Quantum dynamics of proton-coupled electron transfer in model systems

Chemical Physics 302 (2004) 309–322 www.elsevier.com/locate/chemphys

Quantum dynamics of proton-coupled electron transfer in model systems Giovanni Villani

*

Istituto per i Processi Chimico-Fisici, IPCF-CNR, Via G. Moruzzi, 1 I-56124 Pisa, Italy Received 23 February 2004; accepted 28 April 2004 Available online 19 May 2004

Abstract The quantum dynamics of model systems with strong coupling between the electron transfer (ET) and the proton transfer (PT) has been studied: the proton-coupled electron transfer (PCET) process. These reactions are very important in several chemical areas and in biochemistry above all, hence, in this study, the models have been built having in mind biochemical systems. The PCET process has been analyzed in terms of the two constituent phenomena (PT and ET) and we have studied both the situations where the two processes proceed together and where a process (PT or ET) is faster than the other. The risk of the adiabatic approximation for the proton has been underlined since there is the possibility of faster proton transfer despite of the mass disparity of proton and electron. Ó 2004 Elsevier B.V. All rights reserved.

1. Introduction The transfer of proton (PT) involved in the hydrogen bond and the electron transfer (ET) from a donor (D) to an acceptor (A) are very important chemical processes in the biological area and not only. For example, ET reactions play a key role in the bioenergetic pathways of living systems and PT reactions in the DNA systems. It is not an aim of this paper to underline the importance of ET and PT processes since it can be considered acquired. In the last decades the role of the coupling between these two processes has been recognized and these two processes have been studied together: protoncoupled electron transfer (PCET). These reactions are crucial in a wide range of biological and chemical processes: the conversion of energy during photosynthesis and respiration, numerous enzyme reactions, electrochemical and solid state processes. One of the most important example of PCET phenomenon is involved in the conversion of light energy in chemical energy, one of the most important ways of *

Corresponding author. Tel.: +39-050-315-2454; fax: +39-050-3152442. E-mail address: [email protected] (G. Villani). 0301-0104/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2004.04.025

providing energy to life processes. Such conversion process frequently involves a coupling of electron transfer and proton transfer reactions resulting in the pumping of protons across membranes. An instructive example is provided by the reaction center of Rhodobacter sphaeroides [1–4]. Experimental studies have provided rates and kinetic isotopic effects (ratio of the rate with hydrogen to the rate with deuterium) for numerous PCET reactions in solution [5–11]. The large kinetic isotope effect indicates the importance of nuclear quantum effects such as hydrogen tunneling. The physical systems studied exhibit both concerted mechanism (in which the proton and the electron are transferred simultaneously) and sequential mechanisms (in which either the proton or the electron is transferred first). Most of ET reactions in biological systems involve
and therefore they large separations (from 5 to 20 A) have a very weak tunneling coupling between the donor and the acceptor sites. Although, the reactions are very efficient. In order to give an explanation of these experimental dates, even recently [12] mechanisms have been proposed, as the dynamical amplification and the conformational gating. Some years ago, Michell [13] and Onsager [14] suggested that in certain biochemical

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systems long range electron transfer between two redox sites, connected by a network of H-bonds, may occur via a proton assisted mechanism. Now such notion has acquired a wide popularity [15–21]. In the literature [22] someone calls PCET the electron and proton transfer on different centers and hydride transfer the electron and proton transfer on the same center. In other cases [23] the presence of these two processes is called proton assisted electron transfer and the proton transfer process is considered only a help for the more important electron transfer. Here, we use the PCET definition in a general meaning: a strong coupling between the ET and PT processes, hence these two processes are considered of the same importance and are studied at the same level. An immediate question arising in PCET is the following. Let us suppose that the final state of the charge transfer reaction corresponds to the transferred electron and proton. Does the reaction proceed through a consecutive mechanism, electron transfer followed by proton transfer, or does it proceed through a concerted mechanism, where both electron and proton transfer ‘‘together’’? The distinction between these processes is that in the first there two quantum events occur (an electron-tunneling process that is completed and then a proton-tunneling process), while in the last only a quantum event provides a resonant tunneling potential in order to the electron and proton tunnel together. It is worthy noting that, though the concerted reaction mechanism has independent quantum events, the reactions are not independent since, once the electron has been transferred, the energetics have been altered, and this will be reflected in the proton transfer step. The prevalence of one or the other process will depend on the various electronic and vibrational matrix elements, the exothermicities coming from the coupling with the solvent, electronic structure effects, and the reorganization energetics that contribute in determining charge transfer reaction rates. The aim of this paper is a quantum mechanical study of the PCET process by the help of simple models related to biochemical situations. In particular, we have in mind the base pairs in the DNA, where the hydrogen bonds are strictly connected to the electronic part and generate two possible structures: the canonical adenine–thymine and guanine–cytosine base pairs and theirs imino-enol tautomers. This paper can be summarized as follows. In the introduction we have shown the problem and underlined its importance in several contexts put it in the correct frame. In the following, we will describe the models and the methods utilized in this work. The illustration of the results in several ideal cases, but related to physical systems, follows, and, finally, the schematization of the main conclusions.

