Time-dependent quantum wave-packet description of the1pr* photochemistry of pyrrole

Valerie Vallet,*a Zhenggang Lan,a Susanta Mahapatra,b Andrzej L. Sobolewskic and

Wolfgang Domckea

a Chair of Theoretical Chemistry, Department of Chemistry, Technical University of

Munich, D-85747 Garching, Germany. E-mail: valerie.vallet@ch.tum.deb School of Chemistry, University of Hyderabad, Hyderabad 500 046, Indiac Institute of Physics, Polish Academy of Sciences, PL-02668 Warsaw, Poland

Received 26th February 2004, Accepted 1st March 2004First published as an Advance Article on the web 7th July 2004

The photoinduced hydrogen elimination reaction in pyrrole via the conical intersection ofthe 1B1 (

1ps*) excited state with the electronic ground state has been investigated by time-dependent quantum wave-packet dynamics. A two-dimensional model potential-energysurface has been constructed as a function of the NH stretching and the hydrogen out-of-plane bending mode, employing multi-reference ab initio electronic-structure methods. Thebranching ratio of the reactive flux at the conical intersection has been investigated independence on the initial vibrational state of the molecule. The results suggest that lasercontrol of the photodissociation of pyrrole via mode-specific vibrational excitation shouldbe possible.

1 Introduction

As the building blocks of many biologically relevant compounds such as porphyrines and chlor-ophylls, pyrrole and its derivatives play a prominent role in the synthesis of biologically activecompounds, pesticides and organic polymers. The photophysics of pyrrole is therefore a topic offundamental interest. The experimental UV absorption spectrum of pyrrole exhibits a broadintense band centred around 6 eV.1 Interestingly, these strongly absorbing states have a very lowquantum yield of fluorescence. This indicates the existence of very fast nonradiative processeswhich efficiently quench the fluorescence.2

The photophysics of pyrrole involves essentially the four lowest excited singlet states, whichare 1A2(ps*),

1B2(pp*),1B1(ps*) and 1A1(pp*) in order of increasing energy.3 The second and

the fourth excited singlet state, which correspond to valence excitations from p to p* orbitals,are strongly UV absorbing states, which explains the presence of the intense band around 6 eV inthe absorption spectrum. The first and the third excited singlet state correspond to excitations fromthe valence p orbitals of pyrrole to the Rydberg-like 3s orbital. These electronic transitions possessvery small oscillator strengths.4 As discussed by Sobolewski and Domcke,5,6 the 3s Rydberg orbitalof pyrrole has a significant anti-bonding character with respect to theNHbond (s* character). Uponstretching of the NH bond, the 1ps* potential-energy functions intersect the potential functions ofthe valence excited states, resulting in conical intersections along the reaction path for hydrogenabstraction. Due to the presence of these conical intersections between the bright valence states andthe dark 1ps* states, the population of the bright states is transferred to the dark states on a very

DOI: 10.1039/b402979h Faraday Discuss., 2004, 127, 283–293 283

This journal is # The Royal Society of Chemistry 2004

short time scale of �20 fs. This has been confirmed recently by Koppel et al.7 for pyrrole andrelated five-membered heterocycles.

The 1ps* states of pyrrole also cross the S0 ground state, inducing two conical intersections(1A2–S0 and 1B1–S0) along the NH dissociation path, which may provide the mechanism ofinternal conversion to the ground state.5,6 Blank et al.8 and Wei et al.9 have investigated thephotodissociation of pyrrole by photofragment translational spectroscopy. They observed twocharacteristically distinct kinetic-energy distributions of H atoms. They assigned the slow H atomswith a broad velocity distribution to the unimolecular decay process in the electronic ground stateafter internal conversion. They interpreted the fast H atoms as arising from fast direct N–Hbond breaking in excited electronic states, in particular the lowest 1ps* excited state.9

In this paper, we describe our first attempts towards an ab initio based quantum wave-packetdescription of the photoinduced hydrogen elimination and internal conversion dynamics of pyrrolevia the 1ps* states. Multi-reference electronic-structure methods have been employed to identifythe photochemically relevant molecular degrees of freedom, in particular the coupling modes10 ofthe 1ps*–S0 conical intersections. In the present work, we focus on the 1B1(ps*) state. Althoughthis state is higher in energy than the 1A2(ps*) state, its photochemical dynamics is more interestingowing to its similarity with the photochemistry of the lowest 1ps* state of biologically even morerelevant molecules, such as indole, tryptophan and adenine. In the latter systems, the out-of-planeangle of the hydrogen atom of the azine group has been identified as the relevant coupling mode ofthe 1ps*–S0 conical intersection,

