doi.org/10.26434/chemrxiv.7466426.v1

Hydration Structure of Sodium and Potassium Ions with DFT-MDTimothy Duignan, Gregory K. Schenter, Mirza Galib, Marcel D. Baer, Jan Wilhelm, Jürg Hutter, Mauro DelBen, Xiu Song Zhao, Christopher J. Mundy

Submitted date: 14/12/2018 • Posted date: 14/12/2018Licence: CC BY-NC-ND 4.0Citation information: Duignan, Timothy; Schenter, Gregory K.; Galib, Mirza; Baer, Marcel D.; Wilhelm, Jan;Hutter, Jürg; et al. (2018): Hydration Structure of Sodium and Potassium Ions with DFT-MD. ChemRxiv.Preprint.

The ability to reproduce the structure of water around the sodium and potassium ions as determined byexperiment is a key test of the quality of interaction potentials due to the central importance of these ions in awide range of important phenomena. Here, we simulate the Na+ and K+ ions in bulk water using the recentlydeveloped strongly constrained and appropriately normed (SCAN) functional and compare with experimentalX-ray diffraction (XRD) and X-ray adsorption fine structure (EXAFS) measurements to demonstrate that itaccurately reproduces important structural details of the hydration structure of the sodium and potassiumcations. We demonstrate that it performs substantially better than the generalized gradient approximation(GGA) based dispersion corrected revised Perdew, Burke, and Ernzerhof functional (revPBE-D3) and is evenbetter than the random phase approximation level for potassium. Both of these functionals have beendemonstrated to accurately reproduce the structure of bulk water. This improved performance compared withrevPBE-D3 is attributed to smaller fluctuations of the mean error of ion-water cluster binding energies utilizinga novel benchmark for testing functionals.

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Hydration structure of sodium and potassium

ions with DFT-MD

Timothy T. Duignan,†,‡ Gregory K. Schenter,† Mirza Galib,† Marcel D. Baer,†

Jan Wilhelm,¶ Jurg Hutter,¶ Mauro Del Ben,§ Xiu Song Zhao,∗,‡ and

Christopher J. Mundy∗,†,∥

†Physical Science Division, Pacific Northwest National Laboratory, P.O. Box 999,Richland,

Washington 99352, USA

‡School of Chemical Engineering, The University of Queensland, St Lucia, Brisbane 4072,

Australia

¶Department of Chemistry, University of Zurich, CH-8057 Zurich, Switzerland.

§Computational Research Division, Lawrence Berkeley National Laboratory, Berkeley,

California 94720, USA.

∥Affiliate Professor, Department of Chemical Engineering, University of Washington,

Seattle, Washington, USA

E-mail: george.zhao@uq.edu.au; chris.mundy@pnnl.gov

1

Abstract

The ability to reproduce the structure of water around the sodium and potassium

ions as determined by experiment is a key test of the quality of interaction potentials

due to the central importance of these ions in a wide range of important phenom-

ena. Here, we simulate the Na+ and K+ ions in bulk water using the recently devel-

oped strongly constrained and appropriately normed (SCAN) functional and compare

with experimental X-ray diffraction (XRD) and X-ray adsorption fine structure (EX-

AFS) measurements to demonstrate that it accurately reproduces important structural

details of the hydration structure of the sodium and potassium cations. We demon-

strate that it performs substantially better than the generalized gradient approximation

(GGA) based dispersion corrected revised Perdew, Burke, and Ernzerhof functional

(revPBE-D3) and is even better than the random phase approximation level for potas-

sium. Both of these functionals have been demonstrated to accurately reproduce the

structure of bulk water. This improved performance compared with revPBE-D3 is at-

tributed to smaller fluctuations of the mean error of ion-water cluster binding energies

utilizing a novel benchmark for testing functionals.

Introduction

The sodium and potassium ions play a central role in a large range of important industrial

and biological processes. For example, the sodium and potassium ions are considered to

be promising candidates to replace lithium ions in the next generation of energy storage

devices.1–5 Additionally, the flow of potassium and sodium ions through cell membranes is

used to control important biological processes.6 In both of these systems, an important step

is the partial desolvation of the ion as it passes through a small channel. For example,

the ions must desolvate to intercalate into electrode materials or to pass through the ion

pump in the cell membrane. This desolvation can determine the rates and mechanisms

of these processes. A recent demonstration of this point is provided by Peng et al. 7 who

2

demonstrate that interfacial sodium ion diffusion is highly sensitive to the hydration number.