2. Models and systems in study The dynamics of PCET reactions in solution involves three distinct time-scales, generally associated with three different types of degrees of freedom. The electron degrees of freedom comprise the fastest motions, the intermediate time scales correspond to the dynamics of the protons, and the movements of the heavy solvent molecules correspond to the slowest time scales. The ET process is normally treated quantum mechanically and sometimes also the proton degrees of freedom are treated at this level, while the solvent molecules are treated classically (mean field calculations) [24–26] or, rarely, a full quantum model is used [27]. To date, two distinct theoretical formulations for PCET have been proposed. The first was developed by Cukier and co-workers [28,29] and the second was developed by Hammes-Schiffer and co-workers [30–34]. The first approach is in the adiabatic picture and in the evaluation of the rate constant it takes advantage of the mass disparity of proton and electron in finding a restricted tunnel path; the second is a multidimensional analogue of the standard Marcus theory of single ET [35]. Both consider the system and the solvent and they obtain general expressions using remarkable approximations (harmonic approximation, dielectric continuum theory, Golden rule, etc.). Our approach is different: we will concentrate on the analysis of the PCET phenomenon itself and we will leave the study of the changes induced in it by the outside environment for the moment. Following this reasoning, we do not introduce a thermal bath. A thermal bath (or a solvent) can be involved in two different types of interactions with the ET and PT processes: static and dynamical. When the electron and proton transfer rates are slower than the solvent characteristic times these two processes see only a static solvent situation and there are no solvent-dynamics effects. A similar situation can be found when the ET and PT processes are very fast compared to the solventdynamics. This case has been studied by Barbara and co-workers [36,37]. They observe that the characteristic times of the intramolecular ET from the S1 state to the ground S0 state of betaine-30 do not change very much in a wide selection of solvents. Only when the characteristic times of the solvent and of the ET and PT processes are similar, one may expect some solventdynamics effects on the observed ET and PT rates and the process can be solvent-controlled. In the case of static solvent effect, the role of the bath is only in bringing the acceptor and donor states in resonance, hence, we can obtained the same results assuming the degeneration of the donor and acceptor states without an explicit inclusion of the solvent. This is what we do in this paper. On the other hand, the PCET can be essential also in conditions where the systems can be considered

G. Villani / Chemical Physics 302 (2004) 309–322

isolated, for example in the gas phase electron transfer [38–42]. Therefore, we prefer to construct more or less simple models, in order to study this phenomenon. To deal with both PT and ET processes we have used two different kinds of models: a time-dependent simple model and one more complete time-independent. The ingredients of the first model are: a harmonic oscillator, a second time-dependent degree of freedom and two diabatic states for representing the electron on the donor and on the acceptor. The ingredients of the second model are: some anharmonic (fourth degree potential) vibrational degrees of freedom and two (or three) electronic diabatic states with electron on the donor, on the acceptor (and on the bridge). A first general consideration regards the physical meaning of the oscillators: which physical degrees of freedom we want to consider in the study. Two possibilities are generally considered in the literature: the oscillator on the site, in order to represent the site vibrational change due to the presence of the electron, or the oscillator between the sites, as the bond between two atoms in the chemical picture. Here, we have considered both the possibilities. A second general question is related to the physical meaning of the time-dependent oscillator used in the simpler model. It can be seen as a classical description of a set of degrees of freedom not included in the detailed study. If one wants to introduce the bath, it is possible to associate this oscillator to the bath; otherwise, if one wants to study the PCET process itself in an isolated system, it is possible to use the timedependent oscillator to mimic the other degrees of freedom of such system not considered in details. The hydrogen bridge system X–H


Y, with X and Y electronegative atoms (O, N, etc), considered as a molecule, involves three (3N  6, with N ¼ 3) generalized coordinates to be described, i.e. two stretchings and a bending in the local picture. Here, we prefer the description in terms of local modes and not in normal (or in other global) modes, since they describe better the physical PT process, schematized as follows: chemical bond þ hydrogen bondðX–H and H


YÞ # hydrogen bond þ chemical bondðX


H and H–YÞ while such process turns out less obvious in a description with normal modes. Often in the literature (also in the case of local modes) one uses the harmonic approximation in order to arrive a simple solution and to obtain compact expressions for the characteristic times of the system. We have used this approximation in the simpler model. However, if one wants to describe the proton transfer between the two stable positions of the hydrogen bond in a realistic way it is better to remove such approximation and to use a fourth degree potential with two (symmetric or asymmetric) wells. In fact, the aim of

311

this work is precisely the study of the proton transfer between these two minima of potential energy, in connection with the electron passage. If we consider the molecule frozen in the bending, the hydrogen atom moves along the X–Y line; therefore, we can eliminate this coordinate. This is a reasonable approximation because it is demonstrated that the proton moves preferentially along the direction between the two atoms that support the hydrogen bridge. The subsequent problem is to decide whether considering in detail both stretchings or further simplifying the study. Both monodimensional and bidimensional models exist in the literature to describe the hydrogen bridge. In a recent review [43], Nibbering and Elsaesser have shown experimental studies that support the use of a bidimensional model. Some years ago we have shown the importance of a bidimensional picture in order to describe the proton dynamics in the hydrogen bridge of the water molecule [44,45]. Here, we will use a monodimensional oscillator for the hydrogen bridge. Such a simplification surely does not work very well when the two electronegative atoms can move remarkably and this movement alters the potential energy of the hydrogen bridge. This is the case of the water molecule where a monodimensional model for the O–H


