Molecular Dynamics Simulations of PorphyrinâDendrimer Systems

J. Phys. Chem. B 2008, 112, 14779–14792

14779

Molecular Dynamics Simulations of Porphyrin-Dendrimer Systems: Toward Modeling Electron Transfer in Solution Pedro M. R. Paulo,* Jose´ N. Canongia Lopes, and Sı´lvia M. B. Costa Centro de Quı´mica Estrutural - Complexo I, Instituto Superior Tećnico, AV. RoVisco Pais, 1049-001 Lisboa, Portugal ReceiVed: July 31, 2008

We have performed computational simulations of porphyrin-dendrimer systemssa cationic porphyrin electrostatically associated to a negatively charged dendrimersusing the method of classical molecular dynamics (MD) with an atomistic force field. Previous experimental studies have shown a strong quenching effect of the porphyrin fluorescence that was assigned to electron transfer (ET) from the dendrimer’s tertiary amines (Paulo, P. M. R.; Costa, S. M. B. J. Phys. Chem. B 2005, 109, 13928). In the present contribution, we evaluate computationally the role of the porphyrin-dendrimer conformation in the development of a statistical distribution of ET rates through its dependence on the donor-acceptor distance. We started from simulations without explicit solvent to obtain trajectories of the donor-acceptor distance and the respective time-averaged distributions for two dendrimer sizes and diffferent initial configurations of the porphyrin-dendrimer pair. By introducing explicit solvent (water) in our simulations, we were able to estimate the reorganization energy of the medium for the systems with the dendrimer of smaller size. The values obtained are in the range 0.6-1.5 eV and show a linear dependence with the inverse of the donor-acceptor distance, which can be explained by a two-phase dielectric continuum model taking into account the medium heterogeneity provided by the dendrimer organic core. Dielectric relaxation accompanying ET was evaluated from the simulations with explicit solvent showing fast decay times of some tens of femtoseconds and slow decay times in the range of hundreds of femtoseconds to a few picoseconds. The variations of the slow relaxation times reflect the heterogeneity of the dendrimer donor sites which add to the complexity of ET kinetics as inferred from the experimental fluorescence decays. I. Introduction Photoinduced electron transfer is a fundamental process in biological systems, through its role in the primary step of photosynthesis, and also in artificial devices for solar energy conversion, like photovoltaic cells.1-14 The key factor is to use the photon energy to achieve efficient charge separation over long distances, thereby establishing a potential gradient that, in turn, can be converted into chemical energy, in the case of photosynthetic organisms, or into electric energy, in the case of photovoltaic cells. Examples of supramolecular systems that attempt to mimic the antenna effect or the long-range charge separation achieved in natural photosynthesis have been developed, and some of these involve organic dendrimers or dendrimer-like units.15-20 Dendrimers are highly regular and branched polymers, which assume a globular shape in solution with dimensions that are comparable to small proteins (some nanometers in diameter).21-25 The hierarchical growth of these structures and the large number of branching points allow for a controlled functionalization of dendrimers with donor or acceptor units. It is not unusual that one of these functional units is a porphyrin moiety, by analogy with the chromophores found in natural photosynthetic systems.26-31 Alternatively to covalent functionalization is the use of self-assembly strategies, and this also occurs in natural systems, e.g., in the lightharvesting complexes of photosynthetic bacteria. In previous studies, we investigated the noncovalent interactions between oppositely charged porphyrin and dendrimer * Corresponding author. E-mail: [email protected].

molecules in aqueous solution by means of optical spectroscopy and time-resolved fluorescence.32 The strong fluorescence quenching effect observed for a cationic porphyrin (TMPyP) in the presence of negatively charged (PAMAM) dendrimer was assigned to photoinduced electron transfer from one of the multiple tertiary amine groups within the dendrimer structure to the excited-state porphyrin.32c The excited-state decay kinetics of the porphyrin associated with the dendrimer are complex with ultrafast components in the subnanosecond time scale. This behavior was interpreted according to a static-disorder picture, in which case the conformational flexibility of the porphyrindendrimer pair would allow for a spread of donor-acceptor distances, thus leading to a statistical distribution of electrontransfer rates. In this sense, the multiexponential fluorescence decays of the porphyrin-dendrimer pair were fitted with a dispersive kinetics model. Considerable efforts have been dedicated to the investigation of electron-transfer processes, in general, from both experimental and theoretical standpoints.33-36 Electron transfer is an interesting probe reaction for the development and assessment of concepts in chemical reaction dynamics. Issues such as reaction energetics, adiabaticity, and dynamic solvent effects, coupling with intramolecular and medium modes, tunneling, and quantum interference effects have been addressed in the context of ET.37-45 The interdisciplinarity of research on ET is present through concepts borrowed from other fields, e.g., the use of nonadiabatic multiphonon radiationless transition theory to describe the energy-gap law observed in the Marcus inverted region38a,b or the use of the spin-boson model to describe the

10.1021/jp806849y CCC: $40.75  2008 American Chemical Society Published on Web 10/28/2008

14780 J. Phys. Chem. B, Vol. 112, No. 47, 2008 coupling of the medium (thermal bath) to the ET reaction.38c,39,46,47 The progress in computational simulation of chemical systems has introduced powerful possibilities for the theoretical study of ET reactions.48-60 In particular, it made it possible to evaluate “microscopic” parameters used in ET theory, which otherwise were estimated from “macroscopic” measurements, or fitted from experimental results, with the more or less obvious shortcomings involved. Semiclassical approaches have combined molecular dynamics simulations, to describe the low-frequency modes of the medium (solvent or protein motion) coupled to ET, with a quantum mechanical description of the highfrequency modes from intramolecular nuclear degrees of freedom, in the framework of the spin-boson (or dispersed polaron) model.49a,b,53a,b In the high-temperature limit, these studies confirm the validity of Marcus’ theory for ET, which assumes a unidimensional reaction coordinate defined by the energy fluctuation associated with a vertical displacement between the free energy surfaces of the reactant and product states. Molecular dynamics simulations have shown that the free energy profile along this coordinate can be parabolic with a similar curvature for the reactant and product states, as assumed in Marcus’ model.49c,d,50a,51,53a In this paper, we report on the results from a molecular dynamics simulation study of the porphyrin-dendrimer systems for which photoinduced electron transfer was experimentally observed. Computational simulations were used to extract detailed information about donor-acceptor distances and energy parameters from ET theory that contribute to a better understanding of electron transfer in the systems investigated here. The paper is organized as follows: the model employed in the simulations as well as the formalism used for the calculation of ET parameters are outlined in section II; the results from the porphyrin-dendrimer simulations without and with explicit solvent are presented in sections III and IV, respectively; finally, some discussion remarks are given in section V. II. Model and Methods Simulation Details. Molecular dynamics simulations were performed with an all-atom force field based on the OPLS and AMBER force fields, and complemented with DFT calculations to parametrize the atomic charges of the reactant and product sites in the simulated systems. The force field parameters used for the dendrimer were previously given in ref 61, while those used for the porphyrin are presented here as Supporting Information. The initial configurations were prepared by selecting equilibrated dendrimer structures from previous simulations. Two dendrimer sizes were considered here, and these are referred to as generation 2.5 (m.w. 6265.7 Da) and generation 4.5 (m.w. 262 52.1 Da). The porphyrin was added to the simulation box in a position close to the dendrimer’s surface. Two initial configurations were generated for each dendrimer size, as described next. In the case of generation 2.5, the two generated configurations differ in the dendrimer conformation and, therefore, configuration 1a corresponds to a dendrimer with slightly larger radius of gyration and aspect ratio (defined as the ratio between the principal moments of inertia Iy/Ix and Iz/ Ix, where Ix < Iy < Iz) than configuration 1b. In the case of generation 4.5, the dendrimer conformation is the same for both initial configurations, but the position of the porphyrin differs between them. In configuration 2a, the porphyrin is located within a cleft in the dendrimer structure and is pointing outward, while, in configuration 2b, the porphyrin is laying flat on the dendrimer surface. The structures of each initial configuration

Paulo et al.

