Continuum Solvation Models: What Else Can We Learn from Them?

PERSPECTIVE pubs.acs.org/JPCL

Continuum Solvation Models: What Else Can We Learn from Them? Benedetta Mennucci* Dipartimento di Chimica e Chimica Industriale, Universit a di Pisa, Via Risorgimento 35, 56126 Pisa, Italy

ABSTRACT Molecular modeling is nowadays a well-established analytical tool exactly as spectroscopies or other experimental methodologies, and we expect that its impact on many research fields in chemistry, biology, material science, and even medicine will enormously increase in the near future. The real spread and success of this expectation will be strictly linked to different factors, among which a fundamental one will be the capacity of simulations of (supra)molecular systems to include environment effects. It is in fact well-known that molecular responses and processes are strongly affected or, in some cases, completely determined by the surrounding environment (either a solvent, a protein, a membrane, a polymer, or a composite matrix). This Perspective highlights recent achievements and suggests possible future developments of one of the most popular approaches to include environmental effects in molecular calculations, the continuun solvation model.

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t is amazing that a large part of our understanding of the main phenomena which characterize liquids is based on a very simple idea formulated about a century ago; in an enormously complex network of interactions as those characterizing the liquid, we can try to obtain information about chemical and physical properties of the system by just focusing on one component (the solute) while treating the rest as a responsive continuum medium (the solvent). The impact that such a simple idea had on science is immediately evident by recalling the names of some of the researchers who have developed models based on this idea (Born, Debye, and Kirkwood, just to cite a few). Among them, however, there is one who has contributed most to pave the way toward a new way of thinking about solvation; his name is Lars Onsager. His decisive contribution elaborated in 1936 was to substitute the concept of polarization of the solvent with that of the “reaction field”.1 The revolutionary aspect of this new concept is that the description moves from the classical picture based on electric fields to the more “modern” picture based on molecular dipole moments. It is exactly this concept which opened new perspectives for the modeling of solvent effects and made possible its merging with quantum mechanical (QM) descriptions. Continuum solvation models are in fact the ideal conceptual framework to describe solvent effects within the QM approach. In fact, the polarization of the dielectric due to the presence of a solute (i.e., its reaction field using Onsager's language) can be easily translated into a QM language as a proper operator. As a result, the vacuum Hamiltonian which describes the molecular system of interest when isolated becomes an “effective Hamiltonian” (EH), that is, a Hamiltonian for the solute in the presence of a polarized dielectric. As the polarization of the solvent is induced by the solute and the solute itself is polarized back by the solvent, an iterative self-consistent field approach is the straightforward solution to the problem.

r 2010 American Chemical Society

But, SCF is exactly the strategy of the most popular QM methods!

Continuum solvation models are the ideal conceptual framework to describe solvent effects within the QM approach. The perspectives opened by the coupling of quantum mechanics and continuum formulations were elaborated independently by various laboratories in the world, which formulated alternative computational methods.2,3 Among the most popular ones, there is the apparent surface charge (ASC) approach wherein electrostatic interactions with the continuum are modeled by a charge density, σ(s), at the surface of a cavity embedding the solute. Given the solute charge density, an integral equation to determine σ(s) can be formulated based upon solution of Poisson's equation, subject to cavity boundary conditions. Nowadays, it is common to collect under the term Polarizable Continuum Model (PCM) the family of continuum solvation models that use such boundary condition formulation even if the acronym PCM was originally introduced by Tomasi and co-workers to indicate the ASC approach developed in Pisa about 30 years ago.4 The actual PCM approaches differ, one with respect to the other, in terms of the electrostatic expression used to define the ASC density, Received Date: April 20, 2010 Accepted Date: May 4, 2010 Published on Web Date: May 10, 2010

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DOI: 10.1021/jz100506s |J. Phys. Chem. Lett. 2010, 1, 1666–1674