It is therefore very important to build an accurate and detailed understanding of the structure

of solvent around these ions.

Many attempts have been made to simulate the properties of these ions in water.8–21

Classical forcefield molecular dynamics can reproduce a variety structural properties such

as the peak position in the radial distribution function (RDF) but the parameters for the

Lennard-Jones and polarisability interactions have to be explicitly adjusted to do so. These

models also have limited predictive power as new parameters are usually needed to model

the ion in different environments such as inside an electrode material. Hybrid quantum

mechanical/molecular mechanics (QM/MM) is an alternative approach14,21 to the problem.

However, it is unclear how large the QM region needs to be to accurately capture the full

solvent structure. Moreover, significant challenges arise associated with the treatment of the

interface between the two regions. A promising and increasingly widely used approach is

to use molecular dynamics simulations with a full quantum mechanical density functional

theory treatment of the system.22 The dispersion corrected generalized gradient approxima-

tion (GGA) functionals are by far the most widely used in the context of condensed phase

simulation due to their low computational demand. Of these functionals the revised Perdew,

Burke, and Ernzerhof functional with Grimme dispersion correction (revPBE-D3)23–25 has

been demonstrated to accurately reproduce the structure of bulk water.26 However, this

approach often demonstrate a dependence on the specific functional chosen and can fail to

quantitatively reproduce experimental measurements. For example, Galib et al. 20,27 demon-

strated that standard GGA functionals cannot reproduce the experimentally determined

water structure around the sodium cation as determined by x-ray diffraction (XRD) and

X-ray adsorption fine structure measurements (EXAFS).

It is therefore necessary to simulate these ions with quantum density functional theory

(DFT) functionals at higher rungs of the so-called “Jacobs Ladder” of accuracy.28 No signif-

icant change in peak position occurs on moving from the GGA functional PBE to the hybrid

3

functional PBE0,17 indicating that hybrid functionals are unlikely to solve this issue. To this

end, the strongly constrained and appropriately normed (SCAN) meta-GGA functional has

recently been developed. The additional step on Jacobs Ladder has the advantage that the

computational costs are comparable to standard GGA functionals. SCAN has been devel-

oped to satisfy 17 known constraints that a general exchange-correlation functional should

satisfy. It can accurately reproduce binding energies and structures of a variety of molecules

without empirical dispersion corrections.29 Furthermore, Chen et al. 30 and Zheng et al. 31

have used this functional to simulate the structure of bulk liquid water determining a den-

sity of 1.05 g/cm−1 and showing good agreement with experimental oxygen-oxygen RDFs

at room temperature when simulated at an elevated temperature of 330 K. This functional

has also recently been applied to calculate the potential of mean force of the NaCl dimer

in water.32 Additionally, the random phase approximation (RPA) approximation to electron

correlation has recently been implemented for condensed phase calculations.33,34 This level

of theory represents the highest (fifth) rung of the Jacobs ladder and provides an accurate

description of bulk liquid water.34 The RPA method comes with additional computational

costs over standard GGA functionals due to the explicit treatment of electron correlation

and the requirement of larger basis sets. Nevertheless, it is important to quantitatively de-

termine whether these more sophisticated functionals can overcome the limitations of the

GGA based description of ion solvation.