Figure 1. Scheme of the monomer connectivity in a dendrimer of generation (gen) 2.5. The open circles represent tertiary amines (donor groups), and the closed circles represent the terminal carboxylate groups; the dashed lines indicate the concentric layers of monomers that constitute each of the subgenerations (sg). Reproduced with permission from ref 61. Copyright 2007 American Chemical Society.

are shown schematically in Figure 3 (section III) and in more detail in the Supporting Information. We notice that we are considering only two possible conformations for each porphyrin-dendrimer pair without making any assumption about its statistical significance. In view of the many degrees of freedom of macromolecular systems (like those considered here), it is not possible to assume that the configurational space is fully explored within a simulation time interval of nanoseconds. Nevertheless, useful information can still be extracted from MD simulations concerning the ET process and how it is affected, for instance, by conformational changes, dendrimer generation, position of donor sites, etc. In this sense, MD simulations provide, for each configuration considered here, one possible, consistent molecular and structural background where the ET process can be analyzed, and must not be regarded in this casesas usually MD studies aresas a way to derive, by a statistical mechanics algorithm, the equilibrium thermodynamic and structural properties of the system. The simulation box for the simulations without explicit solvent consisted of a cubic cell with a side length of 321.4 Å. The cell volume of 33 200 nm3 corresponds approximately to the concentrations used in experimental studies.32c This issue is relevant for highly charged polymers, like the dendrimers simulated here, because of the dependence of counterion condensation on concentration (or equivalently the volume of the simulation cell) and its effect on the surface charge density. In the absence of explicit solvent, the simulations were done assuming a dielectric continuum with the same relative permittivity of water at room temperature (ε ) 78.4, at 298 K) for the calculation of electrostatic interactions. A large cutoff distance of 50 Å was used for truncation of the long-range potential terms. Bond constraints were applied to bonds involving H atoms using the SHAKE algorithm. This allowed for an integration time step of 1 fs. Other simulation details are described in ref 61. The initial configurations were equilibrated for a period of 100 ps, and then, the simulations were run during a time interval of 1 ns. The atomic positions were recorded every

MD Simulations of Porphyrin-Dendrimer Systems

J. Phys. Chem. B, Vol. 112, No. 47, 2008 14781

Figure 2. Snapshots from simulations without explicit solvent of the porphyrin-dendrimer systems with configuration 1a (A, C) and configuration 1b (B, D). The upper images show the porphyrin (van der Waals scheme) at the surface of the dendrimer (wireframe representation) and its counterions (gray circles). The lower images show the relative positions of the porphyrin (wireframe structure in red) to the tertiary amines (circles: orange, r < 10 Å; green, 10 Å < r < 20 Å; blue, r > 20 Å) of the dendrimer (omitted for clarity).

0.1 ps, and from these, donor-acceptor distances were calculated as the distance separating the nitrogen atom of each tertiary amine group of the dendrimer from the center of mass of the porphyrin. The simulations with explicit solvent were performed using model TIP5P for the water molecules.62 To begin with, two simulation cells were defined with dimensions of 60.0 × 37.3 × 37.3 Å3 (rhombic) and 42.0 × 42.0 × 42.0 Å3 (cubic). The shapes of these cells were chosen to correspond, respectively, to configurations 1a and 1b of generation 2.5. The cubic and rhombic cells were filled with TIP5P water molecules and were subjected to 20 ps of equilibration followed by 30 ps of simulation at constant temperature (298 K) and pressure (1 bar) using a Hoover barostat with time constants of τT ) 1.0 ps and τP ) 4.0 ps. From the simulations without explicit solvent, one example of configuration 1a was sampled and the porphyrindendrimer complex was superimposed to the water simulation cell with rhombic shape in a centered position. The same procedure was done for configuration 1b in relation to the cubic water cell. Next, the water molecules overlapping with the porphyrin-dendrimer complex (i.e., with intersecting van der Waals surfaces of the respective atoms) were erased from the simulation cell, as well as the porphyrin or dendrimer counterions lying outside the boundaries of the water cell. After this procedure, each simulation cell contained 1 porphyrin molecule, 1 dendrimer, 12 sodium counterions, and about 2000 water molecules. These systems were equilibrated for another 50 ps, and the final dimensions of 55.3 × 34.3 × 34.3 Å3 and 40.9 × 40.9 × 40.9

Å3 were reached for the rhombic and cubic cells, respectively. The outcome configurations from the described equilibration steps were assumed as the initial configurations for the simulations with explicit solvent. These simulations were performed in the microcanonical ensemble, i.e., without applying thermostat or barostat schemes. The Ewald sum method was employed for the calculation of the electrostatic interactions with a cutoff distance of 9 Å, and a Verlet radius of 1 Å. This cutoff distance assured that electrostatic interactions from images of the porphyrin-dendrimer complex in neighboring cells (periodic boundary conditions) were negligible. Calculations Applied to Electron Transfer. The reorganization energy of the medium was calculated using a procedure that defines the energy difference between the charge distributions of the reactant and product states as the reaction coordinate for the classical nuclear degrees of freedom coupled to the ET reaction. The exact formalism was introduced by Schulten and co-workers within the framework of a two-state Hamiltonian model that accounts for contributions from both high-frequency modes described quantum mechanically and low-frequency classical modes of the medium coupled to ET.53a,b Here, we outline the features of this model that relate fundamental parameters from ET theory, like the reorganization energy of the medium or the spectral function J(ω), to quantities which can be assessed from molecular dynamics simulations. In this sense, the energy difference ∆E(t) ) EP[q(t)] - ER[q(t)], where ER[q(t)] and EP[q(t)] are the Coulomb energies of the simulated system for the reactant and product states, respectively, describes dielectric fluctuations and relaxation of the

14782 J. Phys. Chem. B, Vol. 112, No. 47, 2008

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Figure 3. Trajectories of the donor-acceptor (D-A) distance for some tertiary amines (the three closest to the porphyrin) calculated from the simulations of the porphyrin-dendrimer systems without explicit solvent: (A, B) configurations 1a and 1b of generation 2.5; (C, D) configurations 2a and 2b of generation 4.5. The schemes above each set of trajectories show the initial configurations of the simulated systems. The porphyrin is depicted in red using spacefill representation, and the dendrimer is depicted in black using wireframe representation.

