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The theoretical evalutation of single-ion solvation free energies, as well as of corresponding derivative thermodynamic solvation parameters, relies on models involving three different levels of resolution: (i) continuum-electrostatics calculations; (ii) classical atomistic simulations; (iii) quantum-mechanical computations. Chapter 3 introduces these three types of modeling approaches, and provides a summary of their strengths and shortcomings. On the low-resolution end, continuum-electrostatics calculations, inspired from the Born model, typically describe the ion as a rigid non-polarizable sphere and the solvent as a continuous medium of infinite extent, linear dielectric response, and homogeneous permittivity. These approaches have a long history and the merit of providing a simple and qualitatively correct framework for intuitive reasoning. However, they neglect the microscopic structure of the solvent molecules and the specific details of ion-solvent interactions, and rely on the ill-defined concept of an ionic radius. In the middle-resolution range, classical atomistic simulations describe the ion in solution as a system of classical point particles (atoms) interacting according to an empirically designed and calibrated potential energy function, called a force field. These methods should in principle be more accurate than continuum-electrostatics approaches, by accounting for the microscopic structure of the solvent molecules. However, in contrast to the latter methods, the averaging over solvent configurations must be carried out explicitly and the considered system is now of finite extent, e.g. liquid droplet or periodic computational box. The latter difference introduces serious methodological issues regarding the choice of boundary conditions, the approximate treatment of electrostatic interactions, and the evaluation of electric potentials based on the sampled configurations. The third issue appears in particular in the form of puzzling inconsistencies affecting the results of calculations involving slightly different potential-evaluation schemes. The origin of these inconsistencies is analyzed in more detail in Chapters 4, 6 and 7. The results of atomistic simulations also depend on the choice of ion-solvent van der Waals interaction parameters, which can be viewed as representing the atomistic analog of the ionic radius of continuum-electrostatics calculations, and suffer from a similar kind of ambiguity. Finally, on the high-resolution end, quantum-mechanical computations describe the ion in solution as a many-particle system characterized by a wavefunction obeying the Schrödinger equation, given a Hamiltonian encompassing Coulombic interactions between all the elementary particles involved. As first-principles approaches, these methods have a bright future and promise to enable the calculation of experimentally-elusive quantities H,svt, H,svt and χsvt based on the most accurate physical model available nowadays, without relying on the specification of ambiguous quantities such as ionic radii or ion-solvent van der Waals interaction parameters. Unfortunately, they have the major shortcoming of being computationally expensive, which results in practice nowadays in severe restrictions concerning the system size, configurational sampling, basis-set size, and treatment of electron correlation.

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