5,6,11 like in the 1B1–S0 conical intersection in pyrrole (the couplingmode of the 1A2(ps*)–S0 intersection of pyrrole, on the other hand, corresponds to a screwing ofthe five-membered ring). We have constructed an accurate two-dimensional (NH stretch coordi-nate, NH out-of-plane angle) ab initio potential-energy surface of the 1B1(ps*) and

1A1(S0) states ofpyrrole. The essential features of the ultrafast nonadiabatic dynamics at the conical intersection areexplored, using standard methods of time-dependent quantum wave-packet propagation.

2 Theoretical framework

2.1 Reduced-dimensionality model

We treat the wave-packet dynamics of pyrrole in a two-dimensional space which is spanned by theNH-stretching coordinate (which tunes the energy gap between the ground and 1ps* states), andthe out-of-plane bending motion of the hydrogen atom of the azine group (which couples the twostates). The other internal coordinates are kept frozen at their ground-state equilibrium values.Some of these modes may play a role as energy-accepting modes in the internal-conversion process.It should be kept in mind that these effects are not included in the present dynamics simulations.

The Hamiltonian of the system is written in a diabatic electronic basis as

H ¼ HN þHel ð2:1Þ

H ¼ TN 1 00 1

� �þ V11 V12

V21 V22

� �; ð2:2Þ

where HN and Hel represent the nuclear and the electronic part of the Hamiltonian operator,respectively. TN is the nuclear kinetic-energy operator. V11 and V22 are the potential energies of thediabatic S0 and 1ps* states, respectively. The diabatic coupling element V12 ¼ V21 describesthe coupling between the two states. It vanishes for planar geometries (C2v) by symmetry.

The system is considered as an effective three-body system, composed of hydrogen, nitrogen andM, the centre of mass of the rest of the rigid pyrrole ring. The nuclear geometry is described interms of Jacobi coordinates (see Fig. 1a). rMN is the distance between M and the nitrogen atom.This distance is kept fixed in the dynamics calculation. r is the distance between the active hydrogenand the centre of mass of the M–N unit, and g is the angle between r~MN and r~. In this set ofcoordinates, the kinetic operator takes the following form

TN ¼ � �h2

2m@2

@r2� �h2

2I

@2

@g2; ð2:3Þ

284 Faraday Discuss., 2004, 127, 283–293

with

m ¼ mH mM þmNð ÞmH þmM þmNð Þ ; mMN ¼ mMmN

mM þmNð Þ ; mM ¼ 4 mC þmHð Þ; 1

I¼ 1

mr2þ 1

mMNr2MN

:

ð2:4Þ

The quantities mH , mN and mM denote the masses of hydrogen, nitrogen, and the four C–H pairs,respectively, and I is the moment of inertia of the total system.

2.2 Ab initio calculation of the adiabatic potential-energy surfaces

The 1B1(1ps*) state has a 3s Rydberg character at the equilibrium geometry, but rapidly acquires H

(1s) character when hydrogen atom dissociates.5,6 This change of character is reflected by a smallbarrier (�0.26 eV) in the 1ps* potential-energy function close to the equilibrium geometry(rNH� 1.25 A). In order to account for the diffuseness of the s* orbital, it is necessary to use ratherextended basis sets on each atom, and in particular on the N–H end. All atoms are described by theaugmented correlation-consistent polarised valence quadruple zeta (aug-cc-pVDZ)12 basis. Onediffuse s function and one set of diffuse p functions were added to the nitrogen, and two diffuse sfunctions and two sets of diffuse p functions were added to the active hydrogen. Their exponentswere derived in an even-tempered manner from the most diffuse s and p functions present in theaug-cc-pVDZ basis by dividing the exponents successively by a factor of 3.0.As a reference for the definition of internal coordinate displacements, the ground-state equili-

