewhere [30,31]. The diabatic free energy surfaces Ui and Uf, take the form 1 Ui(qk,Q)= %' k (qk −qk0i)2 +ui(Q) 2 k 1 Uf(qk,Q)= %' k (qk −qk0f)2 +uf(Q)+DG° 2 k
the (free) energy scale being counted from the minimum of the initial state potential well. The local mode Q is regarded as purely classical. Nuclear tunneling is most conveniently represented by the appropriate nuclear Franck –Condon overlap integrals [2,3], leaving a set of reduced potential surfaces spanned by the low-frequency modes, with DG° modified by the vibrational zero point energy difference of the high-frequency modes. The electronic transition is effected by classical fluctuations of the low-frequency coordinates to the transition configuration. The fluctuations also bring the high-frequency vibrational levels in the initial and final states to resonance, followed by nuclear tunneling along Q between the resonating highfrequency levels. Classical reactive modes contribute to the activation free energy as Q GA = G sol A +G A
(3)
Q where G sol A represents the solvent polarization and G A the molecular degree(s) of freedom. Eq. (3) implies that the reaction Gibbs free energy can be decomposed into separate contributions from the solvent, Dgsol, and the local mode, DgQ, i.e. as
DG° =Dgsol +DgQ
(4)
Determination of Dgsol and DgQ is given below (Eq. (11)) and in Appendix A. The solvent contribution to GA is calculated by the partial symmetry factor hsol characterizing the symmetry of the transition state for the solvent polarization hsol =hsol(Dgsol)=
#U isol/#qk #U isol/#qk −#U fsol/#qk
(5)
U isol and U fsol are the first terms on the right-hand-side of Eq. (2), and the partial derivatives taken at the saddle point of the two potential surfaces spanned solely by the solvent coordinates. This point is uniquely determined once a value of DgQ is chosen. Then 2 sol G sol A =h solE r
(6)
sol r
where E is the solvent reorganization Gibbs free energy. hsol depends linearly on Dgs hsol =
1 Dgs 1+ sol 2 Er
n
The nature of Dgsol and E sol is most clearly appreciated r when the solvent is represented by a single stochastic coordinate [30,31], cf. Appendix A, but corresponding values along individual solvent coordinates can be identified in a multi-dimensional solvent representation [34]. A similar quantity can be introduced for the molecular degree of freedom hQ = hQ (DgQ ) =
(2)
(7)
3327
#ui(Q)/#Q #ui(Q)/#Q− #uf(Q)/#Q
(8)
taken at the crossing point between the two surfaces along Q. hQ (DgQ ) depends on the corresponding driving force DgQ. Q and DgQ, and the slopes of the potential profiles ui(Q) and uf(Q) are uniquely determined by Eqs. (2) and (4) once DgQ is chosen. G Q A can be calculated as GQ A = ui(Q*)
(9)
where the transition configuration Q* is determined by the condition of potential profile crossing ui(Q*)= uf(Q*) +DgQ
(10)
Dgsol and DgQ are uniquely determined by two conditions. One is Eq. (5). The obser6able symmetry factor h and partial symmetry factors at the lowest value of the activation free energy must, secondly, be identical (Appendix A), i.e. h=
n
1 Dgsol 1+ sol = hQ (DgQ ) 2 Er
(11)
These are the values where the activation free energy is minimum which is at the saddle point of the total potential surfaces spanned both by the solvent and local mode coordinates. Eqs. (5) and (11) thus determine the partial driving forces and their contributions to GA. We then proceed to the way the molecular degree(s) of freedom are reflected in the free energy relationships. As noted, this relationship is quadratic for simple charge transfer subject to equilibrium coordinate displacement only, corresponding to a linear dependence of the observable symmetry coefficient on DG° h=
1 DG° 1+ tot 2 Er
n
(12)
is the total reorganization free energy incorwhere E tot r porating both the solvent and local modes. Local modes with features other than equilibrium coordinate displacement can modify this drastically. This applies for example to Morse or Rosen –Morse potentials [35,36], representative of strong chemical bond deformation along stretching and bending modes, respectively. As an immediate illustration of the need for modification of Eq. (12), harmonic motion subject to both equilibrium coordinate displacement and vibrational frequency changes is represented by the following set of equations [37]:
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Q GA = hDG°+ h(1−h)E sol r +wE ri
Q DG°=(2h−1)E sol r −wE ri
w=
i 2 ; f
h(1−h) wh+1−h
(13)
(1−h)2 −wh 2 [wh+1−h]2
(14)
1 2 2 EQ ri = m i (Q0i −Q0f) 2
(15)
Eq. (13) gives the activation free energy in terms of h while Eq. (14) is the equation for h in terms of the Q parameters E sol r , E r , and DG°. These equations show 1. They constitute a parametric relation between the activation free energy and the reaction Gibbs free energy. h is the running parameter. 2. They reduce to Eq. (1) when w= 1, corresponding to pure equilibrium coordinate displacement of the local mode. 3. The following form of h emerges when w 1 h=
1 DG°+wE Q r 1+ 2 E sol r
n
force. Eq. (18) is the correlation from which this value of Q is determined. The running parameter in Eqs. (17) and (18) is the value of Q= Q*, at the transition configuration between the molecular potentials ui(Q) and uf(Q). This parametric representation is entirely general. Practical implementation for the Morse potential, and illustrations of the difference from the harmonic model is given in the next section.
