*Ab initio*molecular dynamics calculations of ion hydration free energies

^{1,a)}, Susan B. Rempe

^{2}and O. Anatole von Lilienfeld

^{3}

### Abstract

We apply *ab initio*molecular dynamics (AIMD) methods in conjunction with the thermodynamic integration or “-path” technique to compute the intrinsic hydration free energies of , , and ions. Using the Perdew–Burke–Ernzerhof functional, adapting methods developed for classical force field applications, and with consistent assumptions about surface potential contributions, we obtain absolute AIMD hydration free energies within a few kcal/mol, or better than 4%, of Tissandier *et al.*’s [J. Phys. Chem. A102, 7787 (1998)] experimental values augmented with the SPC/E water model predictions. The sums of and AIMD , which are not affected by surface potentials, are within 2.6% and 1.2 % of experimental values, respectively. We also report the free energy changes associated with the transition metal ion redox reaction in water. The predictions for this reaction suggest that existing estimates of for unstable radiolysis intermediates such as may need to be extensively revised.

K.L. thanks Tina Nenoff and Matt Petersen for useful discussions. S.B.R. acknowledges funding by the National Institutes of Health through the NIH Road Map for Medical Research. O.A.v.L. acknowledges support from the SNL Truman Program LDRD under Project No. 120209. This work was also supported by the Department of Energy under Contract No. DE-AC04-94AL85000, by Sandia’s LDRD program. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Co., for the U.S. Department of Energy.

I. INTRODUCTION

II. METHOD

A. VASP calculations

B. Visualizing electronic isosurfaces

C. thermodynamic integration

D. thermodynamic integration

E. and thermodynamic integration

III. RESULTS

A. hydration free energy

B. hydration free energy

C.

D.

IV. CONCLUSIONS

### Key Topics

- Nickel
- 52.0
- Free energy
- 34.0
- Density functional theory
- 25.0
- Silver
- 19.0
- Surface charge
- 19.0

## Figures

The binding energies and optimized distances between a molecule and VASP PBE PPs globally scaled by a factor of . [(a) and (c)] ; [(b) and (d)] . The PPs have no core electrons. Dashed lines are cubic spline fits. is meant as a counter example to for gas phase behavior; its behavior in water will not be the focus of this work.

The binding energies and optimized distances between a molecule and VASP PBE PPs globally scaled by a factor of . [(a) and (c)] ; [(b) and (d)] . The PPs have no core electrons. Dashed lines are cubic spline fits. is meant as a counter example to for gas phase behavior; its behavior in water will not be the focus of this work.

for as varies. The bare ion contributions, Ewald corrections, and electrostatic potential shift due to the quadrupole moment have been subtracted. Crosses: , ; circles: , ; triangles, same as crosses but are for SPC/E water. The dashed lines are cubic least-squared fits to the crosses and triangles.

for as varies. The bare ion contributions, Ewald corrections, and electrostatic potential shift due to the quadrupole moment have been subtracted. Crosses: , ; circles: , ; triangles, same as crosses but are for SPC/E water. The dashed lines are cubic least-squared fits to the crosses and triangles.

(a) Integrated changes in electron density, , as a function of spatial coordinate for the various values of . and are the densities for the neutral and the charged systems, respectively. All charged species, have been shifted to . Symbols correspond to actual grid points, the continuous lines are cubic interpolations. [(b)–(d)] Isosurface plots of the electron density difference, (, white , blue ), for , 0.6, and 1.0. Periodic boundary conditions apply; the prominent, eight blue spheres represent the (periodically replicated) changes in densities, and some changes in water dipole moments are apparent too. See Sec. II B for technical details.

(a) Integrated changes in electron density, , as a function of spatial coordinate for the various values of . and are the densities for the neutral and the charged systems, respectively. All charged species, have been shifted to . Symbols correspond to actual grid points, the continuous lines are cubic interpolations. [(b)–(d)] Isosurface plots of the electron density difference, (, white , blue ), for , 0.6, and 1.0. Periodic boundary conditions apply; the prominent, eight blue spheres represent the (periodically replicated) changes in densities, and some changes in water dipole moments are apparent too. See Sec. II B for technical details.