medium upon electron transfer. The quantities ER[q(t)] and EP[q(t)] were evaluated first for configurations q(t) belonging to the potential energy landscape of the reactant state by using the charge distribution of the reactant state to propagate the molecular dynamics trajectory. Every 10 fs, the potential energy term accounting for electrostatic interactions was calculated for the sampled configuration with the charge distribution of the reactant state, ER[q(t)], and of the product state, EP[q(t)]. This procedure was carried for a simulation period of 25 ps. Then, the charge distribution was changed (permanently) to the product state and the system was allowed to evolve to the potential energy landscape of the product state during 5 ps, as dielectric relaxation in response to electron transfer occurs. Afterward, the quantities ER[q(t)] and EP[q(t)] were evaluated for the product state, following the procedure described above, by sampling configurations every 10 fs for a period of 20 ps. From these results, the trajectories ∆E(t) and the respective normalized distributions were calculated. The latter quantity can be identified with the line-shape function Scl(∆E)

of the classical modes coupled to ET, which usually can be described by a single Gaussian function53a

Scl(∆E) )

1

√πΣ

[

exp -

(〈∆E〉 - ∆E)2 Σ

]

(1)

Assuming that the effective energy potentials for reactant and product states are given by two harmonic potentials with similar force constants f and displaced along the reaction coordinate by a quantity ∆, like in Marcus’ model of ET, then the reorganization energy parameter can be identified with

λ ) 1/2 f∆2 ) 1/2(〈∆E〉R - 〈∆E〉P)

(2)

where 〈∆E〉R and 〈∆E〉P are the mean values of ∆E before and after electron transfer, respectively.



The spin-boson model for ET assumes that coupling between electron transfer and thermal motion of the medium is described through an ensemble of linear independent harmonic oscillators, from which only an average quantity, the spectral function J(ω), is required to be known47

J(ω) )

π 2

c

2

∑ mRRωR δ(ω - ωR)

(3)

R

where mR, ωR, and cR are the mass, frequency, and coupling constant of the Rth oscillator. The spectral function J(ω) can be evaluated from molecular dynamics simulations through the relationship

J(ω) σ2 ) ω kBT

∫0∞ C(t) cos(ωt) dt

(4)

which is the cosine transform of the energy-energy correlation function

C(t) )

〈∆E(t) ∆E(0)〉 - 〈∆E〉2 〈∆E2〉 - 〈∆E〉2

(5)

The quantity σ2 appearing in eq 4 is the variance of the energy fluctuations, and it can be identified with the parameter Σ from expression 1. The correlation function of the ∆E trajectories obtained from the simulations and the respective cosine transforms were calculated with routines implemented in FORTRAN. The charge distributions of the simulated porphyrin in the initial and reduced forms (respectively, reactant and product states) were determined from DFT calculations.63 First, the geometries were optimized with the basis set D95V* and the exchange/correlation functional B3LYP. In the initial form, the porphyrin ring is planar, while in the reduced form a slight distortion to a saddle-type conformation occurs (see the Supporting Information). Thus, the vibrational modes of the porphyrin that promote out-of-plane distortion of the pyrrole rings are strong candidates for intramolecular modes coupled to ET. After geometry optimization, the atomic charges were fitted to reproduce the surface of electrostatic potential around the porphyrin molecule using the CHELPG algorithm. A similar procedure was employed for the tertiary amine (donor group) of the simulated dendrimers. In this case, however, only a small segment of the dendrimer structure (containing a tertiary amine group) was optimized with DFT calculations. As expected, the geometry of the tertiarty amine changes from pyramidal to planar around the central nitrogen atom upon oxidation (see the Supporting Information). Again, the vibrational modes of the dendrimer that promote planarization of the tertiary amine groups are strong candidates for intramolecular modes coupled to ET. The atomic charges fitted with the CHELPG algorithm were used, together with the atomic charges from the OPLS force field, to define a set of charge displacements associated with oxidation to the (+1) state of a single tertiary amine within the dendrimer structure. Some illustrative calculations of ET rates, using the parameters obtained from the simulations, were performed with the semiclassical Marcus model for ET33b,c,37b

kET )

[

2πV2 (∆G° + λ)2 exp 4λkBT p(4πλkBT)1/2

]

(6)

where V is the electronic coupling matrix element, λ is the reorganization energy, and ∆G° is the free Gibbs energy change for the ET reaction (other quantities have the usual meaning). We assumed a value of V0 ) 10 cm-1 for the electronic coupling at contact distance, which is small enough to justify the nonadiabatic approximation contained in expression 6 but at the same time large enough to shorten the excited-state decay time of the porphyrin to subnanosecond values, as experimentally observed.32c For the dependence on the donor-acceptor distance r, we assumed V ) V0 exp[-β/2(r - r0)], where r0 is the contact distance and β is the damping constant. Here, we considered a typical value of β ) 1 Å-1 and a contact distance of r0 ) 3 Å. In this case, the donor-acceptor distance was defined as the separation between the positions of the closest carbon or nitrogen atom of the porphyrin ring and the central nitrogen atom of the dendrimer’s tertiary amines.33b For the reorganization energy, we ignored the contribution of intramolecular modes and used only the values of the reorganization energy of the medium estimated from the simulations with explicit solvent (section IV). We assumed a linear dependence of λ with the inverse of the donor-acceptor distance, except at very short distances in which case we assumed that λ levels off at a value of 0.6 eV. In this case, the donor-acceptor distance was defined as the separation between the center of mass of the porphyrin and the position of the nitrogen atom in the dendrimer’s tertiary amines. The free Gibbs energy change was ° estimated from the Rehm-Weller equation ∆G° ) ETEA ° - ∆E0-0 ) -0.389 eV, with the redox potentials, ETMPyP ° ) 1.21 V for triethylamine and ETMPyP ° ETEA ) -0.23 V for meso-tetrakis(4-N-methylpyridinium)porphine, and the respective electronic excitation energy, ∆E0-0 ) 1.83 eV, taken from the literature.32c Born-type corrections were not considered in the illustrative rate calculations. III. Simulations without Explicit Solvent The dendrimer of generation 2.5 contains 30 tertiary amine groups distributed throughout its structure according to the chemical topology shown in Figure 1. Dendrimers grow by selfsimilarity, and thus, in generation 4.5, this number increases to 126 tertiary amines. Each tertiary amine is a potential electron donor for the excited-state porphyrin with different efficiencies according to their relative distance. The distribution of distances, however, is very different from that suggested in Figure 1 due to conformational flexibility of the segment chains that connect the branching points. The simulated dendrimers assume a globular conformation (Figure 2A,B) in solution, which evolves from a prolate ellipsoidal shape (low generations) to a spherical one (high generations) with increasing dendrimer size, due to molecular packing of the exponentially increasing number of branches.21b,23 The exact position of the tertiary amines within the dendrimer structure depends on its conformation (Figure 2C and D). However, on average, there are 7-9 tertiary amines within a distance of 10 Å from the porphyrin in both dendrimer generations. Due to their proximity, these amines should give the highest ET rates and be responsible for most of the fluorescence quenching, while tertiary amines at distances greater than 10 Å should have only a minor effect. For instance, considering only the distance dependence of the electronic coupling V and using a typical value of β ) 1 Å-1 with a contact


Paulo et al.

