«Inner shell definition and absolute hydration free energy of K+(aq) on the basis of quasi-chemical theory and ab initio molecular ...»
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Inner shell definition and absolute hydration free
energy of K+(aq) on the basis of quasi-chemical
theory and ab initio molecular
dynamicsPresented at the 81st International
Article in Physical Chemistry Chemical Physics · April 2004
Impact Factor: 4.49 · DOI: 10.1039/b313756b · Source: arXiv
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Susan B Rempe Dilip Asthagiri Sandia National Laboratories Rice University 103 PUBLICATIONS 1,774 CITATIONS 83 PUBLICATIONS 1,984 CITATIONS
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I. INTRODUCTIONMany of the most characteristic molecular processes in aqueous solutions, including biophysical processes, depend upon electrolytic properties of these phases. Molecular scale understanding of the statistical thermodynamics of ionic species in water is important for understanding those processes. Both theory and simulation of these systems has become much more revealing in recent years. As successive cases of common ions are addressed with more informative tools , idiosyncratic diﬀerences between chemically similar ions, for example, the alkali metals, come to the forefront. This paper considers K+ (aq) in detail, and studies an intrinsic diﬀerence from the cases of Li+ (aq) and Na+ (aq); speciﬁcally, the deﬁnition on the basis of theory and simulation observation of an inner hydration shell.
A satisfactory identiﬁcation of an inner shell makes molecular theory and other subsequent considerations much simpler. For Li+ (aq) [2, 3, 4, 5, 6, 7] and Na+ (aq) [8, 9], such an inner shell is obvious from the most primitive observation, the radial distribution of water (oxygens) conditional on the atomic ion. For K+ (aq) that identiﬁcation is less trivial, but that is the goal of this note. This is likely to be an issue of broader relevance to other ions in solution, for example, HO− (aq) [10, 11].
The more revealing tools hinted at above are ab initio molecular dynamics (AIMD) and the quasi-chemical theory of solutions. K+ (aq) at high dilution has been the subject of a preliminary AIMD calculation . In contrast, a recent simulation study focused on inﬁnite dilution hydration free energies utilizing classical force ﬁeld models . Concentrated KOH aqueous solutions have been studied by Car-Parrinello ab initio molecular dynamics methods, too [14, 15]. A more expansive molecular dynamics study of KF aqueous solution on the basis of a classical model force ﬁeld is presented in .
AB INITIO MOLECULAR DYNAMICSII.
The K+ (aq) model system consisted of 32 water molecules and one K+ ion contained in a cubic box of length 9.865 ˚ and subjected to periodic boundary conditions. With A this box volume, the water density in the system matched the experimental density of bulk liquid water at standard conditions, with no alterations made for the presence of the ion.
A structure obtained from a well-equilibrated system used in a previous study of Na+ (aq)  provided a template for generating initial conﬁgurations for the current investigation.
The sodium ion and 8 nearest water molecules were removed from the former system and replaced with a cluster composed of 8 water molecules surrounding a K+ ion. The geometry of the inserted K+ cluster was ﬁrst optimized in a separate ab initio calculation and, when placed in the simulation box, the cluster became 10-fold coordinated due to the proximity of two other water molecules in the box.
It is the dynamical evolution of this initial conﬁguration, determined within an ab initio framework, that is presented and analyzed here. Alternative initial conﬁgurations generated in an analogous manner using 6 and 9 water molecules were used to conﬁrm that the results were independent of initial conditions, but will not be addressed further.
Ab initio molecular dynamics simulations based on the PW91 gradient-corrected electron density functional description of the electronic structure and interatomic forces were performed on the K+ (aq) system using the VASP program [17, 18]. Valence electrons, which included the semi-core 3p states of K, were treated explicitly, while Vanderbilt [19, 20] ultrasoft pseudopotentials described core-valence interactions for all atoms. The valence orbitals were expanded in plane waves with a kinetic energy cut-oﬀ of 29.1 Ry, and the equations of motion were integrated in time steps of 1.0 fs for a total simulation time of 20 ps.