Pair correlation functions between and the O (solid line) and H (dashed line) sites of molecules. (a) ; (b) . The instantaneous hydration numbers are depicted in the insets.

Pair correlation functions between and the O (solid line) and H (dashed line) sites of molecules. (a) ; (b) . The instantaneous hydration numbers are depicted in the insets.

Logarithm of the probability of instantaneous hydration numbers multiplied by thermal energy, in units of kcal/mol. (a) ; (b) . Squares and dashed lines: ; circles and solid lines: . is determined by counting all water oxygen atoms within 2.08, 2.75, 2.90, and 2.92 Å of the four ions, respectively. These distances are determined by locating the first minimum in the ion-water .

Logarithm of the probability of instantaneous hydration numbers multiplied by thermal energy, in units of kcal/mol. (a) ; (b) . Squares and dashed lines: ; circles and solid lines: . is determined by counting all water oxygen atoms within 2.08, 2.75, 2.90, and 2.92 Å of the four ions, respectively. These distances are determined by locating the first minimum in the ion-water .

[(a) and (b)] Pair correlation functions between classical force field and the O (solid line) and H (dashed line) sites of PBE molecules. (a) ; (b) . (c) for classical force field as varies. Crosses and triangles are for AIMD and classical force field treatments of water in simulation cells. The bare ion contributions, Ewald corrections, and electrostatic potential shift due to the quadrupole moment have been subtracted. The dashed lines are cubic least-squared fits.

[(a) and (b)] Pair correlation functions between classical force field and the O (solid line) and H (dashed line) sites of PBE molecules. (a) ; (b) . (c) for classical force field as varies. Crosses and triangles are for AIMD and classical force field treatments of water in simulation cells. The bare ion contributions, Ewald corrections, and electrostatic potential shift due to the quadrupole moment have been subtracted. The dashed lines are cubic least-squared fits.

Pair correlation functions between and the O (solid line) and H (dashed line) sites of molecules. (a) ; (b) . The instantaneous hydration numbers are depicted in the insets.

for and as varies. The bare ion contributions, Ewald corrections, and global shift in the electrostatic potential due to the quadrupole moment have been accounted for. The dashed lines are cubic least-squared fits.

for and as varies. The bare ion contributions, Ewald corrections, and global shift in the electrostatic potential due to the quadrupole moment have been accounted for. The dashed lines are cubic least-squared fits.

## Tables

hydration free energies using different computational protocols. densities and are in units of and kcal/mol, respectively. Experimental values adjusted for surface potentials and standard state contributions are marked with a dagger (see text).

hydration free energies using different computational protocols. densities and are in units of and kcal/mol, respectively. Experimental values adjusted for surface potentials and standard state contributions are marked with a dagger (see text).

hydration free energies. The asterisk denotes AIMD adjusted for finite simulation cell size and packing effects (see text). Also listed are for plus . The SPC/E results for and contain the packing correction. densities and are in units of and kcal/mol, respectively. Experimental values adjusted for surface potentials are depicted with a dagger; see text for details.

hydration free energies. The asterisk denotes AIMD adjusted for finite simulation cell size and packing effects (see text). Also listed are for plus . The SPC/E results for and contain the packing correction. densities and are in units of and kcal/mol, respectively. Experimental values adjusted for surface potentials are depicted with a dagger; see text for details.

hydration free energies, and hydration free energy changes. densities and are in units of and kcal/mol, respectively. All simulations are based on the PBE functional, except that the formalism with is applied for Ni predictions marked with an . The asterisk denotes adjusted for packing effects. Experimental values adjusted for surface potentials are depicted with a dagger; see text for details.

hydration free energies, and hydration free energy changes. densities and are in units of and kcal/mol, respectively. All simulations are based on the PBE functional, except that the formalism with is applied for Ni predictions marked with an . The asterisk denotes adjusted for packing effects. Experimental values adjusted for surface potentials are depicted with a dagger; see text for details.

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