TABLE 1: Average Values of the Radius of Gyration Rg, of the Principal Moment of Inertia Ix, and of the Aspect Ratios Iy/Ix and Iz/Ix for the Simulated Dendrimers gen. 2.5

config. 1a config. 1b

gen. 4.5 a

config. 2a or 2b

no w/ no w/ no w/

porph.a porph.b porph. porph. porph. porph.

Rg (Å)

Ix (103 amu Å2)

Iy/Ix

Iz/Ix

11.2 ( 0.2 11.9 ( 0.4 10.6 ( 0.1 10.5 ( 0.2 17.6 ( 0.1 17.4 ( 0.2

270 ( 7 243 ( 10 266 ( 12 288 ( 7 4079 ( 59 3628 ( 53

2.0 ( 0.2 2.8 ( 0.2 1.7 ( 0.1 1.5 ( 0.1 1.2 ( 0.0 1.4 ( 0.1

2.2 ( 0.2 3.0 ( 0.2 1.9 ( 0.1 1.7 ( 0.1 1.3 ( 0.0 1.5 ( 0.1

Results from simulations without porphyrin (see ref 61). b Results from simulations with porphyrin (this work).

distance of r0 ) 3 Å, the estimated ET rates (eq 6) for donor-acceptor distances superior to 10 Å are lower by more than 3 orders of magnitude than the rate at contact distance. To evaluate the effect of porphyrin association on the dendrimer conformation, we compare the average radius of gyration and aspect ratio with previous results from simulations of the dendrimer alone (Table 1). The relative characteristics of configurations 1a and 1b for generation 2.5 seem to be preserved upon association with the porphyrin (i.e., the former configuration shows a slightly larger radius of gyration and aspect ratio). Generation 4.5 is always more spherical (aspect ratio closer to unity) than generation 2.5, in agreement with the morphological transition mentioned above. However, the shape of configurations 1a and 2a or 2b becomes more asymmetrical when associated with the porphyrin, as inferred from the increase observed in their aspect ratios. This is due to favorable electrostatic interactions between the dendrimer’s terminal groups (negatively charged) and the porphyrin (positively charged), which compensate for the conformational entropic penalty associated with extension of the branches involved in these interactions. We have also evaluated the effect of porphyrin-dendrimer interaction on the planarity of the porphyrin ring and found that neither the saddle type of distortion nor any other type is particularly favored by comparison to free porphyrin (see the Supporting Information). The trajectories of the donor-acceptor distances in the porphyrin-dendrimer systems with generation 2.5 (Figure 3A and B) show that, within a simulation time of 1 ns, the porphyrin remains approximately in the same site at the dendrimer’s surface (i.e., diffusion of the porphyrin across the dendrimer’s surface is not observed in this time scale) due to the strong electrostatic interactions with its terminal groups. However, significant changes in the donor-acceptor distances can still occur by conformational fluctuations of the dendrimer. For instance, the top trace in Figure 3A (the third from below) shows the effect of a dendrimer branch folding over the porphyrin on the donor-acceptor distance of a tertiary amine belonging to that branch. This folding motion was observed for configuration 1a, and the resultant conformation is shown in Figure 2A. By contrast, no major structural rearrangements were observed for configuration 1b (traces of Figure 3B), although fluctuations of a few angstroms occur for the donor-acceptor distance of some tertiary amines and these can induce variations of almost an order of magnitude in the respective ET rates. Similar behavior was observed in the simulations of the porphyrin-dendrimer systems with generation 4.5 (Figure 3C and D). Again, it was observed that conformational dynamics of the dendrimer branches induce changes in the donor-acceptor distance of some tertiary amines by about a few angstroms but without causing major displacements of the porphyrin position at the dendrimer’s surface. It should be noticed that the absence of hydrodynamic effects accelerates the conformational dynamics in the simulations without solvent.

Figure 4. Distribution of the donor-acceptor (D-A) distances calculated from the simulations of the porphyrin-dendrimer systems without explicit solvent: configurations 1a (closed circles) and 1b (open circles) of generation 2.5 and configurations 2a (closed triangles) and 2b (open triangles) of generation 4.5. The distributions shown here represent the average from all of the tertiary amines in the dendrimer calculated over 104 configurations sampled at equally distributed intervals during the simulation time of 1 ns. The data for generation 4.5 were vertically shifted by two units to make the representation more clear.

From the trajectories of all tertiary amines, the histograms of donor-acceptor distances were calculated over the whole simulation interval for several simulated configurations of the porphyrin-dendrimer systems (Figure 4). This afforded broad distributions of donor-acceptor distances but with local features that are probably reminiscent of the hierarchical chemical structure of the dendrimer. These features are qualitatively similar for both configurations within each dendrimer generation. Furthermore, the superimposition of all distributions shown in Figure 4 reveals that these roughly overlap up to a donor-acceptor distance of about 11-12 Å, which means that the density of tertiary amines in the vicinity of the porphyrin at the dendrimer’s surface is comparable between generations 2.5 and 4.5 (this aspect is not evident from Figure 4 because the results for generation 4.5 are vertically shifted by two units). Including the standard deviation of the individual data points in the distributions reinforces this comparison (see the Supporting Information), and it provides another perspective on the effect of conformational fluctuations in these systems. However, despite the coarse similarity between the distributions of generation 2.5, it is clear that the density of tertiary amines rises up at shorter distances for configuration 1a than for configuration 1b. The overall efficiency of ET in these systems should be rather sensitive to the density of tertiary amines at close separation distances from the porphyrin. To illustrate this aspect, we estimated ET rates from the donor-acceptor distances