σ(s). The most recognizable members of the PCM family are the original formulation also known as D-PCM,4 the integral equation formalism PCM (IEF-PCM) developed by Cances and Mennucci,5 the SVPE (surface and volume polarization for electrostatics) and SS(V)PE (surface and simulation of volume polarization for electrostatics) models developed by Chipman,6 the conductor-like screening model (COSMO) developed by Klamt7 and the related C-PCM by Barone and Cossi,8 and GCOSMO by Truong and Stefanovich.9 Each of these alternative formulations has its pros and cons, which have been discussed many times in the literature (the interested reader is referred to the cited reviews2,3 for a detailed analysis). What is important to recall is that the fundamental aspect which has determined the relative success of a given formulation both in terms of accuracy and applicability is the efficiency and robustness of its computational implementation. As a matter of fact, the numerical solution of the electrostatic equations which all these PCM methods have in common requires that the cavity surface be discretized into finite elements (Figure 1). Upon discretization, the integral equation for σ is replaced by a set of coupled linear equations that determine a set of point charges qi located at the representative points of the surface elements whose interaction with the solute charge density represents the electrostatic part of the solvent effects. As the discretization process generates discontinuities,

singularities, or other numerical instabilities in all of the different formulations of the PCM approach, strategies have been developed to ensure robustness and stability of the results,10-15 and it is exactly the efficiency of these strategies that is the key ingredient which differentiates the various formulations. Indeed, nowadays, the coupling of continuum solvation models and QM methods represents a very popular tool in molecular modeling. In this Perspective, however, we would like to go beyond the status of the art and discuss the possibility for these approaches to continue to represent a source of knowledge about the role that environment plays on properties and behaviors of molecular systems also in the future. Historically, continuum models have based a large part of their success on the low computational cost; this aspect has often made them a much better choice with respect to the alternative family of discrete solvation methods in which an explicit microscopic description of the solvent molecules is maintained (Figure 2). Nowadays, the development of both computer power and algorithm efficiency has made the discrete methods more feasible, and therefore, the original cost/time advantage of continuum models is rapidly decreasing. However, the advantages that still make PCM methods a good choice are not limited to the computational cost; for example, these methods contemporaneously account for long-range electrostatic interactions and polarization effects. Both of these two aspects are not easily available in discrete methods, which in most cases introduce artificial cutoff and/or completely neglect the polarization of the environment. In addition, continuum models automatically give a configurationally sampled solvent effect, and this allows one to avoid statistical analyses based on delicate sampling averages. Such an implicit sampling is assured by the use they make of macroscopic solvent properties such as the dielectric permittivity and the refractive index. These properties also automatically account for time averages and, apparently, should prevent the use of continuum models to describe time-dependent processes giving rise

Figure 1. Example of molecular cavities and relative surface mesh.

Figure 2. Continuum versus discrete solvation methods.


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relaxation of the two parts of the system (the source of polarization and the environment) be coupled from the t = 0 moment (the initial nonequilibrium) to the final re-equilibration at t = ¥. To allow for this time-dependent mutual polarization, a completely new approach to the solution of the electrostatic equation is required. A possible new strategy is that the PCM charges qi become a set of dynamic variables with their own evolution (on the same footing as the nuclear coordinates of the solute). The main difficulty arising from this fully variational scheme is the need for an energy functional which is suitable for a boundary values representation of the electrostatic problem, that is, the one adopted in PCM. This functional also needs to be variational with respect to both solute and solvent degrees of freedom so that, when minimized, the free energy of the system at equilibrium is recovered.21 In the literature, there are examples of functionals; however, in most cases, they are not energy functionals in the sense that their minimization does not lead to the electrostatic free energy, or they are not easily applicable in the context of ASC solvation models.22 This is thus a challenge for the near future; in fact, if such functionals are consistently formulated within the PCM-like approach, a very powerful new tool will be available for a real time-dependent generalization of continuum models. In particular, molecular dynamics applications of PCM approaches will be possible, and due to their computational efficiency, they will become one of the most competitive strategies to perform ab initio MD simulations in solution. For example, Car-Parrinello23 or other Lagrangian procedures24 will be easily extended to PCM solvated systems, with the result of a significant decrease of the total computational time and a more complete description of the electrostatic effect of the solvent. At the moment in fact, these methods when applied to solvated systems use small solute-solvent QM clusters or QM/MM approaches, which cannot properly account for long-range effects and mutual polarization, respectively. Indeed, we can guess that the availability of these real TD-PCM approaches within the most largely diffused ab initio MD (AIMD) packages will generate a fast and large diffusion of QM studies of TD processes in solution exactly as the standard (and static) implementation of PCM methods in softwares such as Gaussian,25 GAMESS,26 or TURBOMOLE,27 has significantly contributed to the enormous diffusion of QM modeling of solvated systems that we have seen in the last years. This completely new formulation of PCM-like approaches is necessarily connected to the previously quoted issue of robustness and stability of the numerical implementation of the model. In particular, when MD simulations are the goal, it is necessary to have algorithms that allow for a reliable exploration of the potential energy surfaces involved. This, when translated into the PCM language, means to ensure the robustness and smoothness of the PCM energy, which is possible only if the already cited discontinuities and singularities originating from the discretization process can be removed. This removal can be possible by introducing a continuos charge formalism.11,14,15 This formalism is smooth as the associated free energy in solution is a continuous function with continuous first and second derivatives of all of the parameters involved in the definition of the model, including