Structural analysis

One of the main the differences between this study and the studies of Chen et al. 30 and Zheng

et al. 31 is the simulation at 300 K rather than the 330K used and using newly optimized

pseudo potentials for the oxygen atom. Herein, we use the same pseudo potentials as was

used in Chen et al. 30 and Zheng et al. 31 for hydrogen. The resulting oxygen-oxygen RDFs

are shown in Figure 1 for both revPBE-D3 and SCAN. The revPBE-D3 functional has been

shown to give near perfect agreement with the experimentally determined RDF.26,35 The

4

structure is significantly more enhanced for the SCAN functional resembling36 the structure

using the GGA of Becke37 and Lee, Yang and Parr38 (BLYP) in addition to the dispersion

correction (D2) put forth by Grimme.39 The origin of the discrepancy in comparison with

previous SCAN studies30,31 is not immediately apparent. The structural effects associated

with quantum nuclear effects have been demonstrated to be quite small and so likely cannot

explain the discrepancy.40,41 Nevertheless, the slightly over-structured behaviour of SCAN is

consistent with the work of Yao 42 who also simulated at 300 K with CP2K43 and CPMD44

demonstrating good agreement between the two basis set approaches, namely gaussian func-

tions and plane-waves. Simulation of 20 ps utilizing SCAN water in a slab configuration

at 300 K using the protocol discussed herein is consistent with the density of 1.05 g/cm3

calculated by Chen et al..30 These results presented here suggest some caution in the use

of SCAN to describe pure water may be appropriate as its RDF and density appear to be

comparable to BLYP-D2 at 300 K in terms of agreement with experimental measurements.26

Table 1: RDF peak positions for different DFT functionals compared with ex-periment demonstrating that the SCAN functional provides closer agreement tothat predicted from X-ray diffraction and EXAFS.20 Units are in A, error is ±0.05 A.

revPBE-D3 SCAN RPA ExperimentNa+ 2.51 2.36 - 2.38K+ 2.98 2.78 2.86 2.79

Figure 2 depicts the RDF for the sodium and potassium ions utilizing both revPBE-D3

and SCAN functionals. Significant differences are clear. As shown in Table 1, the experi-

mental peak position is reproduced much more accurately with the SCAN functional than

with the GGA. The inability to reproduce this peak position has been problematic with

other GGA functionals such as BLYP-D2.20 To the best of our knowledge this is the first

demonstration that the SCAN functional can significantly outperform simpler GGA func-

tionals in reproducing bulk structural properties in the condensed phase. As previously

stated, previous research has demonstrated that SCAN can accurately reproduce bulk wa-

ter structure.30,31 This is also true for the revPBE-D3 GGA functional.26,35 Similarly, the

5

2 3 4 5 6r [Å]

0.5

1

1.5

2

2.5

3

g (r

)

SCANrevPBED3

Figure 1: Oxygen-oxygen RDFs with the revPBE-D3 and SCAN functionals.

0 2 4 6 8r [Å]

0

2

4

6

8

g (r

)

revPBE-D3SCANExp. peak position

(a) Sodium (Na+)

2 4 6 8r [Å]

0

2

4

6

g (r

)

Exp. peak positionrevPBE-D3SCANRPA

(b) Potassium (K+)

Figure 2: RDFs of solvated sodium and potassium ions in water with the revPBE-D3 andSCAN functionals demonstrating that the SCAN functional reproduces the experimentallyobserved peak position much more accurately.

6

Na-Cl potential of mean force (PMF) has been computed with SCAN, however relatively

small differences are observed when compared to other DFT functionals and there are no

comparisons of the outcomes of the aforementioned PMF with experimental results to demon-

strate an improvement. It has already been demonstrated that the PMF calculated with the

BLYP-D2 functional leads to reasonably good agreement with experimental osmotic/activity

coefficients.45

Previous studies have determined that solvation structure of the potassium ion as one

of the most difficult cases to reproduce using GGA functionals.16 Given the good overall

agreement of the solvation structure of the sodium ion with SCAN, it gives rise to the

question of what the level of performance will be for wavefunction methods that include

electron correlation effects. To this end, we have also simulated the potassium ion treating

electron correlation at the level of the random-phase approximation (RPA).34 Due to the

large computational costs of this functional we have not yet attempted the simulation of

the sodium ion with this method. Figure 2 suggests that the RPA functional performs

substantially better than the GGA in comparison with the experimental peak position.

Although the K–O distance is somewhat larger than the experimental value, this may be

due to a lack of statistics. Nevertheless, given that we used the converged GGA simulation

to generate the initial configuration it is clear that the RPA is producing results more in

line with EXAFS experiment. Interestingly the RPA is also more structured than both the

GGA or the meta-GGA. Given the excellent agreement of RPA results for water under bulk

homogeneous conditions,46 overall the performance of the RPA is satisfactory and hints that

properly converged correlated wavefunction methods may well out-perform lower rungs of

Jacob’s Ladder but at a large computational overhead.