brium geometry has been determined at the Møller–Plesset (MP2) level of theory using theGaussian 98 ab initio package.13 The ab initio calculations of the ground and excited state potential-energy surfaces have been performed at the CASSCF level using the MOLPRO package.14 Theactive space involves three p orbitals (1b1� 2b1 , 1a2), two p* orbitals (3b1 , 2a2), and one s* (10a1)orbital. It turned out to be necessary to include three occupied orbitals of s character (8a1–9a1 ,6b2), and three additional virtual orbitals of s* character (11a1� 12a1 , 7b2) to obtain smoothpotential-energy surfaces in the region of the barrier of the 1ps* surface. This active spacecorresponds to a distribution of 12 electrons in 12 orbitals.The N–H distance has been varied between 1.9 a0 and 9.0 a0 , with a high density of points (steps

of 0.1 a0) in the region of the barrier of the 1ps* state and near the conical intersection. The out-of-plane bending angle has been varied from 0� to 70�, in steps of 5�.

2.3 Analytical representation of the diabatic potential functions

We wish to construct an analytical diabatic potential model which should be as simple as possibleand at the same time should reproduce the ab initio potential-energy data within an accuracy of 0.1

Fig. 1 Definition of the Jacobi (a) and internal (b) coordinates for H atom dissociation in pyrrole.

Faraday Discuss., 2004, 127, 283–293 285

eV or better. For the description of the potential-energy surfaces, the intermolecular distance rNH

between nitrogen and hydrogen and the out-of-plane bond angle y of the hydrogen atom arethe most natural coordinates (see Fig. 1b). We construct the potential-energy surfaces in thesecoordinates, while the wave-packet propagation is performed in the Jacobi coordinates.

Details of the choice of the analytical expansions and the fitting procedure are described in aforthcoming publication.15 The S0 diabatic potential function is approximated as

V11 ¼ D1e 1� exp �a1 rNH � r1ð Þð Þ½ �2 þ 1

2k1 rNHð Þy2; ð2:5Þ

with

k1 rð Þ ¼ p1 þ p2rð Þ 1� f1 rð Þð Þ þ p3 exp � r� d1ð Þa1

� �f1 rð Þ; ð2:6Þ

f1 rð Þ ¼ 1

21þ tanh

r� d2b1

� �� �: ð2:7Þ

The 1ps* diabatic potential function is modelled as

V22 rNHð Þ ¼ 1

2V in

22 rNHð Þ þ Vout22 rNHð Þ

� �� 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðV in

22 rNHð Þ � Vout22 rNHð Þ

� 2 þ 4l222

q

þ 1

2k2 rNHð Þy2;

ð2:8Þ

with

V in22ðrÞ ¼ E2

0 þD2;ine ½1� expð�a2ðr� r2ÞÞ�2; ð2:9Þ

Vout22 rð Þ ¼ D2

e þ a3 exp � r� r3ð Þa2

� �; ð2:10Þ

and

k2 rð Þ ¼ 1

2q1 þ q2rð Þ � 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiq3 þ q2rð Þ2 þ 4q24

q: ð2:11Þ

k2(r) is set to zero when r is larger than 2.55 a0 .The ansatz for the non-adiabatic coupling term reads:

V12 ¼ l12ðrÞy; ð2:12Þ

with

l12ðrÞ ¼1

2lmax 1� tanh

r� d12b2

� �� �: ð2:13Þ

Table 1 gives the parameters appearing in eqns. (2.5)–(2.13). Note that the diabatic potential-energy functions are nonseparable in r and y. The adiabatic potential-energy functions acquire anadditional pronounced nonseparability through the conical intersection. The maximum deviationbetween the adiabatic model potential-energy and the ab initio data is 0.05 eV. The resultingdiabatic and adiabatic potentials as functions of the internal coordinates rNH and y are displayed inFigs. 2a and 2b, respectively.

