Ion Solvation Free Energies Research Articles

Equivalence of M- and P-Summation in Calculations of Ionic Solvation Free Energies.

Condensed-phase simulations commonly use periodic boundary conditions (PBCs) to represent the thermodynamic limit. For the vapor to liquid transfer of an ion, the gas/liquid boundary and its associated potential change are then missing. Furthermore, the electric potential and field at a given point are given by conditionally convergent infinite series, for which different summation schemes give different results. Nevertheless, standard simulation protocols can be used to compute experimental quantities unambiguously. In particular, using an auxiliary test particle and a multistep solvation path, we show that particle-based, Ewald, and common molecule-based summation schemes for the potential and field are all essentially equivalent. However, all methods require prior knowledge of the gas/liquid boundary potential to compute ionic solvation free energies using PBC protocols for both force-field and quantum-mechanical models.

Real single ion solvation free energies with quantum mechanical simulation.

Single ion solvation free energies are one of the most important properties of electrolyte solutions and yet there is ongoing debate about what these values are. Only the values for neutral ion pairs are known. Here, we use DFT interaction potentials with molecular dynamics simulation (DFT-MD) combined with a modified version of the quasi-chemical theory (QCT) to calculate these energies for the lithium and fluoride ions. A method to correct for the error in the DFT functional is developed and very good agreement with the experimental value for the lithium fluoride pair is obtained. Moreover, this method partitions the energies into physically intuitive terms such as surface potential, cavity and charging energies which are amenable to descriptions with reduced models. Our research suggests that lithium's solvation free energy is dominated by the free energetics of a charged hard sphere, whereas fluoride exhibits significant quantum mechanical behavior that cannot be simply described with a reduced model.

Concepts and protocols for electrostatic free energies

Electrostatic free energies play an essential role in numerous biomolecular processes occurring in solution. Difficulties arise when the long-range Coulomb interaction is computed for idealised infinite simulation models with periodic boundary conditions. To maintain a neutral simulation box and a finite per-box energy, a neutralising charge density or ‘gellium’ is commonly used, leading to a mean box potential that is constrained to be rigorously equal to zero at all times. Thus, in considering quantities such as ion solvation free energy, the potential drop to move from solvent into the usual, gas phase reference state is missing. In fact, for an infinite molecular system, the electrostatic potential itself is not uniquely defined, but takes the form of an infinite series that is only conditionally convergent. This leads to several possible computational conventions that give different values for the potential and field, all mathematically valid. For experimentally measurable quantities, however, unique results are obtained when sufficiently large simulation boxes are utilised. These concepts are detailed, as well as a fundamental, linear response theoretical framework that provides qualitative understanding of the physical processes involved, especially dielectric relaxation of the environment in response to a new solute charge. Illustrative applications to ligand binding and biomolecular electron transfer are described.

Role of Local Response in Ion Solvation: Born Theory and Beyond.

The nature of ion solvation has drawn the interest of scientists for over a century, yet a thorough theoretical understanding is still lacking. In this work, we focus on the microscopic origins underlying ionic charge asymmetric and nonlinear response contributions to ion solvation free energies. We first derive an exact expression for the charging component of the ionic free energy, the free energy change when the Coulomb interactions between a fixed ion and the solvent are gradually "turned on". We then introduce the concept of a Gaussian test charge distribution, a generalization of the classical electrostatic point test charge that can be used to probe dielectric response in atomically detailed models. This enables the study of a thermodynamic cycle that isolates a linear and charge-symmetric contribution to the free energy that is well-described by Born-model-like dielectric continuum theories. We give a simple physical derivation of the classic Born model that locally relates the induced charge density in a linear dielectric model to the applied ionic charge distribution. The nonlinear response and charge asymmetric contributions to the ion solvation free energy are then examined in the remaining steps of the cycle and compared to classic thermodynamic cycles for this process using computer simulations. The insights provided by this work will aid the development of quantitative theories for the solvation of charged solutes.

Incorporating excluded solvent volume and physical dipoles for computing solvation free energy.