4.2. Rotational reorganization Rotational reorganization of larger molecular fragments, induced by molecular ET, has acquired new importance in ET-induced function of rotaxane [25,29], and other nanoscale supramolecular systems. The potential describing the rotational motion in the initial and final electronic states along the rotational angle q are suitably represented by
(16)
ui(Q) = V0i sin2 q
(19)
h therefore differs from 0.5 (h\ 0.5) for symmetric reactions (DG° =0) as in Eqs. (1) and (12). This can profoundly affect analysis of ligand substitution reactions, chemical and electrochemical dissociative ET (cf. below), and optical bandshape analysis of charge transfer transitions [38]. Other limiting analytical forms emerge when w is large or differs only slightly from unity.
uf(Q) = V0f cos2 q
(20)
with the equilibrium values q0i = 0 and q0f = p/2. Motion of this kind is representative of large-amplitude internal molecular motion, and obviously far from harmonic. This also becomes apparent by the calculation of the activation Gibbs free energy. We introduce first the reorganization energies along the rotational coordinate E qri = ui(q0f) −ui(q0i) =V0i
4. Dissociative ET and rotational reorganization
(21)
E qrf = uf(q0i) −uf(q0f) =V0f
4.1. Dissociati6e ET Theoretical ET frames were extended early to dissociative and associative mechanisms in ligand substitution processes of transition metal complexes [39] and organic nucleophilic substitution reactions [40]. Analogies to ET processes where chemical or electrochemical reduction is accompanied by chemical bond fission were also noted early [41]. Dissociative ET has attracted recent attention [26 –28,42] particularly in organic electrochemistry [16,43]. The molecular potentials here, however, do not accord with the harmonic approximation but arbitrary molecular potentials can be incorporated by the parametric procedure. The parametric relations analogous to Eqs. (13) and (14) are [37] GA = ui(Q)+E sol r
dui/dQ dui/dQ− duf/dQ
DG°=ui(Q)−uf(Q)+E sol r
n
2
dui/dQ+ duf/dQ dui/dQ− duf/dQ
(17)
n
(18)
Eq. (17) is analogous to Eq. (13) and provides the activation Gibbs free energy in terms of the transition configuration of the local coordinate Q at given driving
If the driving force is not large ( DG° B min{V0i, V0f}), the method above may be applied, giving the following equations equivalent to Eqs. (13) and (14), and (17) and (18) [44] GA = Er + h=
V0i V0i + V0f
2
+
V0i V0i + V0f
V0i DG° V0i + V0f
V0i V0i + V0f
V0f + Er
V0f − V0i V0i + V0f
(22) (23)
The symmetry factor is thus constant. Driving force variation results in a relative vertical shift of ui(Q) and uf(Q) while the position of the solvent polarization profiles remains unchanged. The activation Gibbs free energy for symmetric reactions (DG° = 0, V0i = V0f = V0) is 1 1 q GA = E sol r + Er 4 2 This is clearly different from Eq. (1)
(24)
A.M. Kuznetso6, J. Ulstrup / Electrochimica Acta 46 (2001) 3325–3333
1 1 q GA = E sol r + Er 4 4
(25)
Both the absolute value of the molecular reorganization energy and its appearance in the activation free energy thus differ from the harmonic approximation. Outside the limits DG° Bmin{V0i, V0f} the variation of DG° is reflected only in the potential profile shift for the solvent polarization. The activation barrier is GA = V0i +
(Er +DG°−V0i)2 4Er
for DG°\ V0i −Er GA =
V0f −V0i V0f +V0i
(Er +DG°+V0f)2 4Er
(26)
for DG°B Er
V0f −V0i −V0f V0f +V0i (27)
The molecular contribution to GA here either vanishes, Eq. (27), or is constant, V0i (Eq. (26)).