286 Faraday Discuss., 2004, 127, 283–293

2.4 Wave-packet propagation method

The photoinduced dynamics of pyrrole is treated in the time-dependent picture by solving the time-dependent Schrodinger equation with a wave-packet propagation method on a two-dimensionalgrid. The initial wave function C(t ¼ 0) is given, in the Condon approximation, by the initialvibrational state in the electronic ground state. The excited-state wave function is propagated usingthe split-operator (SO) method.16 We have used the fast Fourier transform (FFT) method17 toevaluate the kinetic-energy part of the time propagator. To avoid problems with the finite range ofthe grid, we apply a damping function as defined in ref. 18. The time dependence of the adiabaticelectronic populations is calculated from the expectation values of the projection operators in therespective electronic states.19

We have generated the eigenfunctions of the diabatic S0 state by using the spectral-quantisationmethod detailed in ref. 20. The dissociation probabilities have been calculated using as the time-cumulated flux through a dividing surface r ¼ rflux , which is placed beyond the conical intersection:

PR tð Þ ¼ �h

m

Z t

t0¼0

Im C rflux; g; t0ð Þ @C rflux; g; t0ð Þ@r

� �� �

dt0: ð2:14Þ

The energy-resolved reaction probability is defined in a similar way

PR Eð Þ ¼ �h

mIm C rflux; g;Eð Þ @C rflux; g;Eð Þ

@r

� �� �

; ð2:15Þ

Table 1 Numerical values of the parameters in the diabatic model

V11 V22 V12

E20 ¼ 5.584 eV lmax ¼ 2.4 eV

D1e ¼ 5.117 eV D2;in

e ¼ 8.070 eV d12 ¼ 3.454 aur1 ¼ 1.959 au r2 ¼ 1.922 au b2 ¼ 1.942 aua1 ¼ 1.196 au a2 ¼ 0.882 aup1 ¼ 5.147 eV D2

e ¼ 4.092 eVp2 ¼ �1.344 eV au�1 a3 ¼ 0.091 eVp3 ¼ 0.884 eV r3 ¼ 5.203 aua1 ¼ 0.775 au a2 ¼ 0.774 aub1 ¼ 0.00015 au q1 ¼ 3.818 eVd1 ¼ 3.100 au q2 ¼ �1.219 eV au�1

d2 ¼ 2.696 au q3 ¼ 2.444 eV 1/2

q4 ¼ 0.226 eV 1/2

l22 ¼ 1.669 eV

Fig. 2 Diabatic (V11 ,V22) (a) and adiabatic (b) potential-energy surfaces of the S0 and1ps* states as a function

of the NH stretching coordinate rNH and the H-out-of-plane coupling motion y.

Faraday Discuss., 2004, 127, 283–293 287

using the energy-normalised wave function C(r,g,E), which is obtained by Fourier-transformingthe time-dependent wave packet on the dividing surface:

C rflux; g;Eð Þ ¼ 1ffiffiffiffiffiffi2p

pZ 1

�1C r; g; tð ÞeiEt=�hdt r¼rfluxj : ð2:16Þ

3 Results and discussion

3.1 Vibrational eigenstates of the electronic ground state

We first report the energy spectrum of the diabatic S0 state. An initial Gaussian wave packet isdefined as

C t ¼ 0ð Þ ¼ N exp � r� r0ð Þ2

2s2r

" #exp � g� g0ð Þ2

2s2g

" #; ð3:17Þ

where N is the normalisation constant. The parameters are chosen such that this wave packet isinitially shifted (r0 ¼ 4.55 a0 , g0 ¼ 0.25 rad) with respect to the equilibrium geometry re ¼ 4.15 au,g ¼ 0, with sr ¼ 0.3 au and sg ¼ 0.25 au along r and g, respectively. The grid consists of 128� 128points, ranging from 3 a0 to 6 a0 along r and �2.0 rad to 2.0 rad along g. This grid is large enoughto ensure that the wave function does not reach the grid boundaries during the propagation. Thetotal propagation time is 1.0 ps with a time step Dt ¼ 0.05 fs. No damping function was employedin this calculation.

The energy eigenvalues of the 15 lowest vibrational states and their assignments in terms ofquantum numbers are given in Table 2. The fundamental of the coupling mode is 850 cm�1 whenthe stretching mode is not excited. It decreases to 834 cm�1 for v ¼ 1 of the stretching mode. Thefundamental of the stretching mode is 3602 cm�1. The experimental values (see ref. 21 and refer-ences therein) of the fundamentals of the NH-stretching and out-of-plane bending are 3531 cm�1

and 475 cm�1, respectively. While our diabatic surface reproduces the stretching fundamental withgood accuracy, the calculated bending fundamental is significantly too large. This effect reflects thestrong coupling of the 1B1(ps*) and S0 states via the out-of-plane bending mode: the repulsion ofthe surfaces upon bending flattens the adiabatic S0 surface.