The solvation free energy described using the Born equation depends on the solute charge, solute radius, and solvent dielectric constant. However, the dielectric polarization derived from Gauss's law used in the Born equation differs from that obtained from molecular dynamics simulations. Therefore, the adjustment of Born radii is insufficient for fitting the solvation free energy to various solute conformations. In order to mimic the dielectric polarization surrounding a solute in molecular dynamics simulations, the water molecule in the first coordination shell is modeled as a physical dipole in a van der Waals sphere, and the intermediate water is treated as a bulk solvent. The electric dipole of the first-shell water is modeled as positive and negative surface charge layers with fixed charge magnitudes, but with variable separation distance as derived from the distributions of hydrogen and oxygen atoms of water dictated by their orientational distribution functions. An equation that describes the solvation free energy of ions using this solvent scheme with a TIP3P water model is derived, and the values of the solvation free energies of ions estimated from this derived equation are found to be similar to those obtained from the experimental data.

Hydronium and hydroxide at the air–water interface with a continuum solvent model

The distribution of hydronium and hydroxide ions at the air–water interface has been a problem of much interest in recent years. Here we explore what insights can be gained from a continuum solvent model. We extend our model of ionic solvation free energies and surface interaction free energies to include hydronium and hydroxide. The hydronium cation is attracted to the air–water interface, whereas the hydroxide anion is repelled. If the cavity size parameters required by the model are adjusted to reproduce solvation energies, quantitative agreement with experimental surface tensions is achieved. To the best of our knowledge, this is the most accurate theoretical estimation of this property so far. The results indicate that even if ‘water structure’ is important, its effects can be captured with a relatively simple model. They also contradict the inference from electrophoresis that there is strong hydroxide enhancement at the air–water interface.

ChemInform Abstract: Predicting Accurate Absolute Binding Energies in Aqueous Solution: Thermodynamic Considerations for Electronic Structure Methods

AbstractReview: 49 refs.

Adsorption of Eu3+ and Am3+ ion towards hard donor-based diglycolamic acid-functionalised carbon nanotubes: density functional theory guided experimental verification

Structure, bonding, energetic and thermodynamic parameters of trivalent Eu3+ and Am3+ with diglycolamic acid-functionalised carbon nanotubes (CNT–DGA) in gas and solvent phases are reported in order to understand their complexation and extraction behaviour. The calculation was performed with generalised gradient approximated BP86 density functional and hybrid B3LYP functional using SVP/TZVP basis set. The free energy of extraction, ΔGext, of Eu3+ and Am3+ was computed using the Born–Haber thermodynamic cycle in conjunction with conductor-like screening model solvation approach. The calculated free energy of extraction was found to be exergonic using an explicit cluster water model for hydrating the ions and it was found to be higher for Eu3+ ion over Am3+ ion. From the estimated distribution constant using synthesised CNT–DGA, it is indisputably established that the Eu3+ ion is preferentially extracted over Am3+ ion and hence confirms the acceptance of the explicit cluster model for ion solvation free energy and thermodynamic cycle for the evaluation of free energy of extraction in solution phase. This is perhaps the first of its kind that a combined theoretical and experimental study has been performed on the free energy of extraction of Eu3+ and Am3+ using CNT–DGA.

Incorporation of the TIP4P water model into a continuum solvent for computing solvation free energy

The continuum solvent model is one of the commonly used strategies to compute solvation free energy especially for large-scale conformational transitions such as protein folding or to calculate the binding affinity of protein–protein/ligand interactions. However, the dielectric polarization for computing solvation free energy from the continuum solvent is different than that obtained from molecular dynamic simulations. To mimic the dielectric polarization surrounding a solute in molecular dynamic simulations, the first-shell water molecules was modeled using a charge distribution of TIP4P in a hard sphere; the time-averaged charge distribution from the first-shell water molecules were estimated based on the coordination number of the solute, and the orientation distribution of the first-shell waters and the intermediate water molecules were treated as that of a bulk solvent. Based on this strategy, an equation describing the solvation free energy of ions was derived.

A Continuum Solvent Model of the Partial Molar Volumes and Entropies of Ionic Solvation

Continuum solvent models of electrolyte solutions are extremely useful. However, before we can use them with confidence, it is important to test them by comparison with a range of experimental properties. Here, we have adapted our recently developed1,2 simple continuum solvent model of ionic solvation free energies to calculate the solvation entropies and partial molar volumes of a group of monovalent and monatomic ions. This procedure gives good quantitative agreement for larger ions, and reproduces key qualitative features, such as the shift to positive entropies of solvation for iodide and the shift to negative partial molar volumes for small cations. Small ions require a correction to account for dielectric saturation effects, which brings them also into good agreement with experiment. We argue that this model does not require ad hoc corrections, and uses parameters that have good external physical justification. This work therefore establishes that our continuum solvent model can provide a satisfactory understanding of ionic solvation. It can thus serve as a foundation for improved models that explain and predict more complex ion specific effects.