For the ﬁrst 3 ps of the AIMD simulations, a constant temperature of 300 K is maintained by scaling the velocities. During this time, the K+ immediately sheds half of its 10 coordinating water molecules by expanding into a more relaxed geometry with fewer water ligands, see Fig. 1. When the temperature constraint is removed, the system heats up a little and the number of coordinating water molecules ﬂuctuates between four (4) and eight (8). Note that it takes more than 4 ps for the system temperature to stabilize around a new average value. The structural analyses presented here utilize the simulation data collected between 10 and 20 ps, for which the system temperature averages 340±23 K.
III. QUASI-CHEMICAL THEORY
The discussion of the quasi-chemical approach , and its application to ion-solvation problems (see, for example [2, 22]) has been considered in detail before. Brieﬂy, the region around the solute of interest is partitioned into inner and outer shell domains. For K+ (aq), the inner shell comprises the water molecules directly coordinated with the ion. This domain is treated quantum mechanically. The outer-shell contribution has been assessed using both a dielectric continuum model and classical force ﬁeld simulations (see ). The theory permits a variational check of the inner-outer partition [9, 23], but this aspect has not been pursued here.
The inner shell reactions are:
K+ + nH2 O ⇋ K[H2 O]n +
The free energy change for these reactions were calculated using the Gaussian programs .
The K · [H2 O]n + (n = 0... 8) clusters were geometry optimized in the gas phase using the B3LYP hybrid density functional  and the 6-31+G(d,p) basis set. Frequency calculations conﬁrmed a true minimum, and the zero point energies were computed at the same level of theory. Single point energies were calculated with the 6-311+G(2d,p) basis.
For estimating the outer shell contribution, the ChelpG method  was used to obtain partial atomic charges. Then the hydration free energies of the clusters were calculated using a dielectric continuum model , with the radii set developed by Stefanovich et al.  and surface tessera generated according to Ref. . With this information and the binding free energies for the chemical reactions, a primitive quasi-chemical approximation to the excess
chemical potential of K+ (aq) in water is:
reaction in an ideal gas state, n is the hydration number of the most stable inner shell cluster, and β = 1/kB T. The density factor ρH2 O appearing in Eq. 3.1 reﬂects the actual density of liquid water and its eﬀect is included by the replacement contribution of −nkB T ln(ρH2 O /ρ0 ) = -nkB T ln(1354), where ρH2 O = 1 gm/cm3, and ρ0 = 1 atm/RT. (Discussions on standard states and this replacement contribution can be found in Pratt and Rempe  and Grabowski et al. .) Note that it is in forming Eq. 3.1 that approximations enter the theory, but all these approximations are available for scrutiny and improvement.
Following this procedure, the most probable inner-shell coordination number was found to be four (4). Based on this identiﬁcation, we pursued more reﬁned molecular dynamics calculations to assess the outer-shell contribution using the K[H2 O]4 + cluster. The oxygen and hydrogen atoms of the cluster were assigned the van der Waals parameters of the classical SPC/E  water model. The ChelpG charges were retained for the cluster. This cluster was placed in a bath of 306 SPC/E water molecules and its hydration free energy calculated in the same manner as described in . In contrast to this earlier study, only one box size was used. This was founded on the earlier observation that using diﬀerent box dimensions did not change the hydration free energy much, indicating that ion(aq) ﬁnite size eﬀects are modest. More expansive discussions of outer shell contributions can be found in [2, 21, 23, 32]
Fig. 2 shows the radial density distribution of the water oxygen atoms around the ion, obtained by analyzing the last 10 ps of the simulation using a bin width of 0.1 ˚. The A density of water peaks at only a little over 2.5 times the density of bulk water and the ﬁrst coordination shell is characterized by a broad ﬁrst peak spanning over 1 ˚ in width A with a shallow ill-deﬁned minimum, in contrast to the much narrower and better deﬁned distributions of water around the smaller alkali metal cations, Li+ and Na+ . The area under the ﬁrst peak indicates that the mean number of 5.9±0.9 water molecules coordinate the ion within a distance of r=3.50 ˚. Of more interest is the secondary structure appearing A in the ﬁrst peak, indicating a composite nature of the water distributed around potassium ion.
