Density Functional Calculations for Aqueous Silver Clusters

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B: Liquids, Chemical and Dynamical Processes in Solution, Spectroscopy in Solution

Density Functional Calculations for Aqueous Silver Clusters Containing Water and Nitrate Ligands Roger Charles Baetzold J. Phys. Chem. B, Just Accepted Manuscript • Publication Date (Web): 03 Sep 2019 Downloaded from pubs.acs.org on September 3, 2019

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The Journal of Physical Chemistry

Density Functional Calculations for Aqueous Silver Clusters Containing Water and Nitrate Ligands

Roger C. Baetzold* 4026 W 32nd. Street, Erie, Pennsylvania 16506 United States

Corresponding Author: Roger C. Baetzold, Email- [email protected]

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ABSTRACT: The incorporation of nitrate ion into silver-water aqueous clusters has been examined using PBE0 density functional theory with the SMD solvation model. The Gibbs free energy of solvation and other thermodynamic variables are calculated using the harmonic/ rigid rotor/ ideal gas model at 298.15 K for aqueous solutes including the effects of solute relaxation in water and with London dispersive forces at the D3 level. Free energies of solvation for Ag+ and NO3- were found to agree well with experimental values of -118.2 and -60.1 kcal/mol, respectively calculated using cluster- continuum models with six to eight water molecules and including solute relaxation and London D3 dispersive interactions. An analysis of data of varying cluster size upon calculated free energy is presented. A direct procedure is applied to aqueous clusters such as AgnZ(NO3-)(H2O)5, AgnZ(H2O)5 and (NO3-)(H2O)6 n=1-4; Z=0, +1 in the SMD solvent representation in order to calculate equilibrium constants for nitrate association with silver clusters in solution that includes fully relaxed solutes. The equilibrium structures of the nitrate containing clusters involve one or more bonds from nitrate oxygen to positive silver clusters. Water molecules interact with nitrate through H atoms and overall the structure represents a silver-nitrate cluster with water ligands having similarity to a close ion pair in many aspects. The neutral silver atom is attached to nitrate through H bonded water molecules. The ratio of nitrate containing silver clusters to nitrate free clusters using calculated equilibrium constant of 0.51 l/mol for Ag+ is small in the range of many experiments. Similar values are found for positive silver clusters up to four atoms in size. The resulting procedures were applied to aqueous clusters of Agn(NO3)m+(n-m) that have been previously experimentally studied for silver reduction in aqueous solution. A chain like structure with collinear and bidentate oxygen bonds to silver was found and the equilibrium constants for clustering were determined. A 2 ACS Paragon Plus Environment

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simplified model calculation for the reduction of Ag(H2O)6+ clusters in the presence of silver clusters in aqueous media was studied to understand catalytic effects observed in these systems. The reduction potentials vary with silver cluster size indicating a more favorable reduction caused by the presence of larger silver clusters.

INTRODUCTION Silver clusters in solution and the solid phase are increasingly useful beyond conventional photography1 and include applications such as bactericides2, environmental pollution control agents3 and medical agents.4 Silver clusters have been noted to be particularly reactive with dissolved organic matter5 in aqueous solution. Various studies have probed the fundamental properties of silver particles in aqueous media6-11 in relation to these applications. The existence of cluster growth units larger than monomer has been reported in many studies. A study of aqueous solutions of silver nitrate with citrate reducing agent using small angle X-ray scattering gives evidence for Ag13 as the elementary growth particle which can agglomerate to form larger nanoparticles.7 In the presence of silver clusters, growth mechanisms to form larger nanoparticles often involve effects due to the counterions such as citrate11 or other anions. In the presence of nitrate counterions, the slow reduction of silver ions has given evidence for Ag3(NO3)2+ clusters that may lead to nanoplates of silver.12 Evidence for the presence of aqueous ion pairs and ion clusters of other materials such as alkali chlorides is found in molecular dynamics simulations in the aqueous phase.13 In a separate application the usefulness of models of hydrated silver clusters has been demonstrated in studies explaining



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aspects of aqueous SERS phenomena.14 The hydrated silver cluster Ag(H2O)2+ has found use in computational studies of the hydrogen reduction of Ag+ to form nanoparticles.15 Nucleation of silver particles by reduction of Ag+ in solution to form nanoparticles has been probed by kinetic studies.16-18 Sigmoidal shape growth curves are observed and give evidence for an autocatalytic mechanism of growth. Related studies with the reducing agent hydroquinone19,20 have demonstrated the need for borohydride or photochemical treatment in order to form silver seeds that serve as a catalyst for further silver reduction. Hydroquinone is a reducing agent commonly used in photographic experiments and an early photographic study21 provided evidence that a critical size of four or more silver atoms was required to act as a catalyst for reduction of silver aqueous ions under typical conditions of room temperature and a few minutes time. In this experiment vapor deposited silver on inert support was used as the catalyst for reduction of silver ions which mirrors analogous reactions that take place in silver halide photography, where the photochemical and catalytic steps are not separated. More precise control of cluster size was achieved in experiments using vapor deposited mass selected silver clusters as catalysts for silver ion reduction and gave additional evidence for the critical size concept.22 Meanwhile, a number of charged and neutral nanoclusters of silver have been prepared and studied in aqueous solution using pulse radiolysis.23-27 Their redox potentials were measured and an oscillatory odd-even trend was displayed for the small clusters. These findings were applied as a model for silver photographic development in which the properties of silver clusters in aqueous solution were proposed as a model for silver ion reduction at the solid silver catalyst located at the solution interface.28,29 These studies show the importance of silver nanoparticles in aqueous solution as models for redox reactions. 4 ACS Paragon Plus Environment

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Much attention has been focused on gas phase silver cation clusters coordinated with water molecules. Collinear structures containing multiple ligands beginning with two water molecules coordinated to silver ion through oxygen have been frequently observed in the gas phase. A recent review has collected experimental gas phase studies of Ag(H2O)n+ n=1-6 that determine the bond strengths for successive ligand additions.30 Gas phase experimental studies using photoelectron imaging of silver anion hydrated with one or two water ligands have been reported and the corresponding structures computed with density functional theory (DFT).31 One water ligand interacts through hydrogen with Ag anion or through oxygen with neutral Ag. X-ray absorption studies of hydrated silver ion gave evidence for five and six water ligands in the first coordination sphere with a significant number of collinear bond angles.32 Calculations of gas phase bare silver clusters have a long history and only some recent work that is most relevant to this study is noted. Structures and properties of neutral clusters up to 100 atoms were computed using different methods and show a slow convergence to bulk behavior.33 A detailed study of small clusters using DFT and coupled cluster methods (CCSD) compared several density functionals and gave good support to the PBE0 functional for determining structures and properties of clusters up to four atoms.34,35 More recently a study of silver clusters up to seven atoms with the three charge states has examined geometry and many physical properties of the clusters for a range of density functionals comparing results to experimental and CCSD(T) calculations. A ranking of relative accuracy of functionals was made on this basis.36 The Ag(H2O) cluster has been examined in a range of calculations37,38 including those that include relativistic effects and the PBE0 functional was concluded to reproduce stability of this cluster and a range of other silver clusters with lone pair ligands quite well. 5 ACS Paragon Plus Environment