A key property of interest is the hydration number of the ions. However, this quantity is

not well defined for ions with water molecules that cross between hydration layers. The best

way to understand the ion hydration and its model dependencies is through examination

of the incremental RDFs that separates the distribution into contributions from the closest

7

2 3 4r [Å]

0

1

2

3

g (r

)

(a) Sodium (Na+)

2 3 4 5r [Å]

0

1

2

g (r

)

(b) Potassium (K+)

Figure 3: Incremental RDFs of the sodium and potassium ions in water with the SCANfunctional.

2 3 4 5 6⟨r⟩ [Å]

0

0.05

0.1

σ2

(a) Sodium (Na+)

Figure 4: Analysis of the incremental RDFs for sodium (black) and potassium (red). Eachpoint gives the average position and fluctuation in the position (the Debye-Waller factor) ofone water molecule in the incremental RDF.

8

water molecules. Figure 3 provides this quantity for the sodium and potassium ions using the

SCAN functional. Additional incremental RDFs for the other functionals are provided in the

SI. Figure 4 provides an analysis of the average position and spread of each water molecule

for both ions. There are substantial similarities between the two ions. For instance, it is clear

that both ions have five relatively tightly bound water molecules and a more diffuse sixth

water molecule that occupies the first hydration shell but can often be in the second shell.20

In contrast water molecules beyond the first shell show qualitatively different behaviour for

the two ions. The seventh water molecule for sodium almost always occupies the second

hydration shell and is more localised than the sixth. With potassium the seventh and eight

water molecules bridge the first and second hydration shells blurring the distinction between

hydration layers and are more diffuse than the sixth water molecule. This is consistent with

potassium being classified more as a chaotrope in contrast with the kosomotrope status of

the sodium ion.47 The hydration number defined as the integral of the RDF at the minima

gives values of 5.6±0.2 and 7.5±0.8 with the SCAN functional. The large uncertainty in the

potassium ion coordination number is due to the broad and flat minimum of the RDF and

is the result of the fact that this ion does not have a well defined first hydration layer. These

results are consistent with XAFS and XRD data which gives a sodium hydration number of

5.4 to 5.9.20

EXAFS analysis

Given a detailed description of molecular interactions, an associated ensemble of configu-

rations can be generated. From this ensemble, collective properties and responses of the

system can be determined. In particular, it is possible to generate the X-ray absorption fine

structure signal which is a unique spectral signature associated with the generated ensemble.

(In this work we concentrate on the ”extended” region of the signal, referred to as EXAFS.)

This provides an effective benchmark of our description of molecular interactions, giving us

confidence in its ability to generate accurate ensembles and be used to predict phenomena

9

that are not directly measurable.

EXAFS is an effective probe of the local solvent structure about a solute photo-electron

source. The signal is most sensitive to the solute - nearest solvent distance and its fluctuations

which are measured by the Debye-Waller factor of the solute-solvent vibration, σ2. In Fig. 4,

the lowest five distance with the smallest σ2 contribute the most to the signal. Taking

these and assuming independent single electron scattering events reproduces the qualitative

structure of the signal. EXAFS is sensitive to the balance between solute-solvent and solvent-

solvent interactions that determine the collective solvent structure about the solvent.

In order to connect to experimental measurement, we take configurations from a canonical

ensemble corresponding to the density and temperature of the measurement, determined

from the SCAN description of molecular interaction. For each configuration, we generate an

EXAFS signal using the feff9 code developed by Rehr and coworkers.48 We take the mean

of these configurations to generate a signal corresponding to the ensemble. In Figure 5 we

compare the k2 weighted fine structure, k2 χ(k), vs. k (in A−1), generated in this manner

compared to experimental measurement.20,49

For the case of K+, we recover unprecedented agreement with the measured signal. In

∣χ(r)∣ we recover both the main peak position and width at 2A, but in addition, we recover the

multiple scattering contributions between 3 A and 5 A. The k2χ(k) agreement in amplitude

and frequency from 3 A to 9 A gives us confidence in the consistency between measurement

and simulation.