MD Simulations of Porphyrin-Dendrimer Systems obtained in the simulations of configurations 1a and 1b. First, it was assumed that each simulation of configurations 1a and 1b samples its own subensemble of the overall configurational space of the porphyrin-dendrimer pair. Considering the many conformational degrees of freedom of large flexible molecules, like proteins or dendrimers, it is unlikely that every possible conformation is sampled over short simulation times. Nevertheless, for merely illustrative purposes, it was further assumed that the configurations sampled within each subensemble (1a or 1b) are representative of the distribution of donor-acceptor distances associated with the configurations that belong to that subensemble (this is a weaker condition than to assume ergodicity). This assumption was combined with a static-disorder approach to ET in these systems by considering that, although conformational changes can occur on the time scale of ET, the average ensemble distribution of porphyrin-dendrimer conformations is not perturbed and, furthermore, that fluctuations and dielectric relaxation of the medium coupled to ET occur on shorter time scales, thereby allowing for a nonadiabatic approximation. In this sense, the semiclassical expression of Marcus’ theory for ET (eq 6) was used to estimate the individual ET rate, ki,j ET, for the ith tertiary amine in the dendrimer structure of the jth configuration sampled from a particular simulation trajectory. The overall ET rate of the jth configuration is given N ki,j , over the by the sum of the individual ET rates, kjET ) ∑i)1 ET N tertiary amines of the dendrimer. The distance dependence of ki,j ET is contained in the electronic coupling term V and in the reorganization energy of the medium λ (see section II for further details). The value assumed for the electronic coupling at contact distance, V0 ) 10 cm-1, and the value of the reorganization energy of the medium, λ ) 0.6 eV obtained for the closest donor-acceptor distance (see section IV), were used to determine the adiabacity parameter defined by Rips and Jortner,38c κA ) (4π/p)(V02τL/λ), where τL is the longitudinal relaxation time of the solvent. For the latter quantity, we considered the slow relaxation time of water (∼5 ps) at the surface of PAMAM dendrimers obtained from molecular dynamics simulations.64 This yielded a value of κA ) 0.24, which means that for the closest donor-acceptor distances the nonadiabatic approximation is close to its limit of validity (κA , 1). The histogram of the ET rates kjET for the trajectory of configuration 1a (full trace in Figure 5) shows a distribution with two peaks that can be decomposed in the contributions of conformations before and after dendrimer branch folding (respectively, peaks A and B in Figure 5). Curiously, the distribution of ET rates for the trajectory of configuration 1b (dashed trace in Figure 5) is centered on the same value and has a comparable width to peak A for configuration 1a, i.e., before dendrimer branch folding. This result suggests that the overall efficiency of ET for the porphyrin sitting at the dendrimer’s surface (without folding) is similar for both configurations 1a and 1b, but after folding, some tertiary amines in configuration 1a assume a much more favorable position for ET and the respective average rate for ET increases by about 2-fold. This result illustrates the role of the porphyrin-dendrimer conformation on the efficiency of ET in these systems through its dependence on the distribution of donor-acceptor distances. The rate calculations described above were performed only for the systems with generation 2.5 dendrimer, in which case values of the reorganization energy of the medium λ were calculated from the simulations with explicit solvent. The systems with generation 4.5 are considerably larger, and the same procedure was not applied to calculate λ due to the computational effort involved. However, it is possible to evaluate the


Figure 5. Distribution of ET rates calculated from eq 6 with the donor-acceptor distances obtained from the simulations of the porphyrin-dendrimer systems without explicit solvent: configurations 1a (full trace) and 1b (dashed trace) of generation 2.5. For each of the 104 configurations sampled over the simulation time interval of 1 ns, N the overall ET rate, kjET ) ∑i)1 ki,j ET, was calculated by summing the individual ET rates ki,j of the N ) 30 tertiary amines of the dendrimer ET (see text for further details). The distribution of configuration 1a was decomposed into the contributions of the configurations before dendrimer branch folding (i.e., simulation time t < 200 ps)speak A in gray tracesand after folding (i.e., simulation time t > 600 ps)speak B in gray trace.

effect of dendrimer size on λ using the two-phase model described in section IV. In general, the values of λ estimated with this model (considering the same dielectric permittivities for both generations to isolate just the size effect) are slightly larger for generation 4.5 but within the same range of values determined for generation 2.5. Furthermore, considering that the density of donor sites in the vicinity of the acceptor is comparable for both dendrimer generations (Figure 4), it is not expected that the overall efficiency of ET should vary significantly between generations 2.5 and 4.5. Indeed, the experimental results of fluorescence quenching for the porphyrin-dendrimer systems studied here indicate that the average efficiency of ET for generation 2.5 is only 1.5-fold greater than that for generation 4.5.32c IV. Simulations with Explicit Solvent Simulations with explicit solvent were performed only for the porphyrin-dendrimer systems with generation 2.5. Nonfolded configurations were selected from the simulations without explicit solvent to generate the initial configurations of both types 1a and 1b (snapshots of the initial configurations and simulation boxes with explicit solvent are shown in the Supporting Information). The key parameter evaluated from the simulations with explicit solvent is the energy difference ∆E(t) between the charge distributions of the reactant and product states (i.e., before and after electron transfer) for each sampled configuration. From the total number of 30 tertiary amines (donor sites) in the generation 2.5 dendrimer, we selected 8 of these (for both configurations 1a and 1b) to determine the energy difference ∆E(t) using the procedure described in section II. In the selection criteria, we privileged donor sites at close distance from the porphyrin and, in this sense, we defined a set of 5 tertiary amines at a distance below 10 Å, 2 tertiary amines positioned between 10 and 20 Å, and 1 tertiary amine separated by more than 20 Å.


Paulo et al. be further addressed below). After relaxation is complete, the system explores the potential energy surface of the product state (i.e., after electron transfer) and the energy difference ∆E(t) now fluctuates around the mean value 〈∆E〉P during the final part of the simulation (80-100 ps). The histograms of the ∆E(t) values calculated from the initial and final part of the simulations with explicit solvent (respectively, peaks R and P of Figure 6B) are normally distributed, in accordance with the line-shape function Scl(∆E) expected for the convolution of a large number of classical modes coupled to ET (eq 1). Reorganization Energy of the Medium. The quantities 〈∆E〉R and 〈∆E〉P fitted from the distributions of ∆E(t) values were used to estimate the reorganization energy of the medium, λ (eq 2), associated with ET from each of the eight selected tertiary amines and for both configurations 1a and 1b (Table 2). Interestingly, the λ values found are scattered along a linear trend when represented against the inverse of the donor-acceptor distance (Figure 7). This behavior is reminiscent of the Marcus expression for the reorganization energy associated with ET between two charged spheres (donor and acceptor sites) embedded in a homogeneous dielectric medium37a

λ) Figure 6. (A) Trace of the energy difference ∆E(t) for a simulated porphyrin-dendrimer system showing the instant (t ) 75 ps) in which it was shifted from the potential energy surface of the reactant state to the product state and the subsequent relaxation. (B) Distribution of ∆E values obtained from the trajectories in the potential energy surface of the reactant state (peak R) and of the product state (peak P). The mean values 〈∆E〉R and 〈∆E〉P of the trace shown in part A correspond to the center of the distributions R and P from part B, respectively.

TABLE 2: Values of the Donor-Acceptor Distance (rDA), Reorganization Energy of the Medium (λ), and Relaxation Times (τf, τs) of the Energy Trajectory ∆E(t) for Each of the Tertiary Amines Selected As a Donor Site in the Simulations of the Porphyrin-Dendrimer Systems with Explicit Solvent gen. 2.5

rDA (Å)

λ (eV)

τf (ps)

τs (ps)

config. 1a

5.8 ((0.4) 11.9 ((0.5) 5.9 ((0.3) 8.0 ((0.5) 8.5 ((0.3) 9.6 ((0.4) 25.7 ((0.6) 16.9 ((0.4) 5.7 ((0.4) 7.7 ((0.4) 8.6 ((0.4) 9.5 ((0.4) 9.3 ((0.4) 11.5 ((0.3) 23.1 ((0.4) 17.5 ((0.5)

0.6 1.1 0.6 0.9 0.8 1.0 1.5 1.2 0.7 0.6 0.8 1.0 0.8 1.1 1.4 1.2

0.01 (68%) 0.05 (85%) 0.09 (79%) 0.08 (83%) 0.01 (56%) 0.07 (78%) 0.01 (63%) 0.07 (78%) 0.02 (53%) 0.01 (31%) 0.04 (76%) 0.02 (60%) 0.02 (68%) 0.06 (74%) 0.05 (87%) 0.04 (62%)