Predicting pKa in Implicit Solvents: Current Status and Future Directions

Computational prediction of condensed phase acidity is a topic of much interest in the field today. We introduce the methods available for predicting gas phase acidity and pKas in aqueous and non-aqueous solvents including high-level electronic structure methods, empirical linear free energy relationships (LFERs), implicit solvent methods, explicit solvent statistical free energy methods, and hybrid implicit–explicit approaches. The focus of this paper is on implicit solvent methods, and we review recent developments including new electronic structure methods, cluster-continuum schemes for calculating ionic solvation free energies, as well as address issues relating to the choice of proton solvation free energy to use with implicit solvation models, and whether thermodynamic cycles are necessary for the computation of pKas. A comparison of the scope and accuracy of implicit solvent methods with ab initio molecular dynamics free energy methods is also presented. The present status of the theory and future directions are outlined.

Field-SEA: A Model for Computing the Solvation Free Energies of Nonpolar, Polar, and Charged Solutes in Water

Previous work describes a computational solvation model called semi-explicit assembly (SEA). The SEA water model computes the free energies of solvation of nonpolar and polar solutes in water with good efficiency and accuracy. However, SEA gives systematic errors in the solvation free energies of ions and charged solutes. Here, we describe field-SEA, an improved treatment that gives accurate solvation free energies of charged solutes, including monatomic and polyatomic ions and model dipeptides, as well as nonpolar and polar molecules. Field-SEA is computationally inexpensive for a given solute because explicit-solvent model simulations are relegated to a precomputation step and because it represents solvating waters in terms of a solute’s free-energy field. In essence, field-SEA approximates the physics of explicit-model simulations within a computationally efficient framework. A key finding is that an atom’s solvation shell inherits characteristics of a neighboring atom, especially strongly charged neighbors. Field-SEA may be useful where there is a need for solvation free-energy computations that are faster than explicit-solvent simulations and more accurate than traditional implicit-solvent simulations for a wide range of solutes.

Explicit Solvation Shell Model and Continuum Solvation Models for Solvation Energy and pKa Determination of Amino Acids.

The study of the Explicit Solvation Shell Model (ESS) presented recently [da Silva, E. F.; Svendsen, H. F.; Merz, K. M. J. Phys. Chem. A 2009, 113, 6404.] for calculation of solvation free energy of ions is extended for the study for amino acids. Solvation free energies and pKa of a data set of 10 amino acids is calculated using ESS. The data set of amino acids is selected based on their potential to be regarded as solvents for postcombustion CO2 capture processes. Calculated results are compared against experimental pKa and pKa calculated from PCM, SM8T, and DivCon continuum solvation models. Error estimates of pKa from different models vs experimental pKa data are also given to evaluate the results calculated by different solvation models. This study also involves a comprehensive study of gas phase basicity, proton affinity,ΔGacid0, ΔHacid0, protonation entropy with density functional methods (B3LYP/6-311++G(d,p)) and composite methods (G3MP2B3, G3MP2, CBS-QB3, G4MP2) and their comparison with experimen...

Incorporating the excluded solvent volume and surface charges for computing solvation free energy

Gauss's law or Poisson's equation is conventionally used to calculate solvation free energy. However, the near-solute dielectric polarization from Gauss's law or Poisson's equation differs from that obtained from molecular dynamics (MD) simulations. To mimic the near-solute dielectric polarization from MD simulations, the first-shell water was treated as two layers of surface charges, the densities of which are proportional to the electric field at the solvent molecule that is modeled as a hard sphere. The intermediate water was treated as a bulk solvent. An equation describing the solvation free energy of ions using this solvent scheme was derived using the TIP3P water model.