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In view of this background, earlier and the present computational studies with the PBE0 functional (denoted PBE1PBE) are considered reasonable for hydrated silver clusters in the gas phase and liquid phase.39,40 The earlier studies have focused on the thermodynamic properties of the clusters, their redox properties and structure. Chains of hydrogen bonded water molecules were found to attach to the small neutral clusters through a silver-oxygen interaction and a silver-hydrogen interaction at the ends of the chains. Positively charged clusters have water ligands coordinated to silver principally through oxygen, while negative clusters have principally a silver-hydrogen coordination. As more water molecules attach to the silver cluster more complex structures form through hydrogen bonding. The present work addresses additional issues which are expected to be important in hydrated silver cluster systems and their computational treatment. One Important issue is to begin to evaluate the possible effects of counterions in aqueous solution of these clusters and the degree of association as related to structures and thermodynamic properties of the aqueous cluster. This work undertakes such a study using soluble nitrate ion which is a common counterion in various experiments and has many possible applications in its own right. In the course of this study several computational issues common to hydrated clusters with or without the counterion were evaluated. Dispersion London interactions41 have been shown to improve reliability of DFT calculations in various systems. Hydrogen bonded systems might be particularly prone to these interactions so this matter is explored in this work. Previous calculations39,40 of solution free energy of solvation employed a gas phase structure in a clustercontinuum implicit solvation method.42-50 The effect of solute relaxation in solution on the thermodynamic properties of the cluster is examined in this work. An issue that was examined 6 ACS Paragon Plus Environment

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in lesser detail is the effect of possible anharmonicity in the low frequency vibrations. This was studied for several small silver-water clusters that are sufficiently rigid in structure to facilitate application of the perturbation method VPT251 and are discussed in the Supporting Information. The refined computational protocol is applied to better understand nitrate containing silver-water aqueous clusters and to examine the catalytic role expected for a silver cluster in the presence of mild aqueous reducing agent. METHOD Density Functional (DFT) method and associated thermodynamic calculations were employed within the Gaussian 09 code52 using the PBE0 functional53, sometimes designated PBE1PBE. The jun-cc-pvTZ basis set, derived from the fully augmented aug-cc-pvTZ basis functions54 by removing diffuse functions from H and retaining only s, p, d diffuse functions on O and N atoms55, was employed with the SDD56 basis sets and effective core potential for Ag. In addition, calculations using the cc-pvTZ and 6-31+G(d,p) basis sets for O, N and H atoms were employed. Some other functionals including MO62X, M06L, TPSS, B3PW91, and wB97 were compared to PBE0 for some particular silver-water cluster properties in the Supporting Information. Optimized structures for a range of isomer structures of stable silver-water clusters for each basis set and type of calculation with and without nitrate in the gas phase are first determined. Usually a smaller basis set is used to provide a starting point for optimization using the jun-cc-pvTZ basis. The properties and coordinates of these optimized structures and competing isomers are given in the Supporting Information. A harmonic frequency analysis for each case is employed to assure that no imaginary frequencies are present in all calculations reported in this work and thereby guaranteeing an energy minimum. A range of different 7 ACS Paragon Plus Environment


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starting configurations were examined in order to select the most stable structures. It was found that overall shapes of optimized structures did not change with the different basis sets having only small differences in the coordinates. The Gibbs free energy was calculated using the partition functions appropriate to the rigid rotor/ ideal gas/ harmonic oscillator approximations. The basis set superposition errors (BSSE) between silver cluster and ligand molecules was corrected using the counterpoise method.57 The structures with the most negative free energy were most stable and the energy of hydration at 0 K (ΔE0) and the Gibbs free energy (ΔG⁰) of hydration at 298.15 K were calculated by difference of the comparable quantities for the water cluster molecules and bare silver clusters with or without nitrate ion. Calculations including possible London dispersion effects were examined using the D3 model of Grimme.41 These calculations require a re-optimization of structures in order to determine thermodynamic properties and permits an explicit treatment of dispersive interactions within a given cluster model. It is known that the low frequency vibrations can deviate most from the harmonic behavior of the model employed. In order to examine the magnitude of this effect on energies, calculations using the VPT2 perturbation method51 were undertaken in order to compare thermodynamic properties for smaller silver water clusters for both calculations. The results of this comparison are restricted to semi-rigid clusters and given in the Supporting Information Section. In earlier work40 a method based on the quasi-chemical method58, 59 was employed to treat such effects.


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The solvation model SMD60 contained in the Gaussian 09 code was used with optimized structures in order to simulate the effects of aqueous environments on solutes. The formula43, 44

for the Gibbs free energy of solvation ΔG*solv(S) of solute S in the SMD calculation was

ΔG*solv(S) = Esoln + Gnes - Egas

(1)

where Esoln and Egas are the respective electronic energies of the solute in the presence and absence of the solvent field and Gnes is the sum of non-electrostatic contributions to the Gibbs free energy. The asterisk denotes the same liquid and gas phase standard state. Two levels of calculation are employed for the aqueous terms in eq 1. In the first the solute geometry is fixed at the gas phase structure and in the next the relaxed solute relaxed geometry in employed with SMD. The solvation energies of the silver clusters are calculated in the cluster-continuum method in the cluster cycle which considers clusters of water molecules rather than monomers. This approach uses well known45-50 thermochemical cycles to express the solvation energy ΔG0solv(S) at 1M and 298.15 K liquid standard state and corrects for the change of standard state of 1 mol per 24.46 L at 1 atm in the gas phase to 1 M in the liquid with accounting the concentration of water ligands in the solvent. The formula ΔG0solv(S) = ΔG0 (WnS) + ΔG*solv (WnS) –ΔG*solv (Wn) – ΔGss − RT ln ([55.34]/n)

(2)

Is employed where Wn is a cluster of n water molecules, ΔG*solv(S) is the free energy of solvation of solute S, ΔG0(WnS) is the gas phase free energy of hydration of a cluster of solute S with a cluster of n water molecules , ΔG*solv (WnS) and ΔG*solv (Wn) are the calculated free energy of solvation of WnS and a cluster of n water molecules Wn, respectively and the last two terms are standard state corrections where ΔGss has been calculated as 1.89 kcal/mol at room 9 ACS Paragon Plus Environment