In comparing measured and simulated signals, for the case of Na+, it is still a challenge

to recover the large amplitude of the measured EXAFS signal. The position of the peak

of ∣χ(r)∣ and the interference pattern of Im(χ(r)) appear to match the measured signal.

In addition, the effect Debye-Waller factor (width of χ(r)) appears to be recovered. The

frequency of the oscillations of k2χ(k) are recovered. The ion-water distance and fluctuations

appear to be recovered by the simulation. It is important to note that for Na+ the EXAFS

data is less reliable as for a low-Z atom the fluorescence self-absorption correction is difficult

10

to apply quantitatively.

For Na+ the mean square variation in the distance obtained from the full width at half

maximum of the Gaussian fit can be determined accurately from experimental XRD to be

0.020 A2. This agrees perfectly with the value determined from the simulation data of SCAN.

Discussion and Error Analysis

It is important to understand why this improvement in the structural properties is observed

with the SCAN functional. Obviously the fundamental answer lies in the more accurate

description of exchange and correlation. However, there should also be a discernible im-

provement in the energetics of the system, i.e., the energies computed with SCAN should

more closely agree with high level correlated quantum mechanical calculations than energies

computed with revPBE-D3. Unfortunately, there are various choices for how to compare the

energies for these systems and there is no systematic analysis of which approach is best. If a

reliable method could be identified that did not require high level calculations on very large

systems it would provide a useful tool and serve as a simple and computationally feasible

diagnostic test to determine which DFT functional will be best suited to reproduce struc-

tural details. Currently the only options for this approach rely on qualitative arguments

or to perform the full, computationally demanding simulations. It is not uncommon for

DFT functionals at similar levels of the theory to predict substantially different structural

details.26,32 This is also a challenge in the area of classical force field development where

force field dependence remains a significant issue.

A standard method used in the development of forcefields is to fit the parameters to

reproduce ion-water binding energies calculated with a high-level of quantum mechanical

theory.11,50–52 Similarly, it is common to use ion-water binding energies to benchmark and

compare different levels of DFT theory.51,52 Normally it is the absolute binding energy of a

11

0 1 2 3 4 5 6Radial distance [Å]

0

0.2

0.4

0.6

0.81

1.2

1.4

1.6

1.8

|χ(R

)|

SCAN exp.

0 1 2 3 4 5 6Radial distance [Å]

0

0.1

0.2

0.3

0.4

0.5

|χ(R

)|

SCAN E0 shiftSCANexp.

0 1 2 3 4 5 6Radial distance [Å]

-2

-1

0

1

2

Im|χ

(R)|

SCANexp.

0 2 4 6Radial distance [Å]

-0.4

-0.2

0

0.2

0.4

Im|χ

(R)|

SCAN E0 shiftSCANexp.

1 2 3 4 5 6 7 8Wave number [Å-1]

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

k2 |χ(R

)|

SCAN exp.

1 2 3 4 5 6 7 8 9 10Wave number [Å-1]

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

k2 |χ(R

)|

SCAN E0 shiftSCANexp.

Figure 5: EXAFS calculations compared with experiment for sodium (left) and potassium(right). Experimental data from Ref. 20,49.

12

cluster that is used. This is defined as:

EBind2 = EC+(H2O)n −EC+ − nE(H2O) (1)

Where C+ is the cation. Often only very small clusters are considered and the structures

of the clusters are minimum energy structures or artificial perturbations about it. This is

not the best choice, as the minimum energy structure of an ion-water dimer for instance will

be very different to the typical structures that water molecules take around ions under the

conditions of bulk solvation. Therefore, it is preferable to extract from simulation clusters

of the closest n waters about the ion. Larger water clusters than dimers and trimers will

be more relevant as they will incorporate the desired many-body effects. The definition of

the binding energy given above is not ideal for large cluster sizes, because the error will be

dominated by the error in the water-water interactions. If we are interested in improving

the description of the water structure specifically around the ion a better choice may be:

EBind1 = EC+(H2O)n −EC+ −E(H2O)n (2)

where the E(H2O)n is computed by simply removing the ion and recomputing the energy. This

expression gives the binding energy of the ion to the water not the water-water interaction.