The analysis of Fig. 2 suggests that the ﬁrst coordination shell is actually composed of two sub populations of water molecules, with one subset occupying the region closest to the ion and the other subset situated at a slightly larger distance. In Fig.2, contributions to the radial density distribution of the four (4) water molecules nearest the K+ ion and the next two (2) water molecules have been separated out from the full distribution. Two overlapping, but distinct populations of water molecules are apparent. Note that for r = 3 ˚ where a shoulder appears on the far slope of the principal maximum in g(r), the mean A occupancy is about four (4). This contrasts to the mean occupancy of six (6) in the inner hydration shell given by the conventional deﬁnition above, where the ﬁrst minimum in g(r) establishes identiﬁcation of the inner population.
In the analysis presented next, quasi-chemical theory establishes four (4) as the most probable inner occupancy, in agreement with the ﬁrst subpopulation identiﬁed in the preceding analysis of the dynamical data, and on that basis a quasi-chemical analysis provides a good estimate of the absolute hydration free energy.
Previous simulation results agree qualitatively with Fig. 2, but diﬀerences in resolution , or concentration [14, 15, 16] preclude a more detailed comparison.
B. Quasi-chemical calculations
The hydration free energy of K+, using the classical molecular dynamics calculation for the outer-shell contribution, is −70.5 ± 2.1 kcal/mole for transfer of the solute from 1 mol (ideal gas) to 1 mol (ideally diluted solute) solution. The experimental values are suitably adjusted to account for this choice of standard state. As found before , the agreement between our absolute hydration free energies and the values of Coe and coworkers  (−86 kcal/mole) is poor. The sign and magnitude of this discrepancy is in line with discrepancies already identiﬁed for H+, Li+, and Na+ and further validates our suggestion that the absolute hydration free energies  estimated by Coe and coworkers contain a negative contribution from the potential of the phase.
Nevertheless, solvation free energies of neutral combinations, such as KOH, are thermodynamically unambiguous and can be accessed experimentally. With the hydration free energy of HO−, −123.8 kcal/mole, computed earlier  using the SPC/E model for the outer-shell contribution, the hydration free energy of the neutral combination KOH is −194.3 ± 2.1.
This is in good agreement with the experimental value of −191 kcal/mole by Coe et al., adjusted for our choice of standard states.
We note that for the hydration of HO− (aq), the quasi-component was the tri-hydrated state HO[H2 O]3 −. This choice was conﬁrmed by both ab initio molecular dynamics  and quasi-chemical calculations .
Primitive quasi-chemical theory identiﬁes four (4) as the most probable water/oxygen occupancy of a chemically deﬁned inner shell for the K+ (aq) at inﬁnite dilution. On this basis, that quasi-chemical theory gives a good estimate of the absolute hydration free energy of K+ (aq). These results are consistent with AIMD observations, but the inner shell is less clearly deﬁned by observation of gKO (r) only. In particular, the ﬁrst minimum of gKO (r) is shallow, and the principal maximum shows a distinct shoulder that delineates a second population of two water molecules beyond the inner-most four water molecules. That second set overlaps an anticipated ﬁrst minimum region. These features are distinctly diﬀerent for the corresponding results for Li+ (aq) and, to a lesser degree, Na+ (aq).
Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the US Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. The work at Los Alamos was supported by the US Department of Energy, contract W-7405-ENG-36, under the LDRD program at Los Alamos.