to solvent relaxations. Indeed, time-dependent extensions of these methods have been proposed and successfully applied to specific dynamic processes.

Continuum models automatically give a configurationally sampled solvent effect, and this allows one to avoid statistical analyses based on delicate sampling averages. Nonequilibrium Solvation and Time-Dependent Phenomena. In most cases, the approach used within continuum models to account for a time dependence (TD) is based on the information contained in the frequency dependence of the dielectric function (information either known experimentally or in terms of models such as the Debye model). In fact, if one knows the function ε(ω) it is always possible to obtain the resulting time-dependent polarization using a linear response theory. In general, a further simplification is made by splitting the polarization into distinct parts, each part corresponding to different characteristic times being associated to motions of electrons, atoms, or molecules. Within the continuum framework, the different response times of the various terms constituting the solvent polarization can be (and indeed have been) taken into account using two alternative approaches. Commonly, a separation of the solvent response (e.g., the apparent charges in the PCM scheme) into a fast contribution, associated with the electronic motion, and a slow (or orientational) contribution due to the nuclear and molecular motion is introduced. Alternatively, dynamic and inertial contributions to polarization are not explicitly separated, but they are implicitly taken into account introducing a time-dependent response function. In both cases, possible delays in the solvent polarization due to fast changes in the structure or in the electronic charge distribution of the solute are taken into account and used to simulate solvation dynamics. Historically, these delays, better known as nonequilibrium effects, have been mostly used to describe chemical reactions and electron-transfer processes16 or electronic excitations,17 but they have been used also to simulate vibrational spectroscopies18 and nonlinear optics (NLO) properties.19 More recently, they have been further extended to account for environment effects in excitation energy transfers.20 Most of these extensions, however, are still affected by an intrinsic limit, namely, the relaxation process of the solvent polarization is not “completely” coupled to that of the source of relaxation (a charge or energy transfer, an electronic transition, or another dynamic process), but instead, it is activated by such a source and then allowed to freely (or independently) relax according to the functional form of ε(ω). A correct model would instead require that the


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Figure 3. Left: Example of a position-dependent permittivity function to describe a membrane. Right: Example of a permittivity tensor to describe a nematic liquid crystal.

the position of the atoms of the solute and the presence of external perturbing electric and/or magnetic fields. The continuous description of the ASC density, together with the use of a clever discretization scheme for the solute-solvent interface, ensures that the model is robust in the sense that no Coulomb singularities can arise no matter the conformation assumed by the solute. To summarize, an extension of PCM approaches to real dynamic descriptions is strictly related to a clever definition of the correct energy functional and a parallel efficient implementation of the algorithms which numerically solve the related variational equations. This strict link between the physical nature of the model and its numerical translation is indeed the key aspect of any future successful development of continuum models. More than other methods, continuum models can in fact continue to represent a valid computational strategy only if they combine efficiency with accuracy, and robustness with easiness of use.

