Partial Molar Volumes of Aqua Ions from First Principles - American

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Partial molar volumes of aqua ions from first principles Julia Wiktor, Fabien Bruneval, and Alfredo Pasquarello J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00474 • Publication Date (Web): 06 Jul 2017 Downloaded from http://pubs.acs.org on July 6, 2017

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Journal of Chemical Theory and Computation

Partial molar volumes of aqua ions from first principles Julia Wiktor,∗,† Fabien Bruneval,‡ and Alfredo Pasquarello† Chaire de Simulation à l’Echelle Atomique (CSEA), Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland, and DEN - Service de Recherches de Métallurgie Physique, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, France E-mail: [email protected]

∗

To whom correspondence should be addressed Chaire de Simulation à l’Echelle Atomique (CSEA), Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland ‡ DEN - Service de Recherches de Métallurgie Physique, CEA, Université Paris-Saclay, 91191 Gif-surYvette, France †

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Abstract Partial molar volumes of ions in water solution are calculated through pressures obtained from ab initio molecular dynamics simulations. The correct definition of pressure in charged systems subject to periodic boundary conditions requires access to the variation of the electrostatic potential upon a change of volume. We develop a scheme for calculating such a variation in liquid systems by setting up an interface between regions of different density. This also allows us to determine the absolute deformation potentials for the band edges of liquid water. With the properly defined pressures, we obtain partial molar volumes of a series of aqua ions in very good agreement with experimental values.

The partial molar volume describes how the volume of a solution changes when one mole of a solute is added. This quantity provides valuable information about the molecular and ionic interactions in the solution, 1–4 which is fundamental for a proper understanding of the behavior of electrolytes 5 and of the pressure effect on chemical reactions. 6 While experimental partial molar volumes are well documented for simple ions, 7,8 these quantities are not known for many larger molecular complexes, which are currently under scrutiny as catalysts for water splitting. 9–11 When the experimental characterization is lacking, a modeling approach to accurately predict partial molar volumes is highly desirable. Ab initio molecular dynamics (MD) is currently the method of choice to study the properties of water and aqueous solutions at the molecular level. 12–20 However, such molecular dynamics cannot directly be used for predicting partial molar volumes of ions in solution. This is due to the fact that the pressure is not trivially accessible in charged systems subject to periodic boundary conditions. 21 In such systems, the pressures calculated as derivatives of the total energy with respect to volume suffer from the indetermination of the average electrostatic potential. To achieve physical pressures, it is required to obtain information on the variation of the electrostatic potential upon a change of volume. In the case of semiconductors, this type of information could be acquired from strained superlattice calculations, as 2


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usually done in the evaluation of absolute deformation potentials. 22–24 However, the necessity of applying strain makes this technique inadequate for liquid systems. The pressure artifact due to the electrostatic potential indetermination in charged periodic systems does not only concern molecular dynamics simulations at the ab initio level. It has been observed in the classical MD study of partial molar volumes of single ions, but the origin of the problem was not clearly understood. 25 In classical MD, several sources of energy and pressure artifacts have been documented, and include approximate electrostatic errors, improper summation errors, and finite size errors. 26 However, the pressure artifact addressed in the present study has not been identified, occurs in addition to these other effects, and requires a separate correction. In the present Letter, we determine partial molar volumes of ions in water solution from first principles. We first develop a scheme for evaluating the variation of the electrostatic potential upon a change of volume for a liquid system like water. This piece of information is then used to calculate the absolute deformation potentials of liquid water and to properly define pressures in ab initio molecular dynamics of aqueous solutions. Application to a series of aqua ions demonstrates that partial molar volumes achieved within this scheme can be used to reliably predict experimental values. The pressure of a system containing a charge q can be written as:

P (q) = −

∂E(q) , ∂Ω

(1)

where E(q) is the total energy of the system and Ω its volume. In a charged system, the total energy E(q) depends on the average value of the Hartree potential hvH i. However, the Hartree potential in a periodic system is only defined up to a constant, due to the long-range nature of Coulomb interactions. 22 Therefore, it is conventionally set to zero (zero-average convention) or to a constant that can depend on the volume of the cell. 27–31 Hence, the volume dependence of the total energy, and consequently the pressure, is spuriously defined

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in periodic systems. To overcome this limitation, the convention-dependent energy E ′ (q) needs to be related to a physical energy E(q) having the correct volume dependence. This is achieved by replacing the artificially set potential hvH′ i with an absolute potential hvH i as follows: 21 E(q) = E ′ (q) + q hvH′ i − q hvH i ,