O oscillator is not sufficient and also the O–O degree of freedom must be considered. In fact, in this case the movement of the oxygens modifies considerably the
O–O distance hydrogen bond potential: up to 2.5 A there is a single potential energy well, beyond a double well potential with a barrier proportional to the O–O distance [46]. In contrast, when the two electronegative atoms are part of a large molecule (for example in DNA) their movements are not free (small amplitude) and a monodimensional model is surely more appropriate. Having in mind the applications of these models to the biological context we use this approximation. In any case, I will explore this approximation in a future work in the study of a specific physical system. The electronic part is in our models represented by diabatic states. The choice of this type of electronic states is due to similar considerations than those used in the choice of local modes in the vibrational part. These electronic states have been coupled in the simplest way, i.e. through a constant coupling. Now we describe the details of the two models utilized in this paper. We begin to consider the simpler model, the time-dependent mixed quantum-classical one with two diabatic states. The first state (electron located on the donor) has an harmonic potential with minimum in position x0 and the second (electron on the acceptor) at position x0 . At both electronic states a sinusoidal time-dependent perturbation has been added, that changes the energy with a frequency xY and a maximum amplitude CY . In practical, the Hamiltonian is

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1 p2 2 H11 ¼ KX ðX  x0 Þ þ X þ CY sinðxY tÞ; 2 2lX 1 p2 2 H22 ¼ KX ðX þ x0 Þ þ X  CY sinðxY tÞ; 2 2lX H12 ¼ c:

ð1Þ

The first two terms in H11 and H22 are the potential and the kinetic energy of the X oscillator. The third term in H11 and H22 is the time-dependent oscillator and we assume a parameter Cy that introduces a DE maximum of 2 kcal/mol where an usual barrier in the hydrogen bond is 10 kcal/mol, (for the other parameters see caption of Fig. 1). Hence, the time-dependent perturbation induced by this term modifies remarkably the PES of this system, but it does not change the nature of a double well of the hydrogen bond. Some real systems can be represented by these parameters. For example the hydrogen bond in the water [33,46] and the adenine– thymine base pair where a barrier of 9.73 kcal/mol has been computed [47]. In general, the Hamiltonian (1) is difficult to resolve since it is time-dependent. An approximate way [48] of resolution is introduced: the Schr€ odinger equation with this time-dependent Hamiltonian can be solved with a 1 0.9 0.8 0.7

1

P dia

0.6 0.5 0.4 0.3 0.2 0.1 0 0

1

2

3

4

5

t (arbitrary unity)

Fig. 1. Diabatic electron population of the initial state j1i as a function
The solid line is the case of time for the system with dX–Y ¼ 2:25 A. with xX ¼ 3000 cm1 , Cy ¼ 0 and c ¼ 0:03 a.u.; the other curves, as a function of larger period, are these with 1000xY ¼ 1, 0.2, 0.1 a.u. The time is in arbitrary unit (in this case one arbitrary unit of t  10 fs).

Lanczos technique [49,50] (with Householder reorthogonalization [51] described in previous papers [52,53]) choosing a sufficient small time step (we had found that a time step of 0.0625 fs was appropriated [48]). The time-independent model is described by the Hamiltonian pX2 p2 p2 þ Y þ Z ; 2lX 2lY 2lZ p2 p2 p2 H22 ¼ V42 ðX Þ þ V42 ðY Þ þ V42 ðZÞ þ X þ Y þ Z ; 2lX 2lY 2lZ p2 p2 p2 H33 ¼ V43 ðX Þ þ V43 ðY Þ þ V43 ðZÞ þ X þ Y þ Z ; 2lX 2lY 2lZ H13 ¼ H31 ¼ H23 ¼ H32 ¼ cb ; H12 ¼ H21 ¼ yd ;

H11 ¼ V41 ðX Þ þ V41 ðY Þ þ V41 ðZÞ þ

V4i ðaÞ ¼ aa4 þ ba3 þ ca2 ;

i ¼ 1; 3; ð2Þ

where X, Y, and Z are the proton coordinates and the wells of the potential energy surfaces are the proton binding sites. Here, we assume that the diabatic state j1i is that with the electron on the donor and in all cases studied it is the only populated state at t ¼ 0, the j2i with the electron on the acceptor and the j3i, where present, on the bridge. The X oscillator is on the donor in all the cases. In the D–A case, the Y oscillator is on the acceptor while, in the D–B–A case, the Y oscillator is on the bridge and the Z on the acceptor. In all the cases studied, the initial vibrational state is the ground state of the oscillator. We combined the first two eigenstates of the oscillator only for building the localized state of the symmetric oscillator. The preparation of an initial state among several degenerate or quasi-degenerate states is a general problem but, here, there is not a particular difficulty in building the initial state. In fact for the electronic diabatic part one can imagine an external process that it gives an excess of electron on the donor and we begin to study the ET process only when this state has been just prepared. For the vibrational part in the asymmetric oscillator, as said, the initial state is the ground state of the oscillator. In the case of symmetric potential, the localized state means only that the hydrogen atom is in a part or in the other of the two parts bonded by the hydrogen bridge. For the sake of simplicity we also assume that the three coefficients (a ¼ 0:148863, b ¼ 0, 0.0174963, c ¼ 0:231582, with a barrier of 15 kcal/mol and the asymmetry of the
of the fourth double well 2 kcal/mol and X–Y ¼ 2.6 A) degree potential are independent of the electronic states. The reduced mass (lX , lY , lZ ) is that of the proton. Some example systems described by these parameters can be the iron biimidazoline complexes [54] and the tautomerisation systems of the cytosine, uracil, thymine and their 1-methyl analogues [55]. This model is similar to that utilized by us for the system D–B–A [56], but now the oscillators are not as-