3.2 Wave-packet dynamics at the 1pr*–S0 conical intersection

The main focus of the present work is on the investigation of the effect of mode-specific vibrationalexcitation on the dynamics near the conical intersection. We have employed the same grid along gand the same grid spacing along r as for the generation of the vibrational eigenstates of the S0 state.The grid along r has been extended up to 27.17 a0 , with a total number of 1024 grid points. Theabsorbing function was placed at 26.5 a0 . The wave function is propagated for 300 fs with a timestep Dt ¼ 0.05 fs. The total propagation time is long enough to ensure that the probabilities of the

Table 2 Eigenvalues (En) and assignments (nr , ng) of thepseudospectral peaks of the diabatic ground state

En/cm�1 (nr ,ng) En/cm

�1 (nr ,ng)

2302 (0,0) 7372 (0,6)3152 (0,1) 7572 (1,2)4019 (0,2) 8206 (0,7)4853 (0,3) 8389 (1,3)5704 (0,4) 9023 (0,8)5904 (1,0) 9206 (1,4)6538 (0,5) 9323 (2,0)6738 (1,1)

288 Faraday Discuss., 2004, 127, 283–293

different channels are converged. The probabilities are monitored by the calculation of thetime-cumulated flux and energy-resolved flux on the S0 and

1ps* states before the conical inter-section (r ¼ 5 a0) and at the dissociation limits (r ¼ 9 a0).We use the 15 lowest eigenfunctions discussed in the previous section as initial wave functions,

placing them vertically in the 1ps* diabatic state. For all initial conditions, we observe similarfeatures in the evolution of the excited-state wave function. In all cases, we note very fast, but weakoscillations (Rabi-type oscillations) in the diabatic population of the 1ps* and S0 states, see Fig. 3,which result from the non-zero coupling of these two states in the Franck–Condon region. Due tothe existence of a barrier in the 1ps* potential-energy surface, there are two ways for the wavepacket to escape the potential well in which it is placed by vertical excitation. The high-energy partof the wave packet overcomes the barrier and reaches the conical intersection directly within about10 fs. The low-energy part escapes the potential well by tunnelling, which happens on a longer timescale. These two processes can be distinguished via the profile of the population dynamics and thetime-cumulated flux, see Figs. 3 and 4.When the wave packet reaches the region of the conical intersection, it splits into two parts. One

part follows a diabatic path and moves towards the lower adiabatic dissociation limit, as illustratedby the flux calculated on the lower dissociation channel, see Fig. 4. The other part is transferred toupper adiabatic surface, see Fig. 3. It dissociates directly to the higher adiabatic limit without beingtrapped in the upper adiabatic cone (see Fig. 4). The reason is that the initial mean energy of thewave packet is higher than both dissociation limits. Therefore, the wave packet passes the conicalintersection region only once, as revealed by the monotonous population dynamics in the adiabaticpicture. The probability of internal conversion, that is, return of the system to the electronic groundstate without dissociation, is close to zero in the present model. This finding does not reflect the realsituation for two reasons. First, we have neglected the coupling of the highly excited 1B1(ps*) statewith the lower-lying 1A2(ps*) and

1B2(pp*) excited states. Second, we have neglected intramole-cular vibrational relaxation (IVR) effects, arising from the anharmonic coupling of the photo-chemically active with the many inactive vibrational degrees of freedom. Extensions of the modelare required to account for these effects.In order to shed light on the influence of mode-selective vibrational excitation on the branching

ratio of two dissociation channels, we discuss the flux at the dissociation limits for different initialstates. If we put one quantum of energy into the tuning mode, initial vibrational state (1,0), thewave packet reaches the conical intersection earlier than in the (0,0) case and the populationtransfer occurs on a short time scale, see Fig. 3c. It is noteworthy that the branching ratio does notdepend on excitations along the tuning mode (see Fig. 5(a)).The branching of the reactive flux at the conical intersection depends, on the other hand,