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temperature in previous studies and the last term adjusts for the water cluster concentration at 55.34/n M to the 1M standard state. The ideal gas constant R and temperature T are as usual. Different levels of calculation are employed. In the first level the gas phase optimized structure of the solute is used for the solution geometry. In an improved method the relaxed aqueous geometry of the solute is employed in the liquid phase. The correction due to structure relaxation in solvent is calculated by difference of the solvation energy for gas phase and relaxed aqueous phase geometries of the solute using eq 1. The calculation employs the 631+G(d,p) basis sets on O, N and H atoms. Equilibria are considered using the cluster-continuum model with the 6-31+G(d,p) basis and by completely relaxing reactant and product clusters in the SMD solvent model representation. This direct approximation and its justification have been presented and employed before50,58,61,62,63 in aqueous calculations and applications such as calculation of pKa constants. The difference in Gibbs free energy of product and reactant is calculated to determine the overall free energy change and corrected for BSSE error in order to determine the corresponding equilibrium constant. RESULTS We first turn to the constituent silver-water and nitrate-water clusters as a foundation for a study of properties of silver-water-nitrate clusters. Experimental gas phase bond energies30 for successive addition of water ligand to Ag(H2O)n-1+ for n=1-6 are available at low temperature to reference calculated hydration energy values. The structures of these cation clusters contain collinear oxygen-silver-oxygen bonds and have been calculated with no symmetry restraints to follow the geometry shown in Figure 1. The calculated energy changes at 0 K without zero-point 10 ACS Paragon Plus Environment

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contributions for addition of successive water ligands to the silver ion are shown in Table 1 and compared to experiment. There are two sets of experimental data giving a small range of values in Table 1. The rigid rotor / ideal gas calculations are presented in successive columns beginning with the quasi-chemical method in which low frequency vibrations below 100 cm-1 are replaced by 100 cm-1 vibrations in the vibrational partition function without additional scaling, the harmonic method (harmonic), the perturbative VPT2 method and the harmonic method including dispersive interactions (D3).

Figure 1. Calculated Gas Phase Structure for Ag(H2O)10+ with Numbering to Indicate the Sequence of Water Ligand Additions.



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Table 1. Calculated Energy at 0 K for Successive Water Ligand Additions to Ag(H2O)n-1+ for n=16 Using a Quasi-Chemical Method, the Harmonic Method, the Anharmonic VPT2 Method and the Harmonic Method With Dispersive D3 Interactions Using Jun-cc-pvTZ Basis Sets, the PBE0 Functional and BSSE Corrections with Mean Unsigned Error Versus Experiment (kcal/mol).30 n

Quasi

Harmonic

VPT2

D3

Exp.

1

27.8

30.2

30.2

30.6

31.3−32.7

2

25.5

29.3

29.2

29.8

26.3−30.1

3

11.4

16.0

16.1

17.0

14.8

4

11.4

15.2

15.3

16.3

11.5-13.9

5

8.7

12.2

12.3

13.3

13.6

6

8.6

11.6

8.8

12.7

13.4

MUE

3.6

1.6

2.1

1.6

The values determined with the quasi-chemical method without scaling in the first column in Table 1 show correct trends, but are a few kcal/mol less than experiment. Scaling would be required to obtain better agreement with experiment. The harmonic calculation gives values closer to experiment. The VP2T perturbation method results do not provide significant improved agreement with experiment compared to the harmonic calculation. The dispersion calculation (D3) gives better agreement with experiment than the harmonic approximation for several clusters, but worse agreement for Ag(H2O)2+ and Ag(H2O)3+. Overall, the harmonic calculations with or without dispersive interactions give good agreement with experiment. The mean unsigned error (MUE) is given for each calculation versus the average experimental values. Temperature dependent effects are not tested by this comparison. When the aug-cc12 ACS Paragon Plus Environment

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pvTZ basis is employed in the harmonic approximation, the calculated ligand bond energies are only a few tenths of a kcal larger than with the jun-cc-pvTZ basis. A detailed comparison of the Gibbs free energy of hydration (ΔG⁰) at 298.15 K for forming gaseous silver-water clusters from the corresponding silver cluster and water cluster and corresponding free energies of solvation calculated at different levels is shown in Table 2. The isomer with the most stable free energy of solvation is considered. Detailed structure and property information for competing isomers are contained in the Supporting Information and some have been discussed earlier.39, 40 The lowest free energy gas phase structures of neutral monomer and dimer silver clusters contain chains of water molecules with a terminal Ag-O interaction that also connects to the cluster by a Ag-H interaction at the end of the chain. Neutral silver trimer and tetramer clusters contain water molecules in chain form up to four molecules in length that changes to a preferred three-dimensional structure at larger clusters. Cation silver monomer and dimers contain collinear water molecules as shown in Figure 1. The trimer and tetramer silver cations structures with the maximum number of Ag-O interactions up to four water molecules and for six molecules two three molecule chains are preferred. More detail about these structures and competing isomers is found in the Supporting Information section as well as data for the water clusters. Generally the trends in these Gibbs free energies with cluster size within the same type of calculation are monotonic with few exceptions. Successive additions of water ligands to the most stable silver-water clusters of the same charge give a more positive free energy value. The entropy changes on hydration can play an important role in determining preferred structures.



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Table 2. Calculated Hydration Gibbs Free Energies (ΔG⁰) at 298.15 K, 1 atm for Agn(H2O)m (kcal/mol) Formed from Corresponding Charge States of Agn with H2O Clusters Using PBE0 Functional, SDD + Jun−cc−pvTZ Basis Sets and BSSE Correction at the Harmonic Level With and Without D3 Dispersion. The Free Energy of Aqueous Solvation (ΔG0solv) is Calculated at 298.15, 1M with Gas Phase Solute Structures Followed by Data for Solute Structure Relaxed in Solvent (RL). Entries for Cation Clusters Follow the Neutral Counterparts. cluster

ΔG⁰

ΔG⁰,D3 ΔG0solv

ΔG0solv,RL

ΔG0solv,D3

ΔG0solv,D3+RL

Ag(H2O)

3.5

2.7

0.4

-1.3

-0.6

-2.3

-24.2

-24.6

-104.8

-106.3

-105.2

-106.7

3.9

2.6

2.9

0.5

1.4

-1.0

-46.1

-46.6

-109.3

-111.9

-109.6

-112.2

3.4

2.2

0.0

-0.3

-1.1

-1.4

-54.2

-54.3

-113.2

-115.7

-113.2

-115.7

5.3

4.4

2.3

1.9

0.2

-0.2

-59.5

-58.8

-112.1

-114.7

-112.3

-114.9

7.8

5.8

3.6

3.1

1.6

1.1

-61.9

-62.7

-111.3

-113.9

-112.5

-115.1

10.3

10.1

5.9

3.8

1.9

-0.2

-67.5

-67.5

-113.5

-116.4

-114.4

-117.3

1.2

0.6

-6.2

-6.6

-6.7

-7.1

-14.8

-15.7

-85.1

-86.5

-85.4

-86.8

-0.5

-1.9

-3.7

-4.7

-2.6

-3.6

-28.6

-29.4

-86.1

-88.3

-86.8

-89.0

-3.4

-5.2

-8.9

-9.2

-10.6

-10.9

-34.3

-34.5

-88.7

-91.1

-89.9

-92.3

-2.2

-3.6

-7.0

-8.8

-9.2

-10.9

Ag(H2O)2 Ag(H2O)3 Ag(H2O)4 Ag(H2O)5 Ag(H2O)6 Ag2(H2O) Ag2(H2O)2 Ag2(H2O)3 Ag2(H2O)4