This is also a very important quantity as it is the definition of the energy used in the potential

distribution theorem to determine the solvation free energy of the ion.53

When comparing with multiple structures it is customary to calculate the RMSD in this

binding energy between the DFT functional and a higher level of theory such as MP2. This

is given by:

RMSD =√

1

N∑N

(EMP2Bind1 −EDFT

Bind1)2

(3)

where the sum is over a N frames extracted from the simulation. It should be noted that this

is not the optimal choice if the goal is to accurately reproduce structural properties of the

13

ion in water. The possibility to have large average errors in the total binding energy and still

accurately reproduce the structural details produces a confounding picture for researchers

that are solely concerned with the accuracy of any particular method. This is because the

water structure is actually determined by (mean) forces rather than energies and thus a large

error in the total binding energy will not affect the structural properties if there is not a

large variation in binding energies from structure to structure. Assuming the ions are never

isolated in vacuum, the average error in the total binding energy will be irrelevant in terms

of the structural details. An alternative way of seeing this is that the relative probability

of two states is given by the ratio of the Boltzmann factors which should not be altered by

large constant error terms. That is:

e−βE1+Eerror

e−βE2+Eerror= e−βE1

e−βE2(4)

We suggest that a more effective quantity to examine is the standard deviation of the mean

signed error. This is given by:

SD-ME =√

1

N − 1∑N

((EMP2Bind1 −EDFT

Bind1) − ⟨EMP2Bind1 −EDFT

Bind1⟩)2

(5)

The angle brackets term is the average error in the DFT binding energy compared with

MP2 over the N structures sampled. This expression captures the variation of the error from

structure to structure. If this quantity is zero throughout the entire simulation then the

structural details should be perfectly reproduced regardless of the whether the total binding

energy is accurate or not. Standard basis sets used in condensed phase simulation are

demonstrably far from the complete basis set limit and running simulations at the complete

basis set limit is not computationally feasible. Basis set superposition error is known to be

a substantial issue in clusters and thus is not reasonable to expect DFT-MD methods to

reproduce absolute binding free energies in any case.

When we compute the standard deviation of the mean signed error using the EBind1

14

quantity given above we can see that the error going from revPBE-D3 to SCAN is reduced

by a factor of 7 (5.2 kJmol−1 to 0.7 kJmol−1) for Na(H2O)8 clusters and by a factor of 5

(7.8 kJmol−1 to 1.6 kJmol−1) for K(H2O)8 clusters. The resulting variation with the SCAN

functional is reduced below both thermal noise (2.4 kJmol−1) and “chemical accuracy” (4.2

kJmol−1 ) explaining the satisfactory experimental agreement of structural properties. A

very similar result is also seen for 32 water molecule clusters although a smaller basis set

for the reference MP2 calculations must be used. This reduction is not seen for the RMSD

error where it can actually be slightly larger for SCAN than revPBE-D3. We did not provide

an analysis for the RPA functional due to large memory requirements required to perform

these calculations in the large box sizes used to approximate a non-periodic system. The

supplementary information (SI) provides a detailed comparison of the different methods of

error estimation. It is true that many properties, such as single ion solvation free energies,

are sensitive to the accuracy of the absolute binding free energy not just the variation in

the error. However, once accurate structures are obtained, standard perturbative methods

can be used to estimate absolute binding energies in situations where this quantity is of

importance. See Ref. 53 for an example of this.

In summary, we have demonstrated that the newly developed SCAN functional can ac-

curately reproduce key structural properties of water around the sodium and potassium ions

that simple GGA functionals fail to reproduce. Additionally these properties cannot be ac-

curately reproduced in reasonable computational times using high level RPA functional. We

have argued that the fluctuation of the mean signed error of the ion-water binding energy

should be used to assess the quality of a DFT functional or forcefield and that this quantity

is consistent with the improved performance of the SCAN functional over revPBE-D3.