Strategy using three layers of surface charge for computing solvation free energy of ions

Continuum solvent model is the common used strategy for computing the solvation free energy. However, the dielectric polarization from Gauss's law differs from that obtained from molecular dynamics simulations. To mimic the dielectric polarization surrounding a solute in molecular dynamics simulations, the first-shell water molecule was modeled using a charge distribution of TIP4P molecule in a hard sphere. The dielectric polarization of the first-shell water was modeled as a pair of surface charge layers with a fixed distance between them, but with variable, equal, and opposite charge magnitudes that respond to the electric field on the first-shell water. The water outside the first shell water is treated as a bulk solvent, and the electric effect of the bulk solvent can be modeled as a surface charge. Based on this strategy, the analytical solution describing the solvation free energy of ions was derived, and the values of computed solvation free energy were compared to the values of experiments.

Charging Free Energy Calculations Using the Generalized Solvent Boundary Potential (GSBP) and Periodic Boundary Condition: A Comparative Analysis Using Ion Solvation and Oxidation Free Energy in Proteins

Free energy simulations using a finite sphere boundary condition rather than a periodic boundary condition (PBC) are attractive in the study of very large biomolecular systems. To understand the quantitative impact of various approximations in such simulations, we compare charging free energies in both solution and protein systems calculated in a linear response framework with the Generalized Solvent Boundary Potential (GSBP) and PBC simulations. For simple ions in solution, we find good agreements between GSBP and PBC charging free energies, once the relevant correction terms are taken into consideration. For PBC simulations with the particle-mesh-Ewald for long-range electrostatics, the contribution (ΔG(P-M)) due to the use of a particle rather than molecule based summation scheme in real space is found to be significant, as pointed out by Hünenberger and co-workers. For GSBP, when the inner region is close to be charge neutral, the key correction is the overpolarization of water molecules at the inner/outer dielectric boundary; the magnitude of the correction (ΔG(s-pol)), however, is relatively small. For charging (oxidation) free energy in proteins, the situation is more complex, although good agreement between GSBP and PBC can still be obtained when care is exercised. The smooth dielectric boundary approximation inherent to GSBP tends to make significant errors when the inner region is featured with a high net charge. However, the error can be corrected with Poisson-Boltzmann calculations using snapshots from GSBP simulations in a straightforward and robust manner. Because of the more complex charge and solvent distributions, the magnitudes of ΔG(P-M) and ΔG(s-pol) in protein simulations appear to be different from those derived for solution simulations, leading to uncertainty in directly comparing absolute charging free energies from PBC and GSBP simulations for protein systems. The relative charging/oxidation free energies, however, are robust. With the linear response approximation, for the specific protein system (CueR) studied, the effect of freezing the protein structure in the outer region is found to be small, unless a very small (8 Å) inner region is used; even in the latter case, the result is substantially improved when the nearby metal binding loop is allowed to respond to metal oxidation. The implications of these results to the applicability of GSBP to complex biomolecules and in ab initio QM/MM simulations are discussed.

Coordination state probabilities and the solvation free energy of Zn2+ in aqueous methanol solutions

Coordination state probabilities for the [Zn(H(2)O)(n)(CH(3)OH)(m)](2+) complex in aqueous methanol solutions are calculated as a function of the bulk solution concentration, and the number of methanol ligands, m = 0, 1, ..., 6 with n+m = 6. Zinc ion solvation free energies, which serve to normalize these probabilities, also reproduce the methanol concentration dependence of the experimentally derived free energy of zinc ion transfer from water to aqueous methanol solutions. Coordination state probabilities, p(n, m), are derived by extending quasi-chemical theory of ion hydration to solvent mixtures and mixed ligands. Free energy contributions to p(n, m) include the free energy of forming the mixed-ligand complex in the ideal gas, obtained by quantum chemical calculations, and the solvation free energy of the complex, approximated by a dielectric continuum model. We find that replacing water ligands with methanol ligands preferentially stabilizes methanol-rich complexes in the ideal gas. Conversely, water-rich complexes are stabilized by the solvation free energy contribution, such that the [Zn(H(2)O)(6)](2+) complex is the dominant species in solution for all methanol concentrations considered. Stabilization of the methanol-rich complexes is a consequence of the local coordination chemistry, dominated by the delocalization of charge on the zinc ion, while the stabilization of water-rich complexes is a consequence of favorable ion-solvent electrostatic interactions and smaller dielectric cavities for the water-rich complexes at fixed total charge in the dielectric continuum model. Our analysis also highlights an entropic contribution associated with the reversible work required to remove n water and m methanol molecules from bulk solution to form the [Zn(H(2)O)(n)(CH(3)OH)(m)](2+) complex, which captures the methanol concentration dependence of the solvation free energy of the zinc ion.