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Ag2(H2O)5 Ag2(H2O)6 Ag3(H2O)2 Ag3(H2O)3 Ag3(H2O)4 Ag3(H2O)6 Ag4(H2O)2 Ag4(H2O)4

-37.0

-36.5

-87.6

-90.1

-87.9

-90.4

0.3

-2.8

-5.3

-5.8

-8.6

-9.1

-32.2

-40.2

-84.2

-86.8

-88.8

-91.4

-0.6

-3.4

-7.8

-8.0

-10.5

-10.9

-42.2

-42.4

-88.8

-88.9

-89.2

-89.3

-3.8

-4.2

-5.4

-5.8

-5.8

-6.2

-23.3

-23.9

-79.4

-80.7

-79.8

-81.1

-6.1

-6.5

-10.0

-10.3

-10.3

-10.6

-32.8

-34.1

-85.2

-87.5

-86.9

-89.2

-3.9

-3.7

-7.9

-8.3

-8.0

-8.4

-34.8

-34.2

-82.7

-84.9

-84.3

-86.5

-1.9

-3.4

-7.8

-9.4

-10.4

-11.0

-36.8

-36.1

-84.3

-87.0

-83.8

-86.5

-3.3

-5.9

-7.1

-7.3

-9.3

-9.5

-20.0

-21.9

-75.4

-77.3

-76.3

-78.2

-3.5

-6.5

-10.7

-1.0

-14.3

-14.6

-26.9

-27.2

-76.9

-79.3

-78.7

-81.1

The data in Table 2 show the effects of various approximations on the calculated Gibbs free energies of hydration and solvation free energies of silver-water clusters. Isomers with the greatest stability in aqueous environment are considered with data for other isomers given in the Supporting Information. The London dispersive interactions generally lead to a more stable cluster (-ΔG⁰) and a greater solvation energy (-ΔG0solv) with some small exceptions due to structural changes upon relaxation. The Gibbs free energy of hydration per water molecule within a series of neutral clusters containing the same number of silver atoms increases with size indicating a more stable larger cluster. The calculated aqueous solvation free energies 15 ACS Paragon Plus Environment


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given in Table 2 use the cluster-continuum model expressed by eq 2 and are considered with the solute gas phase or aqueous structure. In the cluster-continuum model the cluster is a representation of the local environment of water molecules near the solute species under study. Relaxation of solute in the solvent represented by the SMD method increases the solvation energy and can amount to several kcal/mol versus the unrelaxed structures. The net relaxation of product minus reactant, as defined in eq 2, has been considered. Neutral clusters containing one silver atom have positive free energy of solution values, but with increasing numbers of silver atoms become negative. Positive silver ion shows more complex trends in the free energy of solvation with cluster size and decreases as the number of silver atoms increase. We turn to the experimental free energy of solvation values at standard state. Experimental data from the Tables of Marcus are employed.64 The absolute solvation free energy values given by Marcus are based upon a proton free energy value of -1056 kJ/mol (-252.4 kcal/mol) before standard state corrections for transfer from gas to liquid. This reference value has undergone revision in more recent work including that of Tissander65 who found -264.0 kcal/mol, Zhan and Dixon66 who calculated -262.4 kcal/mol and Kepp67 who used -262.9 kcal/mol. The Tissander value when including the standard state correction ( ΔGss ) becomes 265.9 kcal/mol has been used by many workers46, 62, 63 and is employed in this work. Using this reference with the data in the Marcus tables yields an experimental free energy of solvation at 298.15 K of -60.1 kcal/mol for NO3- and -118.2 kcal/mol for Ag+. It is noted that an earlier analysis46 reported -118.7 kcal/mol for the experimental solvation free energy of Ag+.


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The Gibbs free energy of solvation of Ag+ at 298.15 K is calculated using eq 2 and is considered for different basis sets with D3 dispersive interactions in Table 3. The structure shown in Figure 1 was followed for successive additions of up to ten water ligands. Even the smaller basis sets provide good agreement with the experimental Gibbs free energy of solvation (-ΔG0solv) for several of the larger clusters. The overall best agreement with experiment is found at clusters with 6-8 water ligands. The solute cluster relaxation in the aqueous environment described by the SMD method provides an increase by a few kcal/mol. The data using the juncc-pvTZ basis set gives optimal agreement with experiment at 6-7 water ligands and then becomes larger at the clusters with 8-10 water ligands. This may indicate that the rather asymmetric larger structure is causing artifacts. The inclusion of D3 interactions slightly increases the solvation energy of the smaller clusters, but a decrease occurs at the larger clusters due to larger interaction energies found in the water clusters relative to the silverwater clusters. This effect gives a good agreement with experiment at the ten water ligand cluster. Excellent agreement with experiment can be found for Ag+ ion free energy of solvation through clusters in the 6-7 water ligand size. Even larger clusters would be required to examine these effects in detail and evaluate convergence with the number of ligands. The mean unsigned error is given for relaxed clusters with six or more water ligands. Caution is always needed because perverse effects such as cancellation of errors could yield a seemingly better agreement with experiment than the method of calculation is entitled to. Table 3. Calculated Ag+ Free Energy of Aqueous Solvation (ΔG0solv(Ag+)) at 298.15 K and Mean Unsigned Error in (kcal/mol) Using the Harmonic Model for Ag+(H2O)n Clusters with Different 17 ACS Paragon Plus Environment


Page 18 of 50

Basis Sets and Indicating D3 Dispersion. Values Are Given Without/With Solute Relaxation in the Solvent. n

6-31+G(d,p)

cc-pvTZ

jun-cc-pvTZ

jun-cc-pvTZ , D3

1

-103.8/-105.3

-104.4/-105.9

-104.8/-106.3

-105.2/-106.7

2

-107.6/-110.2

-108.3/-110.9

-109.3/-111.9

-109.6/-112.2

3

-111.4/-113.9

-109.8/-112.6

-113.2/-115.7

-113.2/-115.7

4

-109.9/-112.7

-109.5/-112.1

-112.1/-114.7

-112.3/-114.9

5

-110.1/-112.6

-109.3/-111.9

-111.3/-113.9

-112.5/-115.1

6

-112.1/-114.6

-111.0/-111.9

-113.5/-116.4

-114.4/-117.3

7

-117.6/-118.2

-113.3/-114.0

-118.3/-118.9

-117.7/-118.3

8

-117.9/-120.3

-112.4/-114.9

-121.0/-123.5

-117.4/-119.8

10

-118.9/-121.0

-111.9/-113.9

-121.6/-123.7

-116.5/-118.6

MUE

2.1

4.5

3.3

0.8

The free energy of aqueous solvation of nitrate is also calculated using the cluster-continuum method used for silver ion. Hydration of the nitrate ions has been studied68-70 by DFT calculations and this work on energy and structure guided our choice of model clusters of (NO3-)(H2O)n where n=1-6 and their shapes. The first three water ligand bind in a bidentate fashion through H to the oxygen atoms in nitrate. The structure may be of equatorial or of near D3h shape. Additional ligands attach to the core D3h or equatorial clusters through hydrogen bonding. No symmetry constraints are employed in these calculations. Figure 2 shows the calculated aqueous phase structures for equatorial (NO3-)(H2O)3 and (NO3-)(H2O)6 which give more stable free energy of hydration in the gas and aqueous phases than the D3h structures. These equatorial structures deviate slightly from planarity and a range of O-H interaction 18 ACS Paragon Plus Environment