less common solvents or to a more complex environment wherein spatial isotropy and homogeneity are lost and/or a composite nature made of very different constituents is acquired. In those cases, a successful application of discrete approaches generally requires both a large experience in the field and specific skills that are not common to the standard user of the computational packages. By contrast, continuum models can, in principle, maintain their simplicity even when applied to the most difficult environments if they are properly reformulated from the very basic equations. In fact, even if continuum models have been originally formulated to study isotropic and homogeneous liquids, their formalism is potentially applicable to more “difficult” environments. Indeed, PCM-like approaches have been already extended to inhomogeneous environments. In particular, two extensions have shown to be quite effective, (i) interfaces between two liquids and membranes28 and (ii) liquid crystalline phases29 (Figure 3). In both cases, the extensions are much easier when the IEF formulation5 of the PCM approach is used; in this formulation, in fact, the same computational framework developed to describe isotropic and homogeneous dielectrics can be easily generalized to two or more dielectrics separated by planar interfaces as well to anisotropic dielectrics. In the first case, a position-dependent permittivity function has to be introduced, whereas in the latter case, the same function becomes a matrix with diagonal values representing the three values of the permittivity along the three axes which characterize the uniaxial or biaxial liquid crystalline phase. Within the IEFPCM framework, all of these cases can be described with exactly the same ASC approach based on the definition of a molecular cavity which is commonly used for standard liquids. The gain in both computational efficiency and ease of use is evident, whereas the limits of the approach remain the same as those that one faces when applying continuum models to standard isotropic and homogeneous environments. Until now, these extensions of continuum models have been almost exclusively used by “expert” researchers who

The strict link between the physical nature of the model and its numerical translation is indeed the key aspect of any future successful development of continuum models.

Complex Environments. Indeed, the enormously larger simplicity of use which has always distinguished continuum models with respect to alternative solvation approaches still represents one of the successful characteristics of these models even if, nowadays, ready-to-use protocols and blackbox-like computational packages are available also for other approaches such as MD simulations in common solvents. The situation however changes completely as soon as we move to


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are also developers of the models due to their character of in-progress approaches. Now, thanks to the robustness acquired by the algorithms and their generalization to different QM methods, the time has come for these extensions to become very effective tools to be easily used also by nonexperts to capture the effects that these “more complex” environments have on properties and behaviors of embedded molecular systems. In particular, the extensions of continuum models to interfaces and membranes represent a valuable approach to predict the changes in the spectroscopic behavior (mostly fluorescence) of probes when immersed in plasma or intracellular membranes and to correlate such changes with the specific structural and polarity characteristics of the membranes. Also, in the case of extensions to anisotropic dielectrics, the interest is primarily of spectroscopic nature as these models can be used to calculate the NMR properties (chemical shieldings and couplings) of selected nuclei of the molecules forming the liquid crystalline phase or of an embedded probe and to use them to obtain information on the order of the phase. Another complex environment that can be effectively described by continuum models is that of metal surfaces or nanoparticles when they act as surface enhancers of the optical properties of nearby molecular systems. The research field of surface-enhanced (SE) spectroscopies is nowadays extremely active due to the enormous potential that these have in many different applications ranging from medicine to technology. Historically, the most famous of these spectroscopies is the SE-Raman spectroscopy (SERS), but nowadays SE-infrared absorption (SEIRA), SE-hyper-Raman scattering (SEHRS), SE-fluorescence (SEF), and many others are also extremely popular. Apparently, polarizable continuum models do not seem to be a proper approach to obtain a reliable description of these effects; however, looking in more detail at the phenomenon, the picture reverts as it can be understood from what follows. There is now a general consensus that the main effect to the overall enhancement of the signal in SE-spectroscopies is the so-called electromagnetic effect; when an electromagnetic field acts on a metal specimen having a curved surface (such as a rough surface, a metal nanoparticle, and so on), it can excite collective resonances of the electron gas in the metal (surface plasmons). Such resonances can create an evanescent electromagnetic field of high intensity, localized in the proximity of the metal surface. A molecule close to such a metallic specimen feels a local field much more intense than the incident one; therefore, its effective response properties will be greatly amplified. Also, in the case of radiation emitted by the molecular system (i.e., in the SEF spectroscopy), resonances in the metal can be excited, and the created plasmons can radiate and enhance the emission from the molecule. However, the collective excitations induced in the metal are also efficiently damped by different mechanisms (such as excitations of electron-hole pairs and then phonons), and the energy originally stored in the excited molecule can be thus dissipated nonradiatively. The luminescence of a molecule near a metal specimen can be either amplified or decreased, depending on which effects prevail (radiative or nonradiative decay). It is clear that all of these phenomena are determined