3329
Eq. (29) determines Q* at given h. Substitution of Q*=Q*(h) into Eq. (30) gives an equation for h. The activation barrier is [37]
n
h 1 GA = h 2E sol r + ui(Q*) − DE 1−h 2
1/2
(31)
Eqs. (29) – (31) can be viewed as a parametric dependence of the activation barrier on the reaction free energy, with Q* as running variable. h can be calculated by Eq. (29) at any value of Q*. Insertion of h(Q*) into Eqs. (30) and (31) then gives GA. Eq. (29) is equivalent to Eq. (13) or (17), Eq. (30) to Eq. (14) or (18). Eqs. (29) –(31) also apply to arbitrary potentials ui(Q) and uf(Q). We now illustrate this general result by specific forms for ui(Q) and uf(Q). The most widely used anharmonic potential is the Morse potential. Intramolecular reorganization along Q in a Morse potential resulting in a displacement of the equilibrium chemical bond length of the Q-mode, Q0i Q0f gives ui(Q) = Di[1− e − k(Q − Q0i)]2;
5. Adiabatic transitions
uf(Q) = Df[1− e − k(Q − Q0f)]2
The activation Gibbs free energy relations above apply when the resonance splitting of the electronic levels in the crossing region of the diabatic potential surfaces, DE, is small [2,3]. Presently this implies that the diabatic potential surfaces are not significantly distorted, i.e. DE5kBT. If DE is large, the process belongs to the adiabatic limit of strong donor –acceptor interaction, and the character of the process different in two respects. Charge transfer is now effected by system motion along the lower adiabatic potential surface (electronic ground state) instead of crossing between the diabatic electronic states representing the reactants and products. The lower adiabatic surface is, secondly, significantly distorted (lowered by the resonance splitting) in the crossing region of the diabatic potential surfaces. In the two-state approximation the adiabatic potential surface is constructed from the diabatic free energy surfaces as 1 U = [Ui +Uf − (Ui −Uf)2 +(DE)2] 2
(28)
The activation barrier is determined by the saddle point on this surface. If we neglect the dependence of DE on the reactive mode coordinates, the equations for the value of Q at the saddle point, Q*, are [37] (1 −h)
dui duf dui/dQ +h =0 or h= dui/dQ−duf/dQ dQ dQ
DG°=(2h−1)E sol r +ui(Q)−uf(Q)−
(2h−1) 2 h(1−h)
(29) DE (30)
(32)
Introducing the new variables X= e − k(Q − Q0i);
X0 = e − k(Q0f − Q0i); and d=
Df DiX 20 (33)
we transform Eq. (29) (1 − h)(1 − X) +hd(X0 − X) =0
(34)
Calculating X and substituting into Eqs. (30) and (31) we obtain
GA = h 2 E sol r + −
d 2E Q DE r − [(1 −h) + hd]2 2 h(1− h)
DE
h(1− h) 2
DG° = (2h− 1) E sol r + +
n
(35) d 2E Q DE r − 2 [(1 −h) + hd] 2 h(1− h)
(1−h)2(d− 1)2dE Q r [(1 −h) + hd]2
n
(36)
where the reorganization energy along Q in the initial state has been introduced 2 EQ r = ui(Q0f) =Di(1− X0)
(37)
Eqs. (35) and (36) give the dependence of GA on DG° with h as running variable. The diabatic limit is reached if terms involving DE are omitted. If the repulsive branches of the initial and final potentials are also identical (i.e. if d= 1), the result of Saveant [26] is recovered. In the limit of large intramolecular reorganization and for identical intramolecular potentials (Di = Df),
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displaced along Q, X0 is small, and Eqs. (35) and (36) reduce to
2 GA = E Q E sol r +h r −
DE 2 h(1−h)
n
−
n
DE
h(1−h) 2 (38)
DE +E Q (39) r 2 h(1−h) As h(1 −h) varies slowly in broad range of h, we can also write Eqs. (38) and (39) in the form
DG°=(2h−1) E sol r −
eff Q 2 eff GA = E Q r +[E r (h) +DG°− E r ] /4E r (h)
