Published: September 15, 2011 r2011 American Chemical Society 3711 dx.doi.org/10.1021/ct200376z | J. Chem. Theory Comput. 2011, 7, 3711–3724 ARTICLE pubs.acs.org/JCTC Polarizable Force Fields and Polarizable Continuum Model: A Fluctuating Charges/PCM Approach. 1. Theory and Implementation Filippo Lipparini* and Vincenzo Barone Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy ABSTRACT: We present a combined fluctuating chargespolarizable continuum model approach to describe molecules in solution. Both static and dynamic approaches are discussed: analytical first and second derivatives are shown as well as an extended lagrangian for molecular dynamics simluations. In particular, we use the polarizable continuum model to provide nonperiodic boundary conditions for molecular dynamics simulations of aqueous solutions. The extended lagrangian method is extensively discussed, with specific reference to the fluctuating charge model, from a numerical point of view by means of several examples, and a rationalization of the behavior found is presented. Several prototypical applications are shown, especially regarding solvation of ions and polar molecules in water. 1. INTRODUCTION The astonishing development of computational resources during recent decades has made possible studies of larger and larger molecular systems together with the computation of accurate and complex physicalchemical properties. Both clas- sical and quantum mechanical (QM) approaches have enor- mously increased either their range of application or their accuracy or both, allowing the study of several processes ranging from folding studies of huge biological systems to extremely accurate computations of spectroscopic parameters for medium- large molecules. Complex systems, like nanostructured ones or solutions and, more in general, what is usually referred to as “Soft Matter” represent, nevertheless, an interesting challenge for theoretical and computational chemistry. The huge dimensionality of such systems, which can be considered microscopic but certainly not molecular, is still far beyond the possibilities of modern compu- tational infrastructures: a computational study of a complex system by means of standard tools is nowadays still unfeasible when a QM treatment of the whole system is required. This can be both a curse and a blessing: the quantity of data arising from the direct study of such a system would be difficult to analyze and even more difficult to translate into chemically understandable information when a local property tuned by the chemical environment is the target of the study. The unfeasibility of “brute force” approaches, on the other hand, is not to be considered as an insuperable obstacle. Chemical intuition is often the way to get a valuable answer at a reasonable price and is the driving force in the definition of focused models, where the target of a study is well-defined and distinguished from the environment, as complex as it might be, that surrounds it. A prototypical focused model may use different levels of theory, from a very sophisticated QM approach to describe the core, to a cheaper one for its closest surroundings, to a classical but still atomistic one for the distant surroundings, to a con- tinuum to describe the boundaries. In this paper, we will focus on the two latter shells and, in particular, on their interface. As the continuum is concerned, the Polarizable Continuum Model (PCM) 1,2 is one of the most successful models, thanks to its generality and its versatility. The PCM represents a solvent, or other more complex matrices 3 such as an anisotropic medium or a weak ionic solution or even a metal nanoparticle, by means of a polarizable, infinite, dielectric medium which surrounds a mo- lecular cavity that accommodates the “solute”. However, when dealing with solvents responsible for specific interactions like hydrogen bonds, a continuous approach may not be sufficient to achieve a correct description of the system: a mixed contin- uousatomistic treatment of the solvent, using molecular me- chanics (MM) to describe the atomistic portion, can be greatly beneficial. 412 On the other hand, the mixed strategy is advanta- geous with respect to a fully atomistic one as the PCM easily takes into account the long-range interactions that would require a huge number of solvent molecules, increasing significantly the computational cost of the simulation, and implcitly includes the statistical average of their configurations. In this paper, we will present a combined PCM/MM descrip- tion using a polarizable force field. The most popular approaches used to include polarization effects in MM include the induced point dipole method, 13 the classical Drude oscillator model, 14 and the fluctuating charges model. 1517 We find the FQ model particularly appealing in view of its strong connection both with quantum mechanics and classical electrostatics: the model is based on concepts, such as atomic hardness and electronegativity, which can be rigorously defined in the framework of density functional theory (DTF); 18,19 on the other, the electronic distribution is represented by effective atomic charges which interact classicaly. There is a strong connection with the formalism adopted by semiempirical meth- ods, like the density functional tight binding 20,21 approach, and the FQ model, for they both treat the electronic polarization with some suitably defined—and QM derived—charges that are made self-consistent; on the other hand, the same strong formal Received: June 6, 2011