Figure 4. PCM representation of a molecular system close to a metal nanoparticle of tetrahedral shape.

by the changes in the response properties of the molecule due to its electromagnetic interaction with the induced polarization of the metal. This polarization will clearly depend on the nature of the metal, the shape and dimension of the metal specimen, the frequency of the electromagnetic fields (either the applied ones or those due to the molecular emission), the relative molecule-metal distance and orientation, as well as the presence of a solvent. Indeed, these are exactly the elements which characterize a PCM approach, the nature of the metal being described by its frequency-dependent permittivity function, the shape and dimension by a proper cavity, the molecule explicitly treated at the QM level, and the solvent treated as usual. Different models can be used to adapt the permittivities experimentally determined on bulk samples to nanoparticles of given shape (in most cases, corrections due to quantum size effects can be accounted for and a nonlocal character can be introduced); these corrections can be easily inserted in the computational code, which at the end will produce for each frequency the effective permittivity which determines the polarization of the metal. In PCM approaches, in the case of metals, such a polarization is approximated in terms of an ASC distribution on the surface of the metal particle which is discretized in terms of point charges30 (Figure 4). These charges will enter into the QM equations to be solved to get either Raman, IR, absorption, emission, or other spectroscopic signals, and they will accordingly modify the specific response of the molecule.31 It is clear that by using these PCM extensions for metal specimen, any particle shape can be simulated by just introducing the proper cavity and the related mapping of the surface in finite elements. Within this framework, the calculations of SE-spectroscopies can be obtained exactly with the same code used for solvated systems and with comparable computational cost. This is, in principle, a real effective approach as it combines an accurate QM description of the molecular probe with a simplified but also physically reliable description of the metal particle, and the two descriptions are coupled. This approach has already shown to be a promising predictive tool to design the probe þ metal system with the desired SE properties. However, as for other extensions of PCM-like methods, until now, these applications have mainly focused on test cases and/or model systems and have been performed by the developers of the methods. The model is now mature to be used by researchers who are interested in the results more than in the method itself; only with a massive application by users not involved in its development, in fact,

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does a method reveals its real potentialities. These applications are quite broad going from the prediction of different SE-spectroscopies of molecular probes in the vicinity of one or more metal nanoparticles of any shape and dimension to SP properties of the metal particle(s). The use of ASC charges to describe the metal polarization in fact can also be used to calculate plasmonic spectra of metal nanoparticles of any shape possible, including the effects of the solvent, and in perspective, it can be generalized to treat also hollow particles (or metal shells) or particles formed by different metals. Nonelectrostatic Effects. From this brief overview, it is evident that the possible future applications of continuum models are numerous and of large scientific impact in terms of the research fields that can be involved and the environments that can be described. In all of these applications, however, continuum models apparently are limited to account for electrostatic interactions. These are surely important (and in many cases dominant), but also, other soluteenvironment interactions can become fundamental to understand a specific molecular behavior, a chemical reaction, or a physical process. The most famous example is the case in which the solute can specifically interact with few molecules of the environment in a strong and persistent way such as in H-bonding systems. In this case, the continuum model alone is necessarily limited. As a matter of fact continuum models using molecular cavities which follow the real geometrical structure of the solute can partially recover the effects of possible anisotropies in the solute-solvent interactions and better describe stronger polarization effects in given regions of the cavity, that is, those corresponding to molecular groups which are more active in their interactions with the environments. These advantages of more refined cavities are however not sufficient to accurately account for all of the aspects related to specific solute-solvent interactions, especially when one is interested in investigating the effects that these interactions have on molecular response properties or spectroscopic signals or when these interactions are accompanied by significant charge transfers between solute and solvent molecules. In all of these cases, the continuum model fails if it is used as it is, while it still represents a powerful approach if it is combined with few discrete solvent molecules (preferably described at the QM level as the solute) to account for all of the specific solute-solvent effects not properly described by the continuum model. Clearly, the computational efforts increase along with the complexity of the description; in fact, what now determines the accuracy and the reliability of the description is the number and the relative positions of the discrete solvent molecules. Both of these issues can be solved using the chemical intuition or, in more difficult cases, by doing a preliminary MD simulation. From such a simulation, one can in fact extract some representative configurations of the first solvation shell(s) and use them to construct the QM “supermolecule” to be combined with the continuum description to account for longer-range (or bulk) effects. As a statistically representative number of different configurations is necessary, the computational cost in some cases can become quite high, but this is an issue which rapidly evolves with the fast