G. Villani / Chemical Physics 302 (2004) 309–322

sumed harmonics and it can mimic the PCET process since the different electronic diabatic populations give the fraction electron sites (D, A and B) and the mean value of the proton packet the movement of the proton among the different binding sites. When the parameter b in Eq. (2) is zero the fourth degree potential describes a symmetric oscillator; if b < 0 the potential has an absolute minimum for a > 0 and vice versa for b > 0. We have labeled for each electronic state ‘‘s’’, ‘‘n’’ and ‘‘p’’ these three cases of potential, respectively. For example, the system snp.pns.psp is

313

in the R potential well and the first excited state in the other, L pffiffiffiffiffiffiffip ffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi 2x0 4 2mH Vb expð 2mVb x0 Þ ð3Þ R h0j1iL ¼ and since the maximum probability of passage between two not degenerate states is PMAX ¼

4C2 DE2 þ 4C2

(where the symbols are the usual meaning) replacing C with the coupling expression (3), we obtain 2

PMAX ¼

4c2 R h0j1iL

DE2 þ 4c2 R h0j1i2L pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 8c2 2mH Vb x0 expð2 2mH Vb x0 Þ p ffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ 2Vb =x20 þ 8c2 2mH Vb x0 expð2 2mH Vb x0 Þ

3. Results and discussion We begin with the Hamiltonian (1) and we put to zero the time-dependent part in a first moment. In this case one obtains a simple solution that can be simplified in an useful equation in the hypothesis of a frequency xx around 3000 cm1 (as it is reasonable for the O–H bond) and a diabatic coupling c of about some hundreds of cm1 . In fact, in this case the effective system becomes a two-state system with a tunneling period Tx ¼ p=C; C ¼ cR h0j0iL with R and L (right and left) those of the state located at x0 and x0 , respectively. Developing the Frank–Condon coupling in the characteristics of the oscillator, we obtain p pffiffiffiffiffiffiffiffiffiffiffiffiffi Tx ¼ c expð 2mH Vb x0 Þ with mH proton mass, Vb potential energy barrier (value of the potential energy at x ¼ 0 when the two wells are at x0 and x0 ) and x0 proton equilibrium position that, in a symmetric system, is dO–O  2dO–H x0 ¼ : 2 For a verification of a two-state model, we also compute the Frank–Condon coupling between the ground state

It is easy to verify that the maximum probability of passage between the two not degenerate states is of the order of 104 –105 for reasonable values of c, DE, Vb and x0 . Considering now the complete Hamiltonian (1) to see which modifications have been introduced by the timedependent term that removes the degeneration of the two diabatic states. In this case, being the Hamiltonian directly time-dependent, there are not in general simple expressions for the characteristic times of this system. However, it is still possible to obtain compact expressions for this time-dependent model in particular conditions. In Fig. 1, we show several cases with different frequencies of the time-dependent term (for the parameters utilized see the figure caption). We note that the transfer of the electron from the donor to the acceptor proceeds step by step since only when the system has two electronic degenerate, or practically degenerate, states the electron can pass in a consistent way, while there are small oscillations when the two diabatic states are too different in energy. If we call d the fraction of time of the electron transfer period when the two electronic states are quite degenerate, we can easy see that the period of transfer is     2Tx  d Ty Ty T ffi Tx þ Int þ1 d ; d ¼a ; 2d 2 2 ð4Þ where Int½x is obviously the integer part of x. In the definition of d we have introduced an ulterior parameter a because one can see that, once defined the characteristics of the proton potential energy (frequency, energy barrier, equilibrium position, etc.), d changes in a simple way with the frequency xY . In the definition of the d parameter, we use the term ‘‘quite’’ degenerate since, as it is evident from the expression of Pmax , the electron transition is significant only in a small range of DE, for an assumed coupling.

G. Villani / Chemical Physics 302 (2004) 309–322

site difference due to the electron presence. In Fig. 2(a) we show the diabatic electron population of the initial state, j1i, and in Fig. 2(b) the mean proton position in

1

0.8

0.6

1

The meaning of Eq. (4) is the following. When the time-dependent oscillator is switched off, T ¼ Tx . Switching on the Y oscillator, the energies of the two diabatic states changes and, in every semiperiod, these states become degenerate. Hence, for a time range d there is a transition between the diabatic states while in a (Ty =2  d) range the diabatic population is practically constant. This means that, in presence of the time-dependent Y oscillator, the period of the X oscillator increases of these time constant steps. The integer number of these constant steps is related to the ratio of Tx and d. The period obtained with the expression (4) is an important result of this paper since it connects explicitly the transition probability of a PCET process to the molecular characteristics. There is a parameter a (or d), but this can be computed or following the first step of this process or by the theoretical considerations on the fraction of time of ET transfer. In fact, the exact value of the parameter d can be obtained both from the transition probability at the time of the first step (P ð1°stepÞ ¼ 0:5 þ 0:5 cosð2  C  dÞ) or from the value of the constant time step in the diabatic population (cos t ¼ Ty =2  d). If one wants to completely avoid the quantum calculations, theoretical consideration with the help of Pmax can give an approximate d value. There are only two limitations. First of all, this expression, is due to a two-state model and, hence, subordinated to the same limitations; second the two times (xx and xy ) must be substantially different. By this expression one can compute in a simple way, and without performing a detailed calculation on the system, the period of an electron transfer in the presence of both proton transfer and time-dependent classical oscillator that mimic other degrees of freedom, inner or external to the system.