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distances of 1.82 Å to 2.21 Å are observed. Smaller clusters with incomplete shells of water molecules contain water molecules at positions close to their complete shell analogues. The cluster continuum method in eq 2 was used to calculate the Gibbs free energy of solvation of nitrate in water at standard conditions. Table 4 contains the gas phase hydration free energy and the aqueous phase values calculated at the harmonic level and include D3 dispersion and aqueous solute relaxation effects along with the mean unsigned error for the equatorial cluster calculations. Some dependence upon cluster size is noted and generally the larger clusters have the most negative values for the aqueous free energy of solvation and agree best with the experimental value of -60.1 kcal/mol. The data calculated with the 6-31+G(d,p) basis are slightly smaller for most clusters than the jun-cc-pvTZ data.



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Figure 2. Calculated Aqueous Phase Structures for (NO3-)(H2O)3 (a) and (NO3-)(H2O)6 (b) in the Equatorial Orientation Calculated with PBE0, Jun-cc-PVTZ Basis in SMD Solvent Model. Table 4. Gibbs Free Energy of Hydration of Nitrate Ion ΔG( gas) and Aqueous Solvation Free Energy ΔG0solv Calculated at STP for (NO3-)(H2O)n Clusters With PBE0 Functional and Jun-ccpvTZ Basis or the 6-31+G(d,p) Basis Denoted G and Mean Unsigned Error in (kcal/mol). Solute Relaxation is Included and D3 Dispersion as Indicated for the Equatorial or D3h Structures. n

ΔG(gas)

ΔG(gas),D3

ΔG0solv

ΔG0solv, D3

ΔG0solv,G

1

-5.4

-6.3

-59.9

-60.7

-57.0

2

-11.9

-13.1

-58.4

-59.4

-58.9

3

-16.0

-17.0

-61.7

-62.2

-59.5

4

-15.3

-17.0

-59.2

-59.9

-57.1

5

-15.1

-16.6

-58.4

-60.1

-56.1

6

-14.8

-17.3

-58.7

-61.1

-55.9

3 D3h

-11.4

-14.1

-56.2

-58.9

-55.8

6 D3h

-12.5

-17.2

-50.0

-54.7

-51.5

1.3

0.8

2.7

MUE

The effects of counterions in aqueous solution on silver-water clusters have been examined by calculations of the properties of hydrated silver clusters with one incorporated nitrate ion by applying the cluster-continuum procedure starting with the optimized gas phase geometry. Several silver clusters containing one nitrate ion and up to five water ligands in the first coordination sphere with the formula Agnz(NO3-)(H2O)5 (n=1-4;z=0,+1) were examined. Only data for the five water ligand clusters is given, although clusters containing fewer water ligands 20 ACS Paragon Plus Environment

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were explored. The harmonic model within the ideal gas/ rigid rotor framework was employed to find the optimized structures reported in this work. Figure 3a shows a cluster containing a silver ion with the corresponding cluster containing a silver atom shown in Figure 3b. Nitrate ion is coordinated to three or more water ligands through hydrogen. The remaining water ligands surrounding the silver part of the cluster in an arrangement depending upon charge. For neutral silver clusters water ligands interact with silver atom through hydrogen. A collinear structure of silver oxygen bonds is found for the silver cation and this type of structure persists in the larger positive silver clusters.

Figure 3. Calculated Gas Phase Structure for Ag+(NO3-)(H2O)5 (a) and Ag(NO3-)(H2O)5 (b). The larger silver units lead to more complex structures. Figure 4a shows Ag2+ with two water molecules bonded in a collinear fashion through oxygen. There is also evidence of interaction through hydrogen bonding with water ligands attached to the nitrate. Each of the two silver atoms interact directly with one nitrate oxygen. Two water ligands are in immediate contact with the nitrate. The cluster containing Ag2 in Figure 4b is an extension of the structure for one neutral silver atom. One of the silver atoms bonds directly to oxygen of nitrate and two 21 ACS Paragon Plus Environment


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water ligands have interactions through hydrogen with the silver cluster and with another water ligand or a nitrate oxygen.

Figure 4. Calculated Gas Phase Structure for Ag2+(NO3-)(H2O)5 (a) and Ag2(NO3-)(H2O)5 (b). Trimer silver units continue the patterns exhibited in the smaller silver clusters and are shown in Figure 5. The silver trimer cation has a water ligand bonded to each silver atom through oxygen, as in the silver cation cluster absent nitrate, and interacts directly with nitrate through one oxygen and the attached water ligands that bond through hydrogen to a nitrate oxygen. A complicated sharing of water ligands between nitrate and silver cluster results. The neutral silver trimer takes a structure that is an extension of the smaller neutral clusters. The cluster is bound directly through one silver atom to an oxygen ion of nitrate ion. One water ligand bonds through hydrogen to the silver trimer.


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Figure 5. Calculated Gas Phase Structure for Ag3+(NO3-)(H2O)5 (a) and Ag3(NO3-)(H2O)5 (b). The tetramer structures shown in Figure 6 maintain the Ag4+ rhombic structure having water ligands bonded terminally through oxygen. This unit interacts with nitrate ion directly through one oxygen ion and additional interactions through the attached water ligands. The neutral Ag4 rhombus interacts with nitrate through one silver atom with oxygen and two water ligands that interact through hydrogen and represents an extension of the pattern found in smaller clusters. Overall, it is noted that the interaction of nitrate containing silver clusters with water ligands is much the same as found in the hydrated silver-water clusters. The structure for the clusters containing positive silver molecules resembles a close ion-pair and a somewhat similar structure is found for the clusters containing neutral silver molecules. It is noted that covalent forces are involved in these treatments that also treat Coulombic forces.