Calculation of Derivative Thermodynamic Hydration and Aqueous Partial Molar Properties of Ions Based on Atomistic Simulations.

The raw ionic solvation free energies calculated on the basis of atomistic (explicit-solvent) simulations are extremely sensitive to the boundary conditions and treatment of electrostatic interactions used during these simulations. However, as shown recently [Kastenholz, M. A.; Hünenberger, P. H. J. Chem. Phys.2006, 124, 224501 and Reif, M. M.; Hünenberger, P. H. J. Chem. Phys.2011, 134, 144104], the application of an appropriate correction scheme allows for a conversion of the methodology-dependent raw data into methodology-independent results. In this work, methodology-independent derivative thermodynamic hydration and aqueous partial molar properties are calculated for the Na(+) and Cl(-) ions at P° = 1 bar and T(-) = 298.15 K, based on the SPC water model and on ion-solvent Lennard-Jones interaction coefficients previously reoptimized against experimental hydration free energies. The hydration parameters considered are the hydration free energy and enthalpy. The aqueous partial molar parameters considered are the partial molar entropy, volume, heat capacity, volume-compressibility, and volume-expansivity. Two alternative calculation methods are employed to access these properties. Method I relies on the difference in average volume and energy between two aqueous systems involving the same number of water molecules, either in the absence or in the presence of the ion, along with variations of these differences corresponding to finite pressure or/and temperature changes. Method II relies on the calculation of the hydration free energy of the ion, along with variations of this free energy corresponding to finite pressure or/and temperature changes. Both methods are used considering two distinct variants in the application of the correction scheme. In variant A, the raw values from the simulations are corrected after the application of finite difference in pressure or/and temperature, based on correction terms specifically designed for derivative parameters at P° and T(-). In variant B, these raw values are corrected prior to differentiation, based on corresponding correction terms appropriate for the different simulation pressures P and temperatures T. The results corresponding to the different calculation schemes show that, except for the hydration free energy itself, accurate methodological independence and quantitative agreement with even the most reliable experimental parameters (ion-pair properties) are not yet reached. Nevertheless, approximate internal consistency and qualitative agreement with experimental results can be achieved, but only when an appropriate correction scheme is applied, along with a careful consideration of standard-state issues. In this sense, the main merit of the present study is to set a clear framework for these types of calculations and to point toward directions for future improvements, with the ultimate goal of reaching a consistent and quantitative description of single-ion hydration thermodynamics in molecular dynamics simulations.

A Fast Solver for a Nonlocal Dielectric Continuum Model

The nonlocal continuum dielectric model is an important extension of the classical Poisson dielectric model, but it is very expensive to be solved in general. In this paper, we prove that the solution of one commonly used nonlocal continuum dielectric model of water can be split as a sum of two functions, and these two functions are simply the solutions of one Poisson equation and one Poisson-like equation. With this new solution splitting formula, we develop a fast finite element algorithm and a program package in Python based on the DOLFIN program library such that a nonlocal dielectric model can be solved numerically in an amount of computation that merely doubles that of solving a classic Poisson dielectric model. Using the new solution splitting formula, we also derive the analytical solutions of two nonlocal model problems. We then solve these two nonlocal model problems numerically by our program package and validate the numerical solutions through a comparison with the analytical solutions. Finally, our study of free energy calculation by a nonlocal Born ion model demonstrates that the nonlocal dielectric model is a much better predictor of the solvation free energy of ions than the local Poisson dielectric model.

Differential Solvation Free Energies of Oxonium and Ammonium Ions: Insights from Quantum Chemical Calculations

We have employed computational protocols to determine the differential solvation free energy in water of oxonium and ammonium ions. We have focused our analysis on pairs of onium ions having the same hydrocarbon content and substitutional pattern (HCSP pairs). In agreement with previous suggestions (Taft, R. W.; Wolf. J. F.; Beauchamp, J. L.; Scorrano, G.; Arnett, E. M. J. Am. Chem. Soc. 1978, 100, 1240), we found that cavitation and van der Waals terms do not contribute to the differential solvation free energy. Moreover, we observe that oxonium ions are more strongly solvated than the analogous ammonium ions even though the former ones have fewer H-bond donor sites. The performance of two different continuum solvation models, PCM and SMD, is discussed.