0.25 (32%) 1.83 (15%) 2.42 (21%) 8.00 (17%) 2.14 (44%) 2.66 (22%) 0.32 (37%) 2.60 (22%) 0.47 (47%) 0.15 (69%) 5.28 (24%) 0.82 (40%) 2.41 (32%) 1.51 (26%) 0.36 (13%) 0.53 (38%)

config. 1b

During the initial part of the simulation (50-75 ps), the porphyrin-dendrimer system explores the potential energy surface of the reactant state (i.e., before electron transfer) and the energy difference ∆E(t) fluctuates around the mean value 〈∆E〉R (Figure 6A). When the charge distribution is shifted from the reactant to the product state (t ) 75 ps), the system relaxes between the respective potential energy surfaces and ∆E(t) shows a decay behavior with a short decay time in the order of tens of femtoseconds and another decay time that varies from hundreds of femtoseconds to some picoseconds (this issue will

(

1 e2 -1 1 1 (ε - εst-1) + 4πε0 op 2aD 2aA r

)

(7)

where εop and εst are, respectively, the optical and static dielectric permittivities of the medium; aD and aA are the radius of the donor and acceptor sites, respectively; and r is the donor-acceptor distance (other symbols have the usual meaning). The Marcus expression (eq 7) was used to estimate λ values with different sets of dielectric permittivities for comparison with the simulation results. For these estimates, it was considered that the donor site in the dendrimer is constituted by the central nitrogen atom of the tertiary amine group and the adjacent methylene groups up to the β-carbons. This gives a donor radius of aD )3 Å, as estimated from Edwards’ method of incremental Van der Waals volumes.65 The donor site is embedded in the organic surrounding provided by the dendrimer structure that contains multiple polar groups, which most likely are involved in the medium response to ET. Therefore, the dielectric permittivities of

Figure 7. Values of the reorganization energy of the medium λ determined from the simulations of the porphyrin-dendrimer systems with explicit solvent: configurations 1a (closed circles) and 1b (open circles) of generation 2.5 (see text for further details). The trend lines shown were calculated with eq 7 using the dielectric permittivities of water (εop ) 1.775; εst ) 78.4), N-methylacetamide (NMA, εop ) 2.031; εst ) 191.3), and N-triethylamine (TEA, εop ) 1.963; εst ) 2.42).



N-methylacetamide (NMA) and N-triethylamine (TEA) were also considered in the estimates (besides those of water).66 These solvents have been used in solvatochromic studies to mimic the organic medium provided, respectively, by the dendrimer branches and branching units.67 For the radius of the acceptor species (the porphyrin), a value of aA ) 7 Å was used, as previously justified.32c We verify that the λ values estimated with the properties of polar media (water and NMA) approximate reasonably well those determined from the simulations (Figure 7), by opposition to the estimates made with the properties of an apolar medium (TEA). Furthermore, the better agreement seems to be verified with the trend estimated for the organic polar medium (NMA), which suggests the role of the organic scaffold of the dendrimer. In particular, the higher polarizability of NMA (εop ≈ 2) should be responsible for the better agreement observed here, since -1 for polar media ε-1 op . εst and thus the static dielectric permittivity has a negligible influence on the predictions made with eq 7. However, the ET reaction occurs in solution and, thus, the role of the aqueous environment surrounding the porphyrin-dendrimer pair should also be taken into account in the dielectric continuum model used to estimate λ. For this purpose, a two-phase model developed for ET at micellar surfaces68 was adapted for the porphyrin-dendrimer systems (Figure 8A). The dendrimer is represented as a spherical region with dielectric permittivities εop,C and εst,C, that differ from those of the surrounding aqueous environment, εop,W and εst,W. The radius assumed for this sphere, R ) 14.5 Å, corresponds to the hydrodynamic radius of the dendrimer estimated from the average radius of gyration obtained in the simulations.61 The donor site (tertiary amine) is represented as a spherical cavity (with radius aD ) 3 Å) embedded in the spherical region that represents the dendrimer organic core, while the acceptor is represented as another sphere (with radius aA ) 7 Å) sitting at the surface. According to the two-phase model, λ can be expressed as the sum of two contributions

λ ) λ0 + λ1 λ0 )

e2 -1 (ε-1 op,W - εst,W) 2 32π ε0 λ1 )

∫∞-ν -ν

e2 (RC - RW) 32π2ε0

D

A

∫ν

C

(8) (ED - EA)2 dV

(ED - EA)2 dV

(9)

(10)

where ED and EA are the vectors of electric displacement from the charge distribution of the donor and acceptor, respectively; Vq are the volumes of the dendrimer core, and of the donor and acceptor cavities, respectively, for q ) C, D, and A; and Rq ) -1 ε-1 op,q - εst,q (expressions for the integrals in eqs 9 and 10 can be found in refs 68 and 69). Due to the three-center correction terms in the two-phase model, λ depends not only on the donor-acceptor distance but also on the angle γ between the vectors that connect the center of the donor and acceptor cavities to the center of the core region (Figure 8A). We assumed the properties of bulk water for the medium surrounding the core region (εop,W ) 1.775 and εst,W ) 78.4), while for the latter we considered the properties of an organic medium with εop,C ) 2 and low polarity, εst,C ) 10, or high polarity εst,C ) 100 (respectively, closed and open symbols in Figure 8B). Using the two-phase model, we estimated λ values for positions of the donor sites close to the dendrimer surface (RD

Figure 8. (A) Scheme of the two-phase model geometry considered for the porphyrin-dendrimer systems. The dark gray circle represents the dendrimer organic core and the light gray circle inscribed in it is the donor site, while the light gray circle sitting outside represents the acceptor. The donor-acceptor distance is given by r ) [(RA - RD cos γ)2 + (RD sin γ)2]1/2. (B) Values of the reorganization energy of the medium λ determined with the two-phase model assuming the properties of bulk water for the surrounding medium and the dielectric permittivities of εop,C ) 2 and εst,C ) 10 (closed symbols) or εst,C ) 100 (open symbols) for the dendrimer organic core (see text for further details). The trend lines obtained with the homogeneous model (Figure 7) for the properties of water and NMA are repeated here for comparison purposes.

) 11.5 Å) and at inner regions of the dendrimer (RD ) 5.5 Å), including its geometrical center (RD ) 0 Å). The values estimated for donor sites close to the surface (squares in Figure 8B) show a better agreement with the simulation results (circles in Figure 7). This probably reflects the open structure of generation 2.5 dendrimers, which is more permeable to the aqueous surrounding, by opposition to the compact structure of higher generations that allows for better defined inner regions. Some dispersion observed in the λ values obtained from the simulations can be ascribed, in the context of the two-phase model, either to different depths (i.e., different RD) of the donor sites in relation to the dendrimer surface or to regions of different dielectric permittivity close to the dendrimer surface. This situation has been considered in micellar systems by defining a shell layer around the core region with intermediate properties between the core and the surrounding medium.68 Although the agreement between the simulation results and the two-phase model is better considering a low-polarity core region (closed squares in Figure 8B), we refrain from making any quantitative conclusion about the polarity of the dendrimer interior due to the scarce data from the simulations (only eight points in the comparison range, r-1 < 0.1 Å-1) and, also, to the approximate character of dielectric continuum models. Alternative approaches to assess the polarity of macromolecules have employed


Paulo et al.