P dia

314

0.4

0.2

0 0

600

2400

3000

1800

2400

3000

1 0.8

3.1. Time-independent model with two diabatic electronic states

3.1.1. Cases 2e–1v As said above, there are two possibilities in the choice of oscillators: on the sites and inter the sites. In this case, obviously, we have done the second choose since there are two sites and only one oscillator. This is the only case in the paper where we assume explicitly this position. In the other cases the results can be associate or to both types of oscillators or to the site oscillators. In this case we assume an asymmetric double well potential energy for the vibrational part in order to underline the

1800 t (fs)

x1

0.6 0.4 0.2 oscillator X

This type of model is time-independent and it includes only the part of the system with ET and PT processes. The system in study is D–A and generally in the literature it is named tunneling model. In the electronic part we have only the state of the donor, j1i, and that of the acceptor, j2i; in the vibrational part, we include explicitly one or two oscillators.

1200

(a)

0 -0.2 -0.4

x2

-0.6 -0.8 -1 0

(b)

600

1200 t (fs)

Fig. 2. Diabatic electron population of the initial state j1i as a function of time (a) and mean proton position (xi ) for the jii diabatic state (b) for the np case and with cd ¼ 0:03 a.u.

G. Villani / Chemical Physics 302 (2004) 309–322

the two electronic states. It is evident the presence of two times: a short one, with small amplitude, due to the electron and a longer one due to the proton. The PT process is modeled by the ET and a strong interaction exists between these two processes with an evident interference. In fact the electronic modulation (due to the change of the diabatic electron population) along the proton transfer curve has a well defined behavior and there are two symmetrical points where this modulation is zero. In the global process we find a complete transfer of both the electron and the proton. If we concentrate the attention around 1500 fs (half period) we see that from the initial condition (1; x0 ) (where the first number is the diabatic state and the second the mean proton position) one arrives to the situation (2; x0 ): both particles have performed a transition. 3.1.2. Cases 2e–2v Now we have the same number of sites and oscillators. This is the typical case of site oscillators and we can test the hydride transfer, both proton and electron transferred on the same site. Physical examples of these reactions can be found in the enzyme liver alcohol dehydrogenase (LADH) [57,58]. We have studied three cases: sp.ps, sn.ns, sn.ps, with the initial proton position localized around x0 . In the first two cases we have assumed that the donor and the acceptor are identical. For example in the case sp.ps we assume that both the donor and the acceptor of the electron have a symmetric oscillator in the presence of the electron and an asymmetric oscillator with an absolute minimum at x0 (oscillator with positive b in Eq. (2) and named p) in the absence of the electron. It is worthy noting that, as previously said, we have also chosen equal parameters in the symmetric and asymmetric potentials independently on the electronic states. In the third case (sn.ps) we differentiate the donor and the acceptor since the asymmetric potential is p for the donor and n for the acceptor, with absolute minimum at x0 and x0 , respectively. In Fig. 3, we show the diabatic population of state 1 2 1 j1i, Pdia ðPdia ¼ 1  Pdia Þ and the mean proton positions in the two oscillators (xi and yi ) for the jiith diabatic state. In the case sp.ps (Fig. 3(a)), at t ¼ 0 x1 of the symmetric oscillator X and y1 of the asymmetric oscillator Y are in the x0 position. In j2i the oscillator X has an absolute minimum of potential energy exactly in x0 and the oscillator Y is a symmetric double well with equal minimums at x0 . In this condition the transition between these two electronic states is complete and the diabatic electron population changes from 1 to 0 and vice versa. The mean proton positions in the symmetric oscillators (x1 and y2 ) remain constant until the electron has passed completely in the other diabatic state (j1i ! j2i for x1 and j2i ! j1i for y2 ) then the proton moves (partially or completely) to the other well and it

315

comes back in a short time. In contrast, the mean proton positions in the asymmetric oscillators (y1 and x2 ) remain constant. Note that the electron period is 80 fs and the proton period is 20 fs, but this does not imply that the proton (faster) has transferred before the electron (slower). Moreover, since the fast proton transition occurs in a practically empty electronic state, if we concentrate the attention only on the populated diabatic state we never find the proton transition. The case sn.ns is substantially different. Now, the initial proton position in the symmetric oscillator X is x0 , but the absolute minimum of X oscillator in j2i is x0 . Moreover, there is a difference of energy of 2 kcal/ mol between these two states since, as said, the asymmetric potential has been obtained by lowering a well and by raising the other of this amount of energy. These two facts hinder the electron transition, as one can see in Fig. 3(b), and they give an electron transition 2 (Pdia ðmaxÞ  0:1), smaller than the pure electronic transition between two states with DE ¼ 2 kcal/mol 2 (Pdia ðmaxÞ  0:3). The mean proton position in the diabatic state j1i remains unchanged while in j2i the proton position catches up the absolute minimum only when the fraction of electron is come back in j1i. Hence, also in this case we find an electron transition followed by a proton transfer although this last transition is faster and in the populated electronic state one cannot find a proton transition. The only difference with the previous case is the uncompleted electron transfer. The case sn.ps (or sp.ns) can be rationalized with the help of the same considerations used for the two previous cases. Instead, this case becomes interesting when the proton state of the symmetric oscillator at t ¼ 0 is the vibrational ground state and, hence, the proton is not localized in a well. In Fig. 3(c), we show the usual quantities for this case. Now the electron transition is not complete (but greater than the sn.ns case) and the mean proton positions, in the symmetric oscillator X in the two electronic states (x1 and x2 ), move towards different wells with a spontaneous localization. This is due to different transition probability of the two wave packets of the proton fraction in x0 and x0 of j1i. The wave packet fraction at x0 finds difficulty in moving in the other electronic state because at that position in j2i there is not the absolute minimum of potential energy. In contrast, the fraction of the proton wave packet centered in x0 makes an easy transition. As a consequence the mean proton position in the symmetric oscillator in j1i is prevalently in x0 and in j2i in x0 . A spontaneous localization in a symmetric double well due to an electron transition in a state with an asymmetric double well has never been discussed in the literature, at our knowledge. In no one of the three studied cases we have found a hydride transfer. This emphasizes the difficulty in simulating the qualitative behavior of the ET and PT processes, underestimated difficulty in the literature.