strongly on the excitation level of the coupling mode, as shown by Figs. 5(b) and 5(c). For the (0,0)initial wave function, most of the wave packet (75%) dissociates to the lower adiabatic limit, while25% of the flux arrives at the upper adiabatic limit. When exciting the coupling mode by onequantum, the flux splits almost equally between the two channels. If we put more energy into thecoupling mode, the wave packet dissociates primarily to the higher adiabatic limit. The branchingratio saturates at 25%/75% for ng ¼ 4. Similar observations can be made for the series (1, ng),ng ¼ 0,4; see Fig. 5(c). In summary, we conclude that excitations along the coupling mode have astrong impact on the behaviour of the wave packet at the conical intersection. This fact can be usedto tune the branching ratio between the two dissociation limits.In these time-dependent wave-packet calculations for a two-mode model of a 1ps*–S0 conical

intersection, we have pointed out that upon excitation of the coupling mode, a significant amountof the population remains in the upper diabatic state and dissociates rapidly. This observationreflects the fact that the excess energy cannot be absorbed by other degrees of freedom. It has beenfound that inclusion of additional modes generally induces a vibrational damping of the modesactively involved in the photodynamics.22 This results in a lowering of the long-time limit of thepopulation of the upper adiabatic state due to vibrational relaxation in the lower adiabaticpotential-energy surface. For our present model, this would imply a reduction of the population ofthe higher dissociation channel and an increased probability for internal conversion. However, thecalculation of the minimal-energy reaction path shows that other degrees of freedom are notstrongly coupled with the NH-stretch coordinate and remain close to their values at the ground-state equilibrium geometry of pyrrole. As a consequence, our findings that excitations of the

Faraday Discuss., 2004, 127, 283–293 289

Fig. 3 Population probabilities of the diabatic 1ps* state and ground state (full lines). The populationprobabilities of the upper adiabatic S1 and lower adiabatic S0 states are shown by the lines with crosses andcircles, respectively, for the initial conditions (0,0) (a), (0,1) (b) and (1,0) (c). The insets display the diabaticpopulation of the 1ps* state for the first 20 fs.

290 Faraday Discuss., 2004, 127, 283–293

coupling mode influence the branching ratio between the two dissociative channels is expected to bequalitatively correct.We have studied in this work, for simplicity, a two-state model, neglecting other low-lying

excited states. Due to the presence of multiple conical intersections between the bright 1pp* statesand the two dark 1ps* states in the Franck–Condon region, the optically prepared wave packetmay have a complicated shape once it arrives in one of the 1ps* states by internal conversion fromthe 1pp* states. By using different initial conditions in our simulations, we wanted to understand

Fig. 4 Time-cumulated flux measured on the upper S1 (lines with crosses) and lower S0 (lines with circles)adiabatic dissociation limits, for initial conditions (0,0) (a), (0,1) (b) and (1,0) (c).

Faraday Discuss., 2004, 127, 283–293 291

the evolution of different components of this possibly complex wave packet. The investigation ofthe wave-packet dynamics at the lower 1A2(ps*)–S0 conical intersection is in progress.

4 Conclusions

As a first step towards a first-principles mechanistic understanding of the 1ps*-driven photo-chemistry of aromatic biomolecules, we have investigated the quantum wave-packet dynamics of atwo-dimensional model of the 1B1(ps*) excited state of pyrrole and its conical intersection with theelectronic ground state. The photochemically reactive NH stretching mode and the vibronicallyactive hydrogen out-of-plane bending model have been taken into account. We have investigated,in particular, the dependence of the branching ratio of the two hydrogen dissociation channels(corresponding to the 2p and 2s pyrrolyl radicals) on the initial vibrational state. A considerabledependence of the branching ratio on the level of excitation of the bending mode has been found.This finding indicates that a control of the photodissociation dynamics of pyrrole via mode-specificIR excitation, as has recently been demonstrated for ammonia,23 should also be possible forpyrrole.

Acknowledgements

V.V acknowledges support by an Alexander von Humboldt research fellowship. S.M acknowledgesthe Alexander von Humboldt Stiftung for the Friedrich Wilhelm Bessel Forschungspreis. Thiswork has been supported by the Deutsche Forschungsgemeinschaft and the Fonds der ChemischenIndustrie.

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292 Faraday Discuss., 2004, 127, 283–293

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