Figure 9. (A) Trajectories of the energy difference ∆E(t) showing the relaxation upon change from the potential energy surface of the reactant state to that of the product state (t ) 75 ps). Traces 2-4 were vertically shifted by 4 units, in relation to the previous one, for better clarity. The solid curves represent a biexponential decay function fitted to each energy trajectory. (B) Energy-energy correlation function C(t) of the energy trajectories ∆E(t) shown in part A of the figure displayed in the same relative order (traces 1-4) and vertically shifted by 0.4 units. From bottom to top, the curves presented here (in both parts A and B) correspond to the tertiary amines at separation distances of 5.8 and 11.9 Å (traces 1 and 2) for configuration 1a and of 5.7 and 11.5 Å (traces 3 and 4) for configuration 1b, respectively (see Table 2).

molecular dynamic simulations to study the response of proteins to external electric fields and could be valuable in the case of dendrimers.70 Molecular dynamics simulations allowed us to extend the determination of λ values to short donor-acceptor distances (r < 10 Å), at which the dielectric continuum models fail to give reasonable estimates due to overlap of the spherical cavities that represent the donor and acceptor sites in these models. In the limit of very short donor-acceptor separations, it is expected that the linear trend observed for the λ values determined from the simulations should level off due to the finite size of the species involved. Dielectric Relaxation. As previously mentioned, the energy difference ∆E(t) describes the dielectric fluctuations and relaxation that accompanies electron transfer in the simulated systems (section II). Next, we address the relaxation of ∆E(t) upon change from the potential energy surface of the reactant state to that of the product state. This change was triggered for each simulation at the instant t ) 75 ps by changing the charge distribution from the reactant to the product state, and the respective energy trajectories ∆E(t) were fitted with a biexponential function in the interval 75-80 ps. In general, a fast component with a decay time of some tens of femtoseconds is always present (Figure 9A and column “τf” in Table 2). Such short times are characteristic of an inertial contribution in solvation dynamics, due to librational motion or free streaming of solvent molecules around the donor-acceptor pair.71 This behavior was identified for water using computer simulations and it was experimentally confirmed with ultrafast time-resolved fluorescence techniques. The slow decay component of ∆E(t) relaxation varies with the tertiary amine selected as the donor site in the simulations (but without any apparent relation to the donor-acceptor separation). In some cases, the slow decay time is around some hundreds of femtoseconds (traces 1 and 3 in Figure 9A). This is the same order of magnitude of the longitudinal dielectric relaxation time of bulk water, τL ) 0.2 ps, which is related to diffusive motion of solvent molecules in

response to charge displacement. In other cases, the slow decay time is around a few picoseconds (traces 2 and 4 in Figure 9A). Similar relaxation times were obtained from molecular dynamics simulations of water in the vicinity of a dendrimer of the same family of those considered in our simulations but of higher generation.64 The dynamics of water associated with the dendrimer’s surface are slowed down due to interactions with the dendrimer, and relaxation times of about 1 and 5 ps were attributed, respectively, to librational and diffusive motions of water molecules at the dendrimer’s surface. These values are comparable with the slow relaxation times of the energy trajectory ∆E(t) obtained in our simulations (column “τs” in Table 2). Longer relaxation times (of approximately 20 ps) were also determined in that study for water trapped inside the dendrimer, but this situation seems less likely for the dendrimers considered in our simulations with explicit solvent due to the open structure of lower generations. The energy-energy correlation function C(t) defined in eq 5 was calculated for the initial part of each energy trajectory ∆E(t), i.e., before electron transfer (50-75 ps). In the limit of linear response theory, the behavior of the correlation function C(t) should be identical to the response of ∆E(t) to the charge displacement involved in electron transfer.53a Indeed, the correlation function C(t) exhibits a fast decay component in the time range of some tens of femtoseconds, followed by a slow decay with a relaxation time of hundreds of femtoseconds (Figure 9B). Although the exact values of the relaxation times differ from those fitted to the energy trajectories ∆E(t), the time scales involved are comparable (Figure 9A). However, the slowest relaxation times of a few picoseconds observed in several ∆E(t) trajectories do not show as an exponential decay component in the correlation function C(t) but rather as more complex long time features. Finally, the spectral density J(ω) that describes the coupling between electron transfer and thermal motion of the medium was evaluated from the energy-energy correlation function C(t) using eq 4. The representation of J(ω)/ω puts in evidence the



Figure 10. Representation of the spectral density J(ω) calculated from the energy-energy correlation functions C(t). The relative order of the spectra shown here, from A to D, corresponds to the same order of traces 1-4 displayed in Figure 9B. The solid traces superimposed to each spectrum represent a moving average with a period of 9 units, except for spectrum B, in which case the solid trace is a calculated spectral function that assumes a Debye model for three dispersion regions with dielectric relaxation times τL1 ) 20 fs, τL2 ) 0.3 ps, and τL3 ) 2 ps (see text for further details).

contribution of modes with frequencies in the range 0-10 ps-1, but in general, the spectra obtained are broad and the contribution of high-frequency modes is non-negligible (Figure 10). We compare the spectral density J(ω) calculated from the simulations with a spectral function that assumes the Debye model for three dispersion regions with dielectric relaxation times τLi (i ) 1-3) 3

J(ω) )

Rηω

∑ 1 + i ωi 2τ i)1

2

(11)

Li

where Ri are relative weights and ηi ) σ2τLi/kBT, with σ2 standing for the variance of the energy fluctuations ∆E (see section II). For the relaxation times τLi, we assumed values in the range of the decay times fitted for the functions ∆E(t) or C(t), i.e., some tens of femtoseconds (τL1 ) 20 fs), hundreds of femtoseconds (τL2 ) 0.3 ps), and a few picoseconds (τL3 ) 2 ps). The values of σ obtained from the simulations are around 0.36 eV, and the relative weights selected for each dispersion component are R1 ) 0.7, R2 ) 0.1, and R3 ) 0.2. The spectral function calculated using eq 11 with this set of parameters (solid curve in Figure 10B) provides a coarse description of the spectral density obtained from the simulations. Dielectric relaxation associated with electron transfer in linear alcohols of long chains is usually described considering three regions of dielectric dispersion with relaxation times that span from hundreds of femtoseconds to tens of picoseconds.72 In the case of the simulated systems, the diversity of relaxation times is attributed to different environments of the solvent molecules in the vicinity of the dendrimer (and also to different motions of solvent molecules, librational or diffusive). However, the dendrimer itself contains multiple polar groups that could also be involved in the medium response to electron transfer. It was

shown from molecular dynamics simulations that relaxation associated with electron transfer in proteins includes contributions due to long-range electrostatic interactions with several polar groups distributed throughout the protein structure.53 A similar situation is plausible for electron transfer at the surface of dendrimers, and the relaxation times determined from the simulations should also include contributions from reorientation of polar groups within the dendrimer structure. The heterogeneity of relaxation behavior in the simulated systems is exemplified by the different relaxation times fitted from the energy trajectories ∆E(t) for the several tertiary amines selected as donor sites (Table 2). The local environment of each tertiary amine must provide its own balance between contributions of polar groups in the organic structure of the dendrimer and contributions of solvent molecules from the aqueous surrounding, besides the role of the interactions between water molecules and the dendrimer’s surface in the relaxation heterogeneity observed in the simulated porphyrin-dendrimer systems. V. Discussion and Concluding Remarks The porphyrin-dendrimer systems studied here are interesting because the donor species (dendrimer) contains multiple donor sites covalently linked throughout its structure, which is flexible and assumes a globular shape in solution with a radius of some tens of angstroms. The acceptor species (excited-state porphyrin), however, is not covalently bounded and interacts with the oppositely charged groups at the dendrimer’s surface. The multiple donor sites are located at different separation distances from the acceptor and therefore contribute with different efficiencies to the overall electron-transfer process, which is responsible for fluorescence quenching of the acceptor. From the simulations, we identify a set of 7-9 donor sites that are at separation distances below 10 Å, and should have a predominant contribution to the overall ET rate. Furthermore, the confor-