15

Computational Details

Ab initio Simulations

For the revPBE and SCAN functionals, Born-Oppenheimer ab initio molecular dynamics

simulations within the NVT (at 300 K) ensemble using periodic boundary conditions are

performed within the CP2K simulation suite (http:www.cp2k.org) containing the QuickStep

module for the DFT calculations.54 The D3 dispersion correction due to Grimme25 was used

for revPBE. A 0.5 fs time step was used. We used a double ζ basis set that has been optimized

for the condensed phase55 in conjunction with GTH pseudopotentials56 using a 400 Ry cutoff

for the auxiliary plane wave basis for the revPBE-D3 simulations and a 1200 Ry cutoff for the

SCAN simulations.32,57 The pseudopotentials for the oxygen atom were reoptimized for the

SCAN functional using the ATOM code of CP2K. A Nose-Hoover thermostat was attached

to every degree of freedom to ensure equilibration.58 The energies were accumulated for ≈

12 ps after 3 ps of equilibration. The sodium and potassium simulations for revPBE-D3 and

SCAN consisted of one sodium ion in a box of water molecules of dimensions 14.33 A3.

For the RPA simulations of potassium the procedure outlined in Ref. 3333,34 was used.

The simulation box consisted of 63 water molecules and a single potassium ion with dimen-

sions of 12.423A3. Basis sets of correlation consistent triple zeta quality, were used including

new basis sets for the potassium. The cutoff is set to 800 Ry. Core electrons are replaced by

pseudopotentials that have been parametrized for the PBE functional. The Kohn-Sham PBE

orbitals were used as input for the RPA calculation. The Monte Carlo (MC) simulations have

been performed employing the same setup as given in Ref. 46, with T = 295 K and p = 1 bar.

The MC efficiency is improved with the presampling of moves, the approximated potential is

calculated using the revPBE-D3 functional. The initial configuration has been equilibrated

with a 15 ps NVT-MD run at the experimental density using the revPBE-D3 level. The

statistics from the MC simulations were accumulated for 6100 frames after equilibration for

500.

16

ORCA59 was used to calculate the cluster energies at the MP2 level of theory. Clusters

of 8 and 32 water molecules were used in the cluster correction calculation with 50 frames

extracted from the 15ps trajectory. The aug-cc-pVxZ basis set was used for the oxygen

and hydrogen atoms.60 Similarly, the cc-pCVxZ basis set was used for the sodium ion61 and

the cc-pwCVxZ basis set for the potassium ion.62 x was either D or Q. Frozen cores were

used for the MP2 calculations. RI-MP2 was used for the 32 water cluster calculations using

automatically generated auxiliary basis functions. For the revPBE-D3 and SCAN cluster

energy calculations CP2K was used with the periodicity none option and a larger cell size

to remove any box size dependence. Otherwise the same parameters, basis sets etc. as the

simulation were used.

EXAFS

The measurements have correction of the multi-electron features. (see Ref. 49). In addition,

a shift in E0 is universally applied to the ensemble, adjusted to fit the first peak at 2A−1 In

Figure 5 we Fourier transform the χ(k) to construct χ(r). This is applied in a consistent

manner to both the measurement and the simulation. χ(r) consists of a distribution of scat-

tering distances that is distorted by the interaction of a photoelectron with the underlying

charge density. This interaction is accounted for in the multiple scattering code, feff. Ex-

perimental measurement of these systems is challenging due to issues associated with Multi

Electron Electron Scattering. In the current work, we carve out clusters of the nearest 10

water molecules to the cation to use as input for EXAFS calculation using default settings of

the FEFF9 code. For the K+ and Na+ systems every 10th configuration was chosen to make

up the ensemble, this gives 3055 configurations for K+ and 3029 configurations for Na+. The

ensemble averaged χ(k) signal is recovered from an average of configurations. Next we take

the Fourier transform of the k2 χ(k) signal to generate the ∣χ(R)∣ and Im[χ(R)] signals.

When compared to simple anions and doubly charged cations, it is has been a challenge

to find the right combination of molecular interaction and statistical mechanical sampling to

17

reproduce the measured EXAFS signals for Na+ and K+ aqueous solvation. It is a challenge

to find a balance between the strong cation-water interaction while maintaining the integrity

of the collective solvent structure, solvent-solvent interaction in the presence of the cation

and disrupting the hydrogen-bonding network, while also describing the long-range, bulk

structure and response of water. The use of a single molecular framework to consistently

describe each of these molecular responses remains a challenge.