−
DE
h(1 −h) 2
(40)
Q DG°=(2h−1)E eff r (h) +E r ;
sol E eff r (h)= E r −
DE
n
(41) 2 h(1−h) Eqs. (40) and (41) show that large intramolecular reorganization gives a constant activation barrier contribution. The latter is decreased by the quantity DE h(1− h)/2, while the solvent part of the reorganization energy is decreased by DE/2 h(1−h). It is worth noting that the model of two identical (but shifted) Morse-like potentials is clearly different from the model of Morse potentials used in Ref. [26] where only the final state repulsive branch was used. Figs. 1 and 2 illustrate numerically the parametric procedure for displaced Morse potentials (Eqs. (35) – (37)) in both the diabatic and adiabatic limits. The
dissociation energy ratio, Di/DfX 20 was in the range 0.5–2, the adiabatic resonance splitting (4 –8)kBT or 0.1– 0.2 eV, and both reorganization (free) energies E sol r and E Q r 20kBT or 0.5 eV. The auxiliary parameter p is zero or unity, and represents the diabatic and adiabatic limit, respectively. The results of the displaced harmonic oscillator model are shown for comparison. Figs. 1 and 2 show 1. All correlations are smooth and appear to resemble one another. However, the absolute activation free energy values depend sensitively on the local mode potential. 2. The activation free energy increases with increasing d. This parameter characterizes the repulsive branches of the Morse potentials in the initial and final states. Small d represents a shallower potential in the final than in the initial state, and vice versa. Smaller values of d therefore give a smaller activation Gibbs free energy (Fig. 1). 3. Harmonic and anharmonic potentials are compared in Fig. 2. The harmonic local mode corresponds to d= 1. The activation free energy for a shallow final state anharmonic mode (d=0.5) is lower than for the harmonic mode (d=1) but higher for a steep repulsive final state branch (d=2). In addition the correlation is notably asymmetric around zero driving force for the anharmonic mode, with larger curvature for negative than for positive DG°. 4. Resonance splitting (DE= 0.1 and 0.2 eV) lowers the activation free energy by approximately DE/2 over most of the driving force range.
Fig. 1. Activation Gibbs free energy dependence on reaction free energy for charge transfer with coupling to a local Morse potential and a harmonic solvent continuum, calculated by Eqs. (32), (35) – (40). Different values of the parameter d = Df/DiX 20 (Eq. (33)). Q Energies in units of kBT, T= 298 K. E sol r = E r = 20. DE= 4. The following parametric variations: 1, d = 2, diabatic limit; 2, d = 2, adiabatic limit; 3, d =1, diabatic limit; 4, d= 1, adiabatic limit; 5, d =0.5, diabatic limit; 6, d =0.5, adiabatic limit. Large d corresponds to strong repulsion in the repulsive branch of the Morse potential, small d to weak repulsion. Large d therefore gives the higher activation free energy.
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Fig. 2. Activation Gibbs free energy dependence on reaction free energy for charge transfer with coupling to a local mode potential and a solvent continuum, calculated as in Fig. 1. Comparison between a strongly anharmonic shallow local Morse potential (d = 0.5) Q and a virtually harmonic potential (d= 1). Energies in units of kBT. E sol r =E r =20. DE=8. 1, harmonic potential (d = 1), diabatic limit; 2, harmonic potential, adiabatic limit; 3, anharmonic Morse potential (d =0.5), diabatic limit; 4, anharmonic Morse potential (d =0.5), adiabatic limit.