Page 24 of 50

Figure 6. Calculated Gas Phase Structure for Ag4+(NO3-)(H2O)5 (a) and Ag4(NO3-)(H2O)5 (b). Table 5 contains the shortest bond lengths found in the optimized structures. Some trends with size are found, but the complex structures can obscure many trends. The shortest Ag+ distance to nitrate O becomes larger with cation silver cluster size because the structure changes from a single bond at Ag+ to more complex bonding at larger silver units. The water ligand oxygen bond to silver in cation clusters remains short at all cluster sizes in the range of 2.16 to 2.29 Å. The shortest Ag-H distance resulting from interaction with water ligands is in the range of 2.5 – 2.9 Å for clusters with neutral or positive silver units. For clusters with neutral silver units, the silver oxygen bond is shorter for oxygen in nitrate than water. 24 ACS Paragon Plus Environment

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Table 5. Shortest Calculated Silver Ligand Bond Lengths (Å) in AgnZNO3-(H2O)5 Gas Phase Clusters Using Jun-cc-pvTZ Basis in the Method Discussed in Text. Nitrate

Water

AgnZ

Ag-O

Ag-O

Ag-H

Ag+

2.168

2.166

2.619

Ag

2.478

3.472

2.803

Ag2+

2.463

2.287

2.822

Ag2

2.271

3.385

2.777

Ag3+

3.125

2.237

2.524

Ag3

2.300

2.784

2.700

Ag4+

2.699

2.265

2.786

Ag4

2.263

3.376

2.867

The electron population of ions in the cluster is calculated with the Atomic Polar Tensors (APT) method71 and given in Table 6 which shows the average charge of the silver atom, nitrate ion and water ligand calculated from the populations in these clusters. A small electron exchange is noted from the silver component of the neutral cluster and to the silver component of the positive cluster. Table 6. The AgnZ(NO3-)(H2O)5 APT Average Charge Is Shown for the Silver Atom, Nitrate and Water Molecule Components. AgnZ

Silver

Nitrate

Water



Ag+

0.749

-0.921

0.034

Ag

0.081

-1.047

-0.007

Ag2+

0.497

-1.030

0.007

Ag2

0.060

-1.088

-0.006

Ag3+

0.329

-1.021

0.007

Ag3

0.045

-1.067

-0.013

Ag4+

0.281

-1.067

-0.012

Ag4

0.020

-1.021

-0.012

Page 26 of 50

The thermodynamic properties of these clusters in aqueous solution at 298.15 K are calculated through optimization of their structure in the aqueous environment represented by the SMD solvation model. The SDD and 6-31+G(d,p) basis sets are employed with the PBE0 functional to determine aqueous equilibrium constants. Consider the aqueous equilibrium of positive silver cluster (AgnZ) and nitrate ion to form the associated cluster AgnZ + NO3- → AgnZNO3-

(3)

n=1-4; Z=0,+1 ΔG03

In the calculation this equilibrium has been represented by the balanced equation using explicit cluster molecules in the continuum model that overall represents the aqueous phase for positive silver clusters by Agn+(H2O)5 + (NO3-)(H2O)6 → Agn+(NO3-)(H2O)5 + (H2O)6

n=1-4 ΔG04

(4)

where the number of explicit water ligands is kept at 5 or 6 in each cluster. A lower degree of hydration is employed to represent the aqueous neutral silver clusters of Agn(H2O) reacting with (NO3-)(H2O)6 in an expression analogous to eq 4 where the associated nitrate neutral silver 26 ACS Paragon Plus Environment

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species and water dimer are the product. The aqueous cluster shape is similar to the gas phase structure and the Ag-O distance to nitrate oxygen has increased from 2.168 Å to 2.210 Å while the solvation energy defined by eq 1 has decreased by 3.2 kcal/mol for the Ag+ associated cluster. In the case of the Ag associated cluster, the Ag-O bond length increases from 2.474 Å to 3.636 Å with a corresponding solvation energy decrease by 4.5 kcal/mol. The Gibbs free energy of each unit in eq 4 is calculated at the optimized structure determined in the SMD solvent model and the overall standard free energy change ΔG04 is calculated. This quantity is converted to ΔG03 by accounting for the (H2O)n product constant concentration of 55.34/n M to give the values in Table 7. Table 7 Standard Free Energy Change (kcal/mol) for Association of Silver Clusters and Nitrate Ion in Equilibria 3 Calculated Using the PBE0 Functional With SDD and 6-31+G(d,p) Basis. The Free Energy Change (eV) upon Reduction in Solution is Also Given. AgnZ

ΔG03

ΔG0(reduction)

Ag+

0.4

-2.87

Ag

2.8

Ag2+

0.6

Ag2

2.0

Ag3+

0.7

Ag3

2.5

Ag4+

-0.1

Ag4

3.4

-3.95

-2.38

-3.28



Page 28 of 50

It is noted that solute relaxation in the aqueous environment is fully accounted for in this calculation. The equilibrium constant for eq 3 is calculated from the free energy change ΔG03 so that the ratio of associated to free silver cluster may be obtained by multiplying the constant by the nitrate concentration for ideal conditions. For example, taking the value for Ag+, the ratio of associated to free silver ion is 0.51C where C is the nitrate concentration. The concentration of associated silver ion with nitrate is small under most experiments and only when nitrate concentrations is 0.1 M and greater would significant association be expected. The magnitude of the free energy change for association is similar for Ag+ and the larger positive silver clusters so that similar association with nitrate is predicted for these cases. The neutral silver atom clusters in solution associate to a lesser degree than for cations as can be calculated from the free energy data in Table 7 and as would be expected on Coulombic grounds. Hydrogen bonding interactions are operating in the neutral silver cluster cases and there is apparently a shallow local minimum. Thus, the complexes formed from the neutral silver atom clusters are probably of short lifetime. The aqueous association of silver ions with multiple nitrate ions was reported12 in studies of nanoplate growth and attributed to the presence of Ag3(NO3)20or Ag3(NO3)2+ clusters as observed by mass spectroscopy of samples taken from aqueous silver nitrate. Additional other small clusters were reported of varying stoichiometry. Computation for this system offers a way to investigate multiple nitrate ions with silver ions. These calculations were applied to the equilibrium in eq 5. nAg+ + mNO3- →

Agn(NO3)m+(n-m)

n=1-3; m=1-4

ΔG05

(5) 28

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The direct method of calculation is employed by optimizing the structures of the small clusters represented in eq 5 in the aqueous phase with the SMD solvation model. This first level calculation is performed without the presence of explicit water ligands near the cluster since doing so requires a more complex calculation and so trends in this data may be more accurate than absolute values. The equilibrium structures calculated for the associated cluster Ag3(NO3)2+ and its reduced form in aqueous solution are shown in Figure 7 where each silver ion is attached to two oxygen ions from the nitrate. The central silver ions is collinearly bonded by oxygen ions in separate nitrate ligands and the terminal silver ions are bonded to oxygens in a bidentate fashion. The aqueous structure of the next larger cluster Ag3(NO3)4- has nitrate ions added to the terminal silver ions converting their bonding to collinear and the next smaller cluster Ag2(NO3)2 has one of the terminal silver ions removed. Such geometric arrangements are found in cluster complexes such as the reported72 dinuclear silver complex of bisdipyridylamine ligand. The Gibbs free energy for forming the Ag3(NO3)2+ cluster from the constituent silver and nitrate ions in solution is calculated to be -8.72 kcal/mol at 298.15 K which corresponds to an equilibrium constant of 2.45x106 M-4 and is sufficient that significant concentrations of this cluster could appear at 0.01 M and greater silver nitrate concentrations. The Gibbs free energy released on reduction by one electron is calculated to be 3.19 eV which corresponds to a reduction potential of -1.09 V in aqueous solution based on the proton free energy of solution of 4.28 eV using calculation methods discussed before.40 The reduced cluster in aqueous solution has one of the bidentate silver ions more distant from the oxygen ions at a closest distance of 3.37 Å versus 2.43 Å in the oxidized form. Table 8 gives data for several of the small related clusters. 29 ACS Paragon Plus Environment