Acknowledgements

This research used resources of the National Energy Research Scientific Computing Center,

a DOE Office of Science User Facility supported by the Office of Science of the U.S. Depart-

ment of Energy under Contract No. DE-AC02-05CH11231 in addition to Pacific Northwest

National Laboratory (PNNL) Institutional Computing resources. TTD, MG, GKS and CJM

were supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy

Sciences, Division of Chemical Sciences, Geosciences, and Biosciences. MD Baer was sup-

ported by MS3 (Materials Synthesis and Simulation Across Scales) Initiative, a Laboratory

Directed Research and Development Program PNNL. PNNL is a multiprogram national

laboratory operated by Battelle for the U.S. Department of Energy. JW and JH are sup-

port by The National Centre of Competence in Research (NCCR) Materials Revolution:

Computational Design and Discovery of Novel Materials (MARVEL) of the Swiss National

Science Foundation (SNSF). XSZ and TTD acknowledge the Australian Research Council

(ARC) funding via project number FL170100101. MDB is supported by the Center for

Computational Study of Excited-State Phenomena in Energy Materials (C2SEPEM) and

by the SciDAC Program on Excited State Phenomena in Energy Materials at the Lawrence

Berkeley National Laboratory, which is funded by the U.S. Department of Energy, Office of

Science, Basic Energy Sciences, Materials Sciences and Engineering Division under Contract

No. DE-AC02-05CH11231, as part of the Computational Materials Sciences Program.

18

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Supplementary Information

Cluster binding energies

The ion cluster binding energies are defined as:

EBind1 = EC+(H2O)n −EC+ −E(H2O)n (6)

25

EBind2 = EC+(H2O)n −EC+ − nE(H2O) (7)

The difference is that EBind2 includes the contribution from water-water interactions as well

as the ion-water binding. It is clear that the minimal SD-ME (1) error for SCAN compared

Table 2: Mean unsigned error in the cluster binding energies and fluctuationsin the mean error. The (1) and (2) indicates whether EBind1 or EBind2 was used.RMSD indicates the root mean squared deviation error estimate, whereas SDME indicates the standard deviation in the mean signed error. The DZ or QZin brackets indicates whether double zeta or quadruple zeta basis sets were usedfor the MP2 level calculation. Units are in kJmol−1.

RMSD(1) SD-ME(1) RMSD(2) SD-ME (2)Na+(H2O)8(revPBE-D3)(DZ) 5.8 5.8 9.4 7.8

Na+(H2O)8(SCAN)(DZ) 9.0 2.2 7.7 4.6Na+(H2O)8(revPBE-D3)(QZ) 8.1 5.2 17.5 4.1

Na+(H2O)8(SCAN)(QZ) 11.0 0.7 24.6 2.9Na+(H2O)32(revPBE-D3)(DZ) 16.3 5.5 81.7 11.5

Na+(H2O)32(SCAN(DZ) 9.3 1.9 23.3 8.4K+(H2O)8(revPBE-D3)(DZ) 10.0 6.3 8.5 7.2

K+(H2O)8(SCAN)(DZ) 9.7 1.9 7.7 4.0K+(H2O)8(revPBE-D3)(QZ) 8.3 7.8 13.2 10.1

K+(H2O)8(SCAN) (QZ) 2.0 1.6 11.9 2.8K+(H2O)32(revPBE-D3)(DZ) 24.4 6.8 54.3 12.2

K+(H2O)32(SCAN)(DZ) 6.9 2.1 20.3 8.1

with revPBE-D3 explains why SCAN has such good structural performance. The RMSD

error for the Na+(H2O)8(QZ) case actually goes up from revPBE-D3 to SCAN demonstrating

that this is not a good estimate of the error if reproducing structural properties is the aim.

It is clear that the error using the double zeta level of theory for the MP2 calculations

shows a less dramatic shift which is not surprising considering there will be significant error

associated with the smaller basis set. 32 water molecule clusters were also computed showing

that the standard deviation in