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G. Villani / Chemical Physics 302 (2004) 309–322 1

1

0.8

0.8

P1

y1

0.6 0.4

0.2

0.2 P dia , X,Y

0.4

0

y2

0

1

1

P dia , X,Y

0.6

P1

x1

-0.2

y2

x2

-0.2

-0.4

-0.4

y 1 ,x 2

-0.6

x1

-0.6

-0.8

-0.8

-1 0

20

40

60

(a)

80

-1

100

0

12

24

(b)

t (fs)

36

48

60

t (fs)

1

P1

0.8

y1

0.6

y2

0.4

x1

0

1

P dia , X,Y

0.2

x2

-0.2 -0.4 -0.6 -0.8 -1 0

20

40

(c)

60

80

100

t (fs)

Fig. 3. Diabatic electron population of the initial state j1i (P1 ) and mean proton position in the donor oscillator (xi ) and in the acceptor oscillator (yi ) for the jii diabatic state: (a) sp.ps case, (b) sn.ns case and (c) sn.ps case and with cd ¼ 0:001 a.u.

3.2. Model with three diabatic electronic states This model includes explicitly a donor (D), an acceptor (A) and a bridge (B), the D–B–A system, largely

studied in the literature and recently studied through a similar method by us [56]. However, in this paper there is an implementation of the previous method with the use of the anharmonic oscillators, necessary to give a

G. Villani / Chemical Physics 302 (2004) 309–322

P2

0.8

y1

0.6 0.4

P1 0.2 P dia , X,Y

3.2.1. Cases 3e–2v The presence of three sites and two oscillators gives two possibilities in the physical meaning of these last ones: inter site oscillators or site oscillators but only on the donor and the acceptor. Both these possibilities are open but we prefer to concentrate on the last one. As in the 2e–2v model, we have studied three cases: sp.ps.ss, sn.ns.ss, sn.ps.ss. Now we have the additional diabatic state j3i (electron in the bridge) and in this state we assume oscillators with a double symmetric well of potential energy. This means that when the electron is not in D or A (is in the bridge), the oscillators in the D and A sites are symmetrical. The main difference between the three and the two diabatic states cases is that now there are two possible processes to transfer the electron from the donor to the acceptor: a direct process D ! A, when this coupling is different from zero, and an indirect one D ! B and B ! A. In all the studied cases, we assume that the direct process, when it exists, is a small perturbation of the indirect process, cd  cb . In this case we have in mind the p-cooperativity in doubly hydrogen bonded systems, as the cyclic arrangement of hydrogen bonds. Fonseca Guerra et al. [59] reported that the contribution of resonance assistance to the total hydrogen bonding stabilization was 13% and 18% for the adenine–thymine and guanine–cytosine base pairs, respectively. Figs. 4 and 5 are analogous to Figs. 3(b) and (c), but with cd ¼ 0:1cb . We have not shown the figures corresponding to the case sn.ps.ss which, as the case with two diabatic states, can be rationalized starting from the other considered cases. In Fig. 4 two times are well evident: that of the electron transfer along the bridge (50 fs) and that of the direct electron passage (500 fs). Now, the transfer from the donor to the acceptor is not complete since a fraction of the electron population remains in the bridge (20%) and the PT and the ET processes are not sequential since these two particles move together. Hence, when the electron population has a maximum in the acceptor (t  250 fs), the proton of the symmetric oscillator in D has been practically completely transferred from a well to the other. In the sn.ns.ss case (Fig. 5) it is evident that now the electron moves to the acceptor more slowly. Also in this

1

0

y2 -0.2

x1 -0.4

x2

-0.6 -0.8 -1

0

150

300

450

600

750

t (fs)

Fig. 4. Diabatic electron population of the jii state (Pi ) and mean proton position in the donor oscillator (xi ) and acceptor oscillator (yi ) for the jii diabatic state. sn.ps.ss case with cd ¼ 0:0001 a.u. and cb ¼ 0:001 a.u.

1

P2

0.8

y1

0.6 0.4

x2

P1

0.2 P dia , X,Y

realistic description of the hydrogen bond. Moreover, in the previous paper three oscillators and three sites (site oscillators) are considered; now we consider also the cases with a number of oscillators different from the number of sites. Generally in the literature this model is named superexchange and the comparison between the tunneling D–A model and the superexchange D–B–A has been largely analyzed. In all the studied cases of this paper, since the mean proton positions in j3i is always complex and oscillating, we avoid showing explicitly these figures.

317

y2

0

x1

-0.2 -0.4 -0.6 -0.8 -1

0

1200

2400

3600

4800

t (fs)

Fig. 5. Idem to Fig. 4 but for the sn.ns.ss case.