speed-up of the machines. Indeed the solvated supermolecule approach, in which the solute plus those solvent molecules that more strongly interact with that are treated as a QM supermolecule immersed in a continuum dielectric, has shown to represent a very effective and accurate approach anytime the solute-solvent interactions are dominated by few specific ones (such as H-bonding with few solvent molecules). The description becomes much trickier when the shortrange interactions are not easily treatable by introducing a few solvent molecules placed in specific active sites but rather they involve the whole inner shell(s) in a dynamic way. This is the case of dispersion interactions. The description of solute-solvent dispersion interactions represents a very delicate issue in both discrete or continuum models. Being a purely QM phenomenon, any approximation in terms of classical models (either based on LennardJones or other semiempirical formulations) is necessarily limited. Continuum models introduced dispersion effects into their set of interactions long ago, combining them with the exchange-repulsion effects. However, the solutions proposed are generally represented by semiempirical corrections to the solvation free energy, and they are based on the cavity surface and/or some parameters specific of the solvent such as the surface tension.32 Few conditions appeared so far to propose some more refined descriptions which can be translated into a QM language and included directly in the solute effective Hamiltonian together with the electrostatic effect.33 The advantages of the QM description are evident; dispersion and repulsion effects can in fact be introduced directly in the calculation of the electronic density and all of the related quantities (the energy but also response properties and spectroscopic signals). In addition, once introduced into the Hamiltonian, they can be extended to electronic excitedstate calculations and taken into account in the simulation of photochemical and photophysical processes. In the future, proposals in this directions will surely appear because for the new applications that we have sketched above to be really attractive, a more complete picture of the environmental effects beyond the electrostatic-only description is required. In fact, this latter can be reasonably correct for electronic ground states, but it has to be expected that dispersion (and repulsion) effects can play an important role in excitation processes, and they can significantly affect properties and behaviors of excited states. In summary, polarizable continuum models will continue to represent a very important tool for our understanding of molecular phenomena and processes of systems embedded in “active” environments. In the near future, their main applications will move from standard liquids to much more complex environments and from electrostatic to other solutesolvent interactions. In this Perspective, I have listed some of these new fields of possible application, but I have surely forgotten some others. In any case, we have to expect that continuum models will last for a long time as they remain one of the most efficient approaches to combine accurate QM descriptions of (supra)molecular systems with simple but

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physically stated descriptions of environments of various complexity. (5)

We have to expect that continuum models will last for a long time as they remain one of the most efficient approaches to combine accurate QM descriptions of (supra)molecular systems with simple but physically stated descriptions of environments of various complexity.

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AUTHOR INFORMATION (8)

Corresponding Author: *E-mail: [email protected].

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Biographies Benedetta Mennucci is Associate Professor at the Department of Chemistry of the University of Pisa. Her research focuses on the development of theoretical models to account for environmental effects in the quantum mechanical description of properties and processes of molecular systems. For more information on her research, see http://benedetta.dcci.unipi.it.

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ACKNOWLEDGMENT The ERANET project NanoSci-ERA within

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the EU FP6 is acknowledged for the support of part of this research. A further acknowledgment for financial support goes to Gaussian Inc.

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Continuum Solvation Models: What Else Can We Learn from Them?

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