6. Concluding remarks Large-amplitude nuclear dynamics and coordinate displacement along strongly anharmonic local nuclear modes is important in a range of known, and non-traditional molecular and supramolecular systems. Rotational reorientation in rotaxanes [25,29], and chemical and electrochemical dissociative ET [26 – 28,41– 43] were noted. Dissociative ET is mostly associated with alkylhalide reduction, but is also encountered in biological macromolecular function. An example is the reduction of the Type II/III trinuclear copper cluster in the blue oxidase ascorbate oxidase [45] by long-range intramolecular ET from the Type I centre [46]. Other cases are chemical and biological proton and atom group transfers [2,3] where bond breaking rests on stretching and bending which by far exceed the amplitudes of zero-point energy vibrations. These cases are natural targets for molecular charge transfer theory. In quantitative approaches none of them, however, accord with the limiting rate constants based on displaced harmonic oscillator models, but the present report shows that contemporary charge transfer theory including new achievements [2,3,8] offers appropriate other means. The contribution of large-amplitude intramolecular reorganization along arbitrary local mode potentials to the observable activation free energy was addressed. A general procedure was first provided where the dependence of the activation free energy on the reaction free energy is recast in parametric form. The running parameter is either the symmetry factor, or the local mode coordinate at the transition configura-
tion. Corresponding values of the activation and reaction Gibbs free energies can next be calculated for all values of the running parameters. The formalism extends, moreover, both to the diabatic and adiabatic limits of molecular charge transfer. Incorporation of large-amplitude local mode reorganization in general has to resort to a numerical procedure, but the parametric form is straightforward, and reduces to analytical form in representative cases. In addition to the harmonic form this applies, particularly, to rotational reorganization (Eqs. (22) – (27)) and specific cases of harmonic vibrational frequency changes and anharmonic potentials. The numerical analysis (Figs. 1 and 2) shows that free energy relationships retain qualitatively similar form for rather different local mode potentials. This is one reason for the apparent successful use of the simplest harmonic model also in cases when the validity conditions of this model are significantly violated. The apparent accordance is, however, misleading. The character of the free energy relationships is thus dominated by local mode reorganization when these modes dominate over the harmonic environmental modes. This modifies the curvature of the correlations, symmetry coefficients etc. The parameters and values of the activation free energy are also sensitively correlated in spite of the smooth character of all the correlations. The sensitivity of the activation Gibbs free energy to the local mode potentials subject to large-amplitude reorganization, and the attractive simplicity of the parametric procedure, therefore warrant broader use of this formalism in new experimental data analysis.
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Acknowledgements Financial support from the Danish Technical Science Council (grant No. 9801629, Russian Foundation for Basic Research (grant No. 00-03-32239) and the EU programme INTAS (grant No. 99-1093) is acknowledged.
Appendix A. Decomposition of the reaction free energy into solvent and intramolecular terms [3] Decomposition of DG° according to Eq. (5) is at first arbitrary and corresponds to all different trajectories across the dividing surface between the potential surfaces in the reactants’ and products’ states. The solvent part of the activation Gibbs free energy is 2 sol G sol A =h solE r =
hsol =
2 2 (E sol (E sol r +Dgsol) r +DG°−DgQ ) = ; sol sol 4E r 4E r
n
1 Dgsol 1 DG°−DgQ 1+ sol = 1+ 2 2 Er E sol r
n
(A1)
and corresponds to crossing at an arbitrary point of the dividing surface or equivalently, crossing at the saddle point of the potential surfaces spanned solely by the solvent coordinates at arbitrarily chosen Dgsol. The total activation Gibbs free energy is GA =
(E sol r +DG°−DgQ ) +ui[Q*(DgQ )] 4E sol r
(A2)
The value of Dgsol or DgQ for which GA is minimum is given by
n
#GA 1 DG°−DgQ #ui[Q*(DgQ )] =0 or 1+ =
hQ #DgQ 2 #DgQ E sol r (A3) The obser6able symmetry factor is, from Eq. (A2) h=
dGA 1 DG° = 1+ sol Er dDG° 2
n
(A4)
Eqs. (A1), (A3) and (A4) thus show that the three symmetry factors are identical. Together with Eqs. (5), (9), (10) and (A2) this provides the general procedure for the free energy relations and for separate determination of Dgsol and DgQ.
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