(2)

where the convention-dependent quantities are emphasized with primed symbols. This implies that the physical pressure of a charged periodic system can be obtained through:

P (q) = P ′ (q) −

q ∂ hvH i q ∂ hvH′ i + , Ω ∂ln Ω Ω ∂ln Ω

(3)

where P ′ (q) = −∂E ′ (q)/∂Ω. Here, ∂ hvH′ i /∂ln Ω results from the volume dependence of the average potential hvH′ i and vanishes in the case of zero-average convention. The term ∂ hvH i /∂ln Ω describes the variation of the absolute potential upon a change of volume. This derivative is well defined and can be calculated, even if hvH i is undetermined in calculations of periodic systems. The MD simulations in this work are performed in the canonical N V T ensemble, with the electronic structure and the atomic forces calculated within the self-consistent Kohn-Sham approach to density functional theory (DFT). We use the suite of codes in the quantumespresso package. 30 Core-valence interactions are described through norm-conserving pseudopotentials. 32 To account for van der Waals interactions, the revised Vydrov and Van Voorhis (rVV10) nonlocal density functional 33,34 is used in the simulations. In the rVV10 density functional, the empirical parameter b is set to 9.3, as this value gives an equilibrium density density of water (0.9945±0.0022 g/cm3 ) 35 in close agreement with experiments. We set the target temperature to 350 K, to obtain a frank liquid-like behavior. We note that the choice of the temperature in the study of partial molar volumes is not critical, as these quantities vary only slightly between 300 and 375 K. 8 The temperature in our MD simulations is controlled by a velocity rescaling thermostat. We set the integration time step to 4


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0.48 fs. The valence wave functions are expanded in a plane-wave basis set defined by cutoff of 85 Ry. We checked that an increase of the cutoff to 150 Ry results in a systematic increase of the pressure by about 0.3 kbar. The systematic nature of this residual effect cancels out in the evaluation of volume differences between systems with and without solute. Nuclear quantum motions associated to the H atoms are here neglected on the basis of the same argument and of their relatively small effect on the pressure of liquid water. 36 To correct the pressures in charged cells [see Eq. (3)], we need to determine the derivatives ∂ hvH′ i/∂ln Ω and ∂ hvH i/∂ln Ω. The first derivative depends on the convention used for the average electrostatic potential and amounts to −0.52 eV for the setup used in this work. In the case of solids, the second derivative is found through strained superlattice calculations. This technique consists of generating an interface between regions of the same material under compressive and tensile strain and monitoring the change of the potential. 22–24 In the case of semiconductors, such an interface can be generated by simply displacing the atomic planes, but this approach cannot be applied to a liquid system. To address the case of liquid water, we generate a supercell containing two regions, one with higher and one with lower density. We use a 64-molecule cell with lattice parameters a = b = 9.89 Å and c = 19.78 Å, and introduce an external step-like potential of 0.001 eV in the left and −0.001 eV in the right half of the cell. Upon the molecular dynamics evolution, this potential leads to an average difference in electron charge densities of about 10% between the two regions. The term ∂ hvH i/∂ln Ω can then be evaluated by considering the variation of the Hartree potential upon electron charge density. However, the potential averaged over a MD trajectory lasting 30 ps still does not reach a plateau in the bulk regions, as can be seen in the top panel of Fig. 1. This problem can be overcome through the observation that a linear relation subsists between the potential and the charge density for small deviations from equilibrium (see Fig. 1). This allows us to calculate the derivative in Eq. (3) as: R ′ R ′ v (z)dz v (z)dz − ∂ hvH i ∂ hvH i H RR H , =− = −¯ ρ LR ∂ln Ω ∂ln ρ ρ(z)dz − R ρ(z)dz L 5


(4)