14790 J. Phys. Chem. B, Vol. 112, No. 47, 2008 mational flexibility of the porphyrin-dendrimer pair allows for changes in the distribution of donor-acceptor distances (or equivalently in the density of donor sites close to the acceptor) that can affect the overall ET rate. This aspect was illustrated with the calculation of ET rates for the simulated porphyrin-dendrimer systems using the donor-acceptor distances and the reorganization energies of the medium obtained from the simulations. Using a nonadiabatic model (semiclassical Marcus expression, eq 6) and assuming a static-disorder picture, it was possible to calculate statistical distributions of ET rates for ensembles of porphyrin-dendrimer configurations sampled from the simulations (Figure 5). In particular, it was possible to relate a specific feature of the porphyrin-dendrimer conformation to the efficiency of ET in these systems through its effect on the calculated distribution of ET rates. This refers to folding of a dendrimer branch over the porphyrin, as it was verified in the simulations with generation 2.5 dendrimer of configuration type 1a. The folding motion displaces some donor sites into positions closer to the acceptor, making ET more efficient by approximately 2-fold. The width of the calculated distributions is related to fluctuations of the position of the porphyrin at the dendrimer’s surface and of the tertiary amines within the dendrimer structure, due to branch dynamics without major structural reorganization of the dendrimer conformation (i.e., with no folding). For these illustrative rate calculations, we assumed a value of 10 cm-1 for the electronic coupling V0 at contact distance, which was considered small enough for the nonadiabatic approximation of eq 6 but still large enough to quench the fluorescence lifetime of the porphyrin to subnanosecond values. Using the adiabacity parameter κA defined by Rips and Jortner,38c we estimated a value of κA ) 0.24 for the closest donor-acceptor distance that is actually in the limit of the nonadiabatic regime. This value of κA was calculated for a slow longitudinal relaxation time of the medium of about 5 ps. From simulations with explicit solvent, we confirmed that relaxation times of a few picoseconds are observed in the response of the energy difference ∆E(t) to the charge displacement involved in electron transfer (Table 2). If slower relaxation times occur in these systems, these must be on time scales much longer than tens of picoseconds covered in our simulations. In that case, electron transfer would be controlled by solvent dynamics and conformational relaxation, with complex kinetics in time scales ranging from a few tens of femtoseconds to hundreds of picoseconds or more (furthermore, we anticipate a transition from a nonadiabatic regime for donor sites at large separation distances to an adiabatic regime for donor sites at short distances). This hypothesis provides a better explanation for the broad distribution of quenching rates experimentally observed for the porphyrin-dendrimer systems than merely the dependence of the ET rate on the donor-acceptor distance assumed in the nonadiabatic approach. Still, some issues prevent us from making more definite conclusions regarding this subject. First, the distributions of donor-acceptor distances obtained here are limited only to a few configurations of the porphyrin-dendrimer pair and the effect of the relative orientation of the donor-acceptor pair on the ET rate is not considered in the nonadiabatic model. Both features could contribute to widening the distributions of ET rates calculated from the simulations. Furthermore, the simulations with explicit solvent do not reach the range of hundreds of picoseconds necessary to evaluate possible conformational dynamics coupled to ET in this time scale. From the experimental perspective, the ultrafast components of fluorescence quenching in the systems studied are below the

Paulo et al. time resolution of the measurements performed in ref 32c, which excludes comparison with the fast relaxation times observed in the simulations. The protonation of tertiary amines (that can occur under nonbuffered conditions, in particular for amines close to the dendrimer’s surface) makes these unable to be active donor sites. Since the number and position of the protonated amines for a certain pH condition follows a statistical distribution,73 this effect can contribute to widening the distribution of ET rates in the real systems. The reorganization energy of the medium (λ) for the simulated systems is the main result obtained from the simulations with explicit solvent. The values calculated for λ are in the range 0.6-1.5 eV and follow approximately a linear trend with the inverse of the donor-acceptor distance (Table 2 and Figure 6). This behavior motivated a comparison with dielectric continuum models. In particular, a two-phase model was adapted to take into account contributions from both the organic medium provided by the dendrimer scaffold and the aqueous medium surrounding it. The better agreement was found considering the properties of a low-polarity organic medium for the dendrimer region. However, the two-phase model fails to give predictions for very short donor-acceptor distances because it approximates the donor and acceptor sites to spherical cavities and for nonspherical molecules this imposes a nonrealistic limit to the shortest distance attainable. This is not a problem for the simulation method used here, and we are able to calculate reorganization energies of the medium at very short donoracceptor separations. The values of λ obtained for the porphyrindendrimer systems are comparable or slightly larger than those generally obtained for electron transfer in proteins in aqueous solution.53,74,75 Even though these are similar systems, in the sense that ET proteins are also organic macromolecules with a donor or acceptor site and several polar groups in their structures, the difference is that the acceptor in the porphyrindendrimer systems sits at the macromolecule’s surface and therefore the contribution of the aqueous environment to the reorganization energy is more significant. The dielectric relaxation and fluctuations that accompany electron transfer in the simulated systems were evaluated from the trajectory of the energy difference ∆E(t) and from the energy-energy correlation function C(t). The relaxation times depend on the donor site selected for evaluation of the functions ∆E(t) and C(t), but in general, there is always a fast decay time with some tens of femtoseconds and a slow one with hundreds of femtoseconds up to a few picoseconds. The changes in the slow relaxation time for different donor sites reveal the heterogeneity of the local environment provided by the dendrimer for the ET reaction, in terms of the relative contributions from different types of water (free or surface associated) or from the dendrimer’s polar groups. Acknowledgment. This work was financially supported by CQE IV and in part by the project POCI/QUI/57387/2004. Partial financial support by the Fundaçaõ Calouste Gulbenkian is also gratefully acknowledged. P.M.R.P. acknowledges the postdoc grant SFRH/BPD/25141/2005 from Fundaçaõ para a Cieˆncia e a Tecnologia. The authors thank Dr. L. F. Veiros for all the support given in the realization of DFT calculations for the determination of atomic charges for the simulated molecules. Supporting Information Available: Parameters of the force field employed for the porphyrin in the simulations. Snapshots of the initial configuration of the simulated porphyrin-dendrimer systems. Optimized geometries (D95V*/B3LYP) of the porphyrin (dendrimer segment) in the initial and reduced (oxidized)

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