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Table 8. Calculated Aqueous Gibbs Free Energy of Association (kcal/mol) for Clusters of Silver and Nitrate ions at STP Using PBE0 Functional, SDD and 6-31+G(d,p) Basis Sets and the SMD Solvation Model. Cluster

ΔG05

Ag(NO3)2-

-4.72

Ag2(NO3)+

-6.55

Ag2(NO3)2

-7.36

Ag3(NO3)2+

-8.72

Ag3(NO3)4-

-10.69

Figure 7. Calculated Equilibrium Aqueous Structure for Ag3(NO3) 2+ in (a) and Ag3(NO3) 20 in (b) Using the SDD and 6-31+G(d,p) Basis, PBE0 Functional and SMD Method. 30 ACS Paragon Plus Environment

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DISCUSSION An objective of this work has been to evaluate computational protocols that can apply to the aqueous silver clusters with and without nitrate counterions. This process entailed testing various approximations built from the PBE0 functional of density functional theory and the SMD solvation model. The jun-cc-pvTZ basis set mainly used for light atoms with SDD pseudopotentials and basis functions on silver gives gas phase values for bond energies of successive water ligand additions to silver ion in good agreement with experiment at very low temperature within the harmonic/ rigid rotor/ ideal gas approach. Dispersion interactions represented by Grimme’s D3 model also gave excellent agreement with experiment. The perturbation VPT2 anharmonic method used for silver-water gas phase clusters at 298.15 K gave larger hydration energies a few kcal/mol larger than the harmonic method and was of more limited application and so is discussed in the Supporting Information. The method was not stable for the large clusters and this limited its scope in these studies. The SMD solvation model used for calculating the Gibbs free energy of solvation was employed using the standard parameterizations across various basis functions even though the original parameters for various molecules were determined using the 6-31+G (d, p) basis functions. Good agreement within 1 kcal/mol was found with the experimental Gibbs free energies of solvation for NO3and Ag+ with this protocol in the cluster-continuum method with six to eight water ligands and each basis set used in this work. The addition of London dispersive interactions through the Grimme D3 method was studied and yielded more stable Gibbs free energies as would be expected. The dispersion calculation was applied cautiously with the SMD solvation calculations over concern that the parameters may not be entirely consistent between the two methods, 31 ACS Paragon Plus Environment


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but excellent agreement for Ag+ solvation free energy of solution was found. Moderately sized clusters containing up to ten ligands were employed in these cluster-continuum method comparisons though even larger clusters would be needed to investigate complete convergence of solvation energy with size, but this task might also be subject to some computational artifacts. Positively charged silver-water clusters have a preferred gas phase structure derived from silver ion collinearly bound to two water molecules that are in turn H bonded to more water ligands and this framework is a starting point for aqueous calculations. This preferred structure also persists for Ag2+ clusters obtained by substitution for Ag+. The collinear motif also continues for larger clusters such as Ag3(H2O)6+ and Ag4(H2O)6+ . Data are given to allow a comparison of various isomers in the Supporting Information. Neutral silver clusters have structures determined in large part by H bonding effects. Chains of H bonded water ligands are observed in many cases for the silver atom. The chain is attached to silver through one oxygen atom and is terminated by attachment to silver through a H atom. A preference for two chains three molecule in length is observed for Ag2(H2O)6 clusters versus a single six ligand chain length and is studied in detail in the Supporting Information. The degree of association of aqueous nitrate ligand with silver-water clusters can be calculated from the free energies calculated in aqueous solution and shown in Table 7 from the equilibrium constant –ΔG = RTln(Keq). The values are used to evaluate the equilibrium constant at 298.15 K. The calculated ratio of silver-water-nitrate to silver-water clusters for Ag+ is small at concentrations of nitrate in the 0.01M range of typical photographic and many other


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experiments and larger nitrate concentrations would be required to increase the ratio. For neutral silver atom clusters the ratio is even smaller. The reduction potential of the associated nitrate clusters at standard conditions can be calculated from the calculated free energy of reduction in the aqueous phase, free energies of solvation and the proton free energy of solvation of 4.28 eV as discussed before40. These values are given in Table 9 alongside a comparison to the corresponding Agn(H2O)5+ clusters. Table 9. Reduction Potentials (V) Calculated for Agn(NO3-)(H2O)5 and Agn(H2O)5+. n

Agn(H2O)5+

Agn+(NO3-)(H2O)5

1

-1.53

-1.41

2

-0.24

-0.30

3

-1.89

-1.90

4

-0.80

-1.00

Bulk,exp.

0.799

We observe that the presence of associated nitrate cluster does not significantly shift the reduction potentials for silver ion cluster reduction for the n=1-4 silver clusters. It is reasonable to expect that the associated clusters are not involved in the reduction of silver ion in many experiments. It is noted that the values of the reduction potential in Table 9 and values for silver clusters with various numbers of water ligands reported earlier39,40 for aqueous silver clusters agree reasonably well with the values estimated earlier by Henglein23 of -1.8, -0.2, 0.9,-0.2V for Agn n=1-4, respectively, but differences show up for the three and four silver atom clusters. The Ag3(NO3)2+ cluster with multiple nitrate ions described above has the reduction 33 ACS Paragon Plus Environment


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potential of -1.08V which is slightly more easily reduced than the Ag(H2O)5+ or Ag(NO3)(H2O)5 cluster. Overall the other multiple nitrate clusters have reduction potentials in the range of the other hydrated clusters. The calculation of free energies of association to form structures resembling close ion pairs of silver ion and nitrate ion using cluster-continuum models yields values that are reasonable. No experimental data are available for the silver-nitrate system, but experimental and calculated values have been presented73 for 1:1 electrolytes such as NaCl, KCl and other alkali halide close ion pairs in aqueous solution. Typical values range from 0.2 to 0.8 m-1 for the equilibrium constant. This range encloses the values calculated for silver-nitrate close ion pairs in this work. Until experiment is available this value is the closest comparison to experiment that is available. Recent molecular dynamics calculations have shown ion-pair formation of alkali ion and nitrate in aqueous solution with evidence for bidentate and single oxygen coordination to K+ cations.74 Experiments12,17,18 show that mild reducing agents do not reduce silver ion in solution for a period of time under particular conditions and that a catalytic center must be present to initiate reduction in the form of a seed particle of silver. The calculated reduction potentials of small silver cations such as those shown in Table 9 are sufficiently negative such that hydroquinone, having an experimental reduction potential1 of 0.70 V, could form only a small concentration of silver atom in accord with the need for catalyst. A simplified model has been considered in order to shed light on the role of a silver cluster located on a solid substrate at the aqueous interface which participates in silver ion reduction from the aqueous phase. The present approximation allows consideration of temperature 34 ACS Paragon Plus Environment