6000

G. Villani / Chemical Physics 302 (2004) 309–322

case an about 20% of the electron fraction oscillates in the bridge and, hence, only a fraction of the electron population arrives to the acceptor. Now all the times of these processes are evident: the two electronic times are 50 and 6000 fs, while the proton time is 500 fs, but the PT and ET processes are completed at the same time. Note that in the asymmetric double well oscillator in a first moment the well at higher energy is populated and after this population decades to the absolute minimum of the potential energy with damped oscillations. 3.2.2. Cases 3e–3v These are the more complex cases dealt by us in this paper and they are also time-computer expensive. Unlike the previous cases now there is a vibrational part also on the bridge, modified by the presence of the electron on this site. We have studied two cases: symmetric and asymmetric oscillator in the site with the electron. In particular, we have studied the snn.nns.nsn (and spp.pps.psp) and nss.ssn.sns cases. Also in these cases there are two possible channels to let the electron move from the donor to the acceptor (directly and along the bridge) and also in these cases we have studied systems where the direct coupling is only a perturbation (cd ¼ 0:1cb ) of the indirect channel. In all cases we have studied also the case with cd ¼ 0, in order to evaluate the effect of the switch on of the small direct coupling. In Fig. 6 we show the snn.nns.nsn case with only the indirect donor–acceptor coupling (along the bridge). It is evident that only a small fraction of the electron moves away from the donor (10%) and the electron population of the acceptor is practically zero. This is a consequence of the initial location of the proton (x0 ) in the symmetric oscillator on the donor since at that position, both in the acceptor and in the bridge, there is not the absolute minimum in the potential energy curve and, as in the previous cases, this condition prevents the electron transfer between the diabatic states. In fact, if we consider the spp.pps.psp case, where the absolute minimum of energy in the other diabatic states corresponds at the initial proton position (x0 ) in j1i, the electron passes practically completely (figures non-reported). In Fig. 6, the proton position in j1i is practically constant because the fraction that comes back in the donor (and that it could modify the proton position) is small and generates only a small modulation of x1 . In contrast, the behavior of the proton positions in the diabatic state j2i in the donor and acceptor sites, x2 and z2 is interesting. In the case of x2 the proton (partial) transfer between the two wells of the asymmetric oscillator is evident, with the stabilization of the proton preferentially around the absolute minimum. On the contrary, the proton position in the symmetric oscillator in the acceptor (z2 ) moves from a well to the other with small oscillations. The switch on of the small direct

1 0.8

P1 y 1 ,z 1 ,y 2

0.6 0.4 0.2 P dia , X,Y,Z

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P2

0

x2

-0.2

z2

-0.4

x1

-0.6 -0.8 -1

0

200

400

600

800

1000

t (fs)

Fig. 6. Diabatic electron population of the jii state (Pi ) and mean proton position in the donor oscillator (xi ), bridge oscillator (yi ) and acceptor oscillator (zi ) for the jii diabatic state. snn.nns.nsn case with cd ¼ 0 and cb ¼ 0:001 a.u.

coupling (Fig. 7) modifies completely the situation of the ET, but less that of the PT process. Now at the small electronic oscillations, with a scale of tens fs, one adds a periodic electron transfer with a large time of 6 ps: the electron moves practically completely from the donor to the acceptor and 80% comes back (a fraction of 10% of electron population oscillates in the bridge). The proton position in the symmetric oscillators has a qualitatively similar behavior than the case without the direct D–A coupling (with a more o less complete passage) and with a characteristic time of 1500 fs, while the proton position in the asymmetric oscillator in D reaches the absolute minimum with damped oscillations. Both these two behaviors are correlated to the ET process. Through the comparison of Fig. 7 with Fig. 5, we deduce that both the ET and PT processes are similar in these two cases. This means that the addition of the vibrational part in the bridge does not modify substantially the situation. Through the comparison between Figs. 6 and 7, it would seem that the transfer of the electron in the case with both the direct and the indirect donor–acceptor coupling is due to the switch on of the direct channel. It can be seen in two ways that the situation is not so easy. First of all an increase of the direct coupling (Fig. 8) does not help the electron transfer, but

G. Villani / Chemical Physics 302 (2004) 309–322 1

P1

0.8

P2 y 1 ,z 1 ,y 2

0.6 0.4

P dia , X,Y,Z

0.2

z2

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x1

-0.6 -0.8 -1 0

1200

2400

3600

4800

6000

t (fs)

Fig. 7. Idem to Fig. 6 but with cd ¼ 0:0001.

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gives a situation similar to that without direct coupling. Moreover, if we remove the degeneration of the bridge there are interesting results. In Fig. 9 we show the case where the bridge energy is 0.01 a.u. higher than that of the couple donor/acceptor. In this case in 1 ps practically there is not electron transfer, but the proton has a well evident periodic dynamics in the diabatic state j1i and, for the small fraction present, in j2i and even in j3i (not reported). Hence, this means that the switch on of the direct coupling modifies the proton and electron transfer not by adding an active channel but by creating an interference, which can be positive or negative (in the case of Fig. 7 it is positive), between the two ET channels. This effect has been already put in evidence in the literature [56,60]. Recent results show that such situations can occur for large flexible molecules [61] or for the case where there is an additional functional group in the bridge compound [62] and they are very sensitive to the coupling values and to the energy states. In these cases, a complete calculation is essential. This interference is also underlined in general and named dynamical amplification [12]. After, we have studied in detail the nss.ssn.sns case in four alternatives: (a) without direct coupling, (b) with direct coupling, (c) with not degenerate bridge (DE ¼ 0:01 a.u.) and only indirect coupling and, finally, (d) with not degenerate bridge and direct and indirect couplings. For all cases the proton in the symmetric

1 1

0.0

P1 0.8

0.9

y 1 ,z 1

0.6 0.4

0.0003

0.0001

0.7

0.2 P dia , X,Y,Z

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P dia

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z2

y2

0

x1

x2

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0.5

0

200

400

600

800

1000

t (fs)

Fig. 8. Diabatic electron population of the initial state j1i as a function of time for the cases with cd ¼ 0:0, 0.0001 a.u. and 0.0003 a.u., indicated in figure.