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where L and R stand for the left and right sides of the supercell, vH′ (z) and ρ(z) are the Hartree potential and the charge density averaged over the x and y directions parallel to the interface, and ρ¯ is the average charge density of water in the cell. In Eq. (4), we could use vH′ rather than vH , as the undetermined part of the potential cancels out from the final expression. We remark that, given the linear relationship between potential and charge density, the left and right integration domains do not need to be restricted to bulk regions. Each of the domains is therefore taken as large as possible and to correspond to half of the supercell to fully benefit from the available statistics. Additionally, to verify the general validity of the present approach, we consider the case of silicon and yield a potential lineup ∂ hvH i/∂ln Ω which differs by only 0.06 eV from that achieved with the conventional strained superlattice technique. Figure 1 (bottom panel) shows the cumulative running average of the potential lineup ∂ hvH i/∂ln Ω as obtained through Eq. (4). The value of the potential lineup is stable during the last 10 ps of the simulation and amounts to 3.82±0.24 eV, where the statistical error of 0.24 eV is estimated by performing a blocking analysis. 37 To further support this estimate, we address the potential lineup for crystalline water in the hexagonal structure of ice Ih . We obtain values of 3.84 eV in the [0001] direction and 3.67 eV in the [1¯100] direction, not far from the value calculated for liquid water. We remark that the present technique for identifying pressure corrections can also be employed in the case of classical molecular dynamics simulations, which are subject to the same artifacts. We emphasize that the potential lineup calculated here is required to correct the convention-dependent pressure P ′ in our setup, but does not carry an intrinsic physical meaning. To achieve a physical and thus transferable quantity, we focus on the absolute deformation potential ai of a state i with energy ǫi : 22

ai =

∂ǫi . ∂lnΩ

6


(5)

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Figure 1: Top panel, the evolution of the Hartree potential vH′ (red solid line) and the electron charge density ρ (blue dashed line) across the supercell partitioned in regions of low (left) and high (right) density of water. The potential is averaged over xy planes and over a MD trajectory of 25 ps after an equilibration period of 5 ps. Red dashed lines represent the average values of the potential in each half of the cell. Bottom panel, the cumulative running average of the potential lineup ∂ hvH i/∂ln Ω.

To calculate deformation potentials, we additionally perform two separate MD simulations of liquid water in cubic supercells, with lattice parameters differing by ±5 % from equilibrium. A duration of 10 ps is sufficient to achieve energy levels of the valence band maximum (VBM) and conduction band minimum (CBM) converged within 0.005 eV. By combining the variation of these energy levels upon strain with the potential lineup ∂ hvH i/∂ln Ω, we obtain the absolute deformation potentials of the valence and the conduction band, aVBM and aCBM , together with the deformation potential of the band gap agap (Table 1). The errors for aVBM and aCBM originate almost entirely from the statistical error of 0.24 eV on the potential lineup. As seen from the deformation potentials, the band gap of water increases 7



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slowly with density. The change in band gap comes mostly from the increase of the CBM energy, since the VBM energy remains almost unaffected. Absolute deformation potentials are not dependent on conventions and can thus be used to extract the potential lineup in different setups without repeating explicit interface calculations. 21 Table 1: Absolute deformation potentials of the valence (aVBM ) and conduction band (aCBM ) for liquid water, and relative deformation potential of the band gap (agap ). aVBM aCBM agap

0.07±0.24 eV −0.31±0.24 eV −0.38±0.03 eV

For illustration, we determine partial molar volumes for the aqua ions Cu2+ , Cs+ , Br− , and Cl− . For each ion, we perform MD runs at two different volumes, one being the equilibrium volume of water and the other one accounting for the experimental partial molar volume of the ion. 7,8 We use cubic cells containing 64 water molecules and a single ion. In all runs, after an initial equilibration period of 5 ps, we collect statistics for 10 ps. We note that the partial molar volumes could alternatively be obtained by imposing the corrected pressure to vanish in a N pT simulation. 38 However, the adopted procedure allows us to compare the corrected and uncorrected results and to better control the errors on the final volumes. Representative structural configurations of the first solvation shell around the Cu2+ , Cs+ , Br− , and Cl− ions are shown in Fig. 2. These structures correspond to the average coordination number. For Cu2+ the first coordination shell is very well defined and the coordination number of 5 is preserved during the whole trajectory. For the other ions, the instantaneous coordination number fluctuates within a range given by ±2. However, we verified that there is no sign of correlation between the average coordination number and the pressure. Additionally, we observe that the average ion–oxygen distance and the coordination number generally do not depend significantly on the volume of the simulation cell (Table 2). Overall, the structural properties of the first solvation shell of the ions considered here agree with previous reports. 13,14,20,39 8


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(a) Cu2+

(b) Cs+

(c) Br−

(d) Cl−

Figure 2: Representative configurations showing the local arrangement of water molecules in the first solvation shell around the Cu2+ , Cs+ , Br− , and Cl− ions.