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dependent effects and evaluation of thermodynamic properties, but considers silver clusters only in the aqueous phase such that the aqueous polarization replaces the polarization due to solid substrate. The hydrated silver ion in solution is represented as Ag(H2O)6+ and the aqueous silver cluster is represented as Agn(H2O)m where different degrees of water ligand association with the neutral silver cluster were considered since it is not clear what degree of hydration should be appropriate in cluster-continuum calculations representing the silver cluster in aqueous solution. It is considered that the hydrated silver ion is reduced in the presence of the silver cluster in the aqueous phase, but no detailed mechanism is assumed. The molecular units are relaxed in the aqueous phase represented by the SMD solvation model using the procedure with jun-cc-pvTZ basis sets described before. The free energy change is computed for the reaction 6. Agn(H2O)m + Ag(H2O)6+ → Agn+1(H2O)m

+ (H2O)6

(6)

The calculated change in free energy for eq 6 is given in Table 10 versus the silver cluster size for different degrees of silver cluster hydration. One conclusion from this data is that the reaction becomes more exergonic in the presence of the silver clusters. There is a clear trend in the data that also shows the odd – even size oscillations where the less favorable values for even clusters are slowly shifting to more exergonic as silver cluster size increases and this effect could be related the critical size concept explained earlier. However, this argument would require more analysis to put on a stronger foundation. The degree of hydration of the silver cluster has only a minor effect on results. The associated silver-water-nitrate cluster Ag(NO3)(H2O)5 was also considered as a source for silver ions in reaction 6. The calculated values are 0.05 eV more positive than the values in Table 10 for Ag(H2O)6+ and do not indicate a 35 ACS Paragon Plus Environment


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preferred reactivity of the nitrate containing cluster. Calculations are underway to evaluate hydrated Ag3(NO3)2+ as a source for silver ions.

Table 10. Gibbs Free Energy Change (eV) for Aqueous Silver Ion Reduction Using Ag(H2O)6+ in the Presence of Agn(H2O)m in Eq 6 at 298.15 K with the PBE0 Functional, Jun-cc-pvTZ basis and SMD Solvent Model. n

ΔG(m=1)

ΔG(m=2)

ΔG(m=4)

0

-2.84

-2.71

-2.66

1

-4.47

-4.54

-4.66

2

-3.54

-3.60

-3.52

3

-4.44

-4.45

-4.51

4

-3.64

-3.97

-3.97

5

-4.80

-4.82

-4.81

6

-4.03

-4.04

-3.90

CONCLUSIONS 1. The protocol for treating aqueous silver-water-nitrate clusters in aqueous solution is based upon the PBE0 functional of Density Functional Theory and the SMD solvation model. Structures of the solute are optimized in the gas phase and then allowed to relax in the aqueous representation. This procedure yields accurate solvation free energies for silver ion and nitrate ion at 298.15 K using a cluster containing 6-8 water ligands with several basis sets. An additional computational effect investigated was the inclusion of 36 ACS Paragon Plus Environment

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dispersive interactions which stabilized gas phase silver-water and water clusters by a few kcal/mol and can change the free energies of solvation for some cluster containing many water ligands. 2. Positively charged silver-water clusters have an optimized structure with collinear silver oxygen bonds for the first two ligands attached to silver ion with subsequent ligands attached by hydrogen bond interactions to the first two ligands. A similar structure is found for Ag2+ and larger positive clusters display this motif. Silver atoms bond to one water molecule through oxygen and then form chains of hydrogen bonded ligands terminating with a bond through hydrogen to the silver atom. Clusters of four atoms of silver have lower symmetry structures dominated by the chains of hydrogen bonded water molecules. 3. The structure of nitrate containing silver-water clusters is controlled by bonds between oxygen ions of nitrate with silver. Positive silver clusters contain water ligands bonded through oxygen that may also attach to the nitrate. Neutral silver clusters contain water ligands bonded through hydrogen to a silver atom that also bond to the nitrate ligand. 4. The free energies of silver-nitrate clusters with one nitrate ion were determined by structure optimization in the SMD solvation model and used to calculate free energies of association and equilibrium constants for nitrate association with silver clusters. The free energy of association is calculated to be 0.4, 0.6, 0.7, -0.1 kcal/mol for Agn+ positively charged silver cluster for n=1-4, respectively and becomes more positive for neutral silver clusters. The concentration ratio of silver nitrate close ion pair structure to free silver ion is 0.51 multiplied by the nitrate concentrations for Ag+ with similar values 37 ACS Paragon Plus Environment


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for the other positive clusters. Neutral silver clusters associate with nitrate in aqueous solution to a lesser degree than the positive counterparts. All of these results for clusters beyond Ag+ are more significantly viewed in term of their trends due to their complexity and the details of the procedure. The reduction potentials of the associated ion pairs is similar to the corresponding hydrated silver ion clusters. 5. The structure of Agn(NO3)m+(n-m) aqueous clusters where n=1-3 and m=1-4 were calculated since some of this series have been implicated in previous nucleation experiments to form silver nanoplates. Their structure contains collinear silver oxygen bonds and bidentate oxygen bonds from nitrate to terminal silver ions. An equilibrium constant for formation of this complex from aqueous silver and nitrate ions is calculated. 6. A simplified aqueous model of catalytic silver ion reduction in aqueous solution based upon aqueous Ag(H2O)6+ in the presence of Agn(H2O) where n=1-6 was computationally evaluated. The free energy change upon reduction to silver atoms is more exergonic in the presence of silver clusters. The trend in free energy change is to more negative values with increasing silver cluster size. Both trends are consistent with a catalytic role for the silver cluster. ASSOCIATED CONTENT Supporting Information Co-ordinates and thermodynamic properties of nitrate-water, silver−water and silver-waternitrate clusters including data for various isomers at the harmonic/rigid rotor level are given for


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the calculations using the PBE0 functional with SDD pseudopotential and basis on Ag and juncc-pvTZ or 6-31+G(d,p) basis on O, N and H. Data using the VPT2 anharmonic calculation is given. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION *E-mail: [email protected] Notes The author declares no competing financial interest. ACKNOWLEDGEMENTS No external funding was involved in this work. REFERENCES (1) James, T. H. The Theory of the Photographic Process, 4th ed.; Macmillan Publishing Co., Inc.: New York, 1977. (2) Sotirou, A. A.; Meyer, A.; Knijnenburg, J. T. N.; Panke, S.; Pratsinis, S. E. Quantifying the Origin of Released Ag+ Ions from Nanosilver. Langmuir 2012, 28, 15929-15936. (3) Batley, G. E.; Kirby, J. K.; Mclaughlin, M. J. Fate and Risks of Nanomaterials in Aquatic and Terrestrial Environments. Acc. Chem. Res. 2013, 46, 854-862. (4) Eckhardt, S.; Brunetto, P. S.; Gagnon, J.; Priebe, M.; Giese, B.; Fromm, K. M. Nanobio Silver: Its Interactions with Peptides and Bacteria, and its Uses in Medicine. Chem. Rev. 2013, 113, 47084754.



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TOC Graphic


Density Functional Calculations for Aqueous Silver Clusters

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