-1 0

200

400

600

800

1000

t (fs)

Fig. 9. Idem to Fig. 7, but with the energy bridge higher than the donor/acceptor of 0.01 a.u.

G. Villani / Chemical Physics 302 (2004) 309–322

oscillators at t ¼ 0 is not localized. In the case (a) (figures not reported), since there is not an unfavorable initial proton position, the electron passes from the donor to the acceptor also without the direct coupling, but with large oscillations. Since the proton in the symmetric oscillator at t ¼ 0 is not localized and the arrival oscillators are asymmetric, the proton fraction localized in correspondence of the absolute minimum of the diabatic arrival states passes more easy than the fraction localized in the other well. Hence, one obtains a spontaneous proton localization in the symmetric oscillators, as already underlined in other cases. Obviously this localization disappears upon time for the overlap of the proton transfer between the two potential wells. The switch on of the direct coupling (case (b)) separates clearly the two ET times and reduces the electronic oscillations. A 30% of the electron fraction oscillates fast in the bridge and the PT process is in this case in complete synchrony with the ET. As an example, in Fig. 10 we show the electron and proton parts of the diabatic state j2i of this case (b). In the case with not degenerate bridge, case (c) (figures not reported), one finds a shortening of the ET period (from 2800 to 2000 fs), the oscillations due to ET along the bridge disappear almost completely, but the ET transfer is not complete. Switching on the direct coupling in this last case, case (d), the ET period remarkably increases (10 ps), with a very evident periodicity (Fig. 11). In contrast, the mean

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P1

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x1

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y1 0.2 P dia , X,Y,Z

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z1

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4800

(a)

9600

12000

9600

12000

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0.2 P dia , X,Y,Z

7200 t (fs)

y2

0

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x2

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-1

-0.4

0

(b)

z2

-0.6

2400

4800

7200 t (fs)

Fig. 11. Idem to Fig. 10, but with the energy bridge higher than the donor/acceptor of 0.01 a.u.: (a) quantities of state j1i; (b) of state j2i.

-0.8 -1 0

600

1200

1800

2400

t (fs)

Fig. 10. Idem to Fig. 7, but for the nss.ssn.sns case.

3000

proton positions in the symmetric oscillators oscillate with characteristic periods and they are substantially delocalized, but in correspondence of the complete electron transfer they become localized.

G. Villani / Chemical Physics 302 (2004) 309–322

4. Conclusions In this paper, we have studied in detail the quantum dynamics of a proton-coupled electron transfer (PCET) process with the possibility of a transfer among some diabatic states for the electron and a transfer between different potential energy wells (equal or different) of the same oscillator for the proton. Two different models have been utilized and several different physical cases can be studied with the help of these models. As in every model study, the present results depend on a number of approximations and assumptions. I want to recall the two principal assumptions used for building the models introduced here: (a) I have not directly considered the solvent and (b) the hydrogen bond oscillator has been assumed monodimensional. Someone can see these assumptions as a severe limitation, but I do not agree with this idea since these models can mimic many physical systems in real situations. In fact, the dissipative effect (due to the solvent) is of course important in real systems, but at larger times than these studied in this paper, where I have only analyzed the ultrafast dynamics of PCET systems. Only at these larger times, the periodic or quasi-periodic behavior of the dynamics is broken by the irreversible processes. On the other hand, a monodimensional oscillator for the hydrogen bond is appropriated in studying a frozen (due to a large mass involved) system, like those of biological interest. The main aim of this paper is to study the possibility of the decomposition of the PCET complex process in terms of proton and electron movements. In general this is possible only in particular cases. In fact, although the proton mass is largely greater than the electron mass it is not absolutely true that the electron is faster than the proton, in general. Hence, the adiabatic approximation [28,29] for these two particles in this transfer process appears dangerous. In this study, we have found some interesting results that can be schematized as follows: 1. In the model with PT and ET quantum processes and a time-dependent oscillator (classical picture) for describing inner or external degrees of freedom of the system, it is possible to find compact expressions of the characteristic times in reasonable approximations (Eq. (4) and Fig. 1). 2. When the donor and the acceptor of the electron are the electronegative atoms that form the hydrogen bond, it is reasonable to consider one oscillator for the vibrational part. This oscillator can be assumed to have an asymmetric double well potential energy due to the presence or not of the electron on the corresponding site. In this case there is a faster ET process that modulates a slower PT one, but the two processes are strictly connected (Fig. 2). 3. In the case of two diabatic states (only tunneling D– A) there is a sequential PT of the ET process, but the

321

fast proton transfer occurs only in the empty diabatic state (Figs. 3(a) and (b)). 4. When there is a transfer between a symmetric oscillator and an asymmetric one, a spontaneous proton localization can be found (Fig. 3(c)). 5. The contemporary presence of direct and indirect (through bridge) donor–acceptor electron transfer can generate a periodicity at a long time (Figs. 5, 7 and 11). 6. The contemporary presence of direct and indirect donor–acceptor electron transfer can generate constructive and destructive interferences (Fig. 8). The models of this paper are very useful in the study of the PCET process itself. A number of physical systems can be studied by these models and some examples have been given in this paper. In general, when a physical system has a strong coupling between the PT and ET processes these simple models can be utilized for a qualitative analysis and for giving previsions without a complete study.

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