In Fig. 3, we give the evolution of the cumulative running average of the corrected pressures. The illustrated results refer to simulations with cells accounting for the experimental partial molar volume. In all cases, the pressure converges to a value close to zero within 10 ps, indicating that the experimental partial molar volume is consistent with the simulations. To examine the dependence on starting configuration, we perform two independent runs at the same volume for the Cl− ion, finding average pressures which agree with each other within the statistical errors. The final corrected pressures for all runs are given in Table 2.

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Figure 3: Cumulative running average of the corrected pressures for the Cu2+ , Cs+ , Br− , and Cl− ions during the last 10 ps of the MD simulations. The pressures displayed pertain to the simulations with cells accounting for the experimental partial molar volume.

We obtain the partial molar volumes from the pressures of the two runs applying a linear interpolation (or extrapolation) to zero pressure. The resulting relaxation volumes are given in Table 3, where they are compared with experimental values. 7,8 We show both the volumes V obtained using the corrected pressures P and the volumes V ′ resulting from the convention-dependent pressures P ′ . For all the considered ions, the partial molar volumes V are found in very good agreement with the experimental data, whereas the volumes V ′ show a clear disagreement. We stress that the volumes V ′ are arbitrary and other pseudopotentials or other conventions for the electrostatic potential will generally produce different results. The validity of the present scheme for the calculation of partial molar volumes critically relies on the control and accounting of the generated errors. The errors given in Table 3 result from the standard error propagation formula 41 and include statistical errors on the pressure calculated via the blocking analysis, the errors originating from the potential lineup, and the error on the equilibrium density of water. 35 In principle, there are also finite-size effects on the total energies of charged supercells, 42,43 which imply additional corrections to the pressure. 44 However, due to the large dielectric constant of liquid water, 45 these corrections affect the partial molar volumes by less than 1 cm3 and can be neglected. The errors of 2 cm3 on the experimental partial molar volumes are taken from the discussion in Ref. 7 10


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Table 2: The lattice parameter a of the cubic simulation cell, the ion–oxygen distance dI–O , the coordination number CN, 40 and the corrected pressure P . For the present setup, the equilibrium volume of 64 water molecules corresponds to a = 12.46 Å. Averages are obtained from 10 ps of MD simulation, after an initial equilibration period of 5 ps. Ion Cu2+ Cu2+ Cs+ Cs+ Br− Br− Cl− Cl− Cl−

a (Å) dI–O (Å) CN P (kbar) 12.46 2.05 5.1 −0.97±0.16 12.33 2.05 5.0 −0.24±0.13 12.46 3.11 8.6 0.68±0.15 12.50 3.15 9.2 0.02±0.21 12.46 3.36 6.4 1.23±0.17 12.56 3.36 6.0 −0.39±0.19 12.46 3.15 6.2 0.71±0.19 12.54 3.15 6.0 −0.59±0.11 12.54 3.15 5.9 −0.31±0.21

Table 3: Comparison of calculated and experimental partial molar volumes. The volumes V ′ are obtained using the uncorrected pressures P ′ . The volumes V result from the use of the corrected pressures P . Experimental results are taken from Refs. 7 and. 8 Ion Cu2+ Cs+ Br− Cl−

V ′ (cm3 /mol) −379 ± 114 −51 ± 25 100 ± 15 89 ± 20

V (cm3 /mol) −42 ± 14 17 ± 6 26 ± 6 19 ± 6

Vexpt. (cm3 /mol) −36 ± 2 16 ± 2 31 ± 2 24 ± 2

To conclude, we addressed the calculation of partial molar volumes of ions in water solution. This result was achieved through the proper determination of pressures in ab initio molecular dynamics of charged systems subject to periodic boundary conditions. For this purpose, we developed a technique to calculate the potential lineup and the absolute deformation potentials for a liquid system. We observed a very good agreement between the calculated and measured partial molar volumes, indicating that this property can be successfully predicted from first principles.

Acknowledgement We thank A. Baldereschi, G. Miceli, and R. Resta for useful discussions. Financial support 11



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is acknowledged from the Swiss National Science Foundation (SNSF) (Grant No. 200020152799). This work has been realized in relation to the National Center of Competence in Research (NCCR) “Materials’ Revolution: Computational Design and Discovery of Novel Materials (MARVEL)” of the SNSF. We used computational resources of the Swiss National Supercomputing Centre and of the Ecole Polytechnique Fédérale de Lausanne.

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Graphical TOC Entry ion solvation

∆Volume

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Partial Molar Volumes of Aqua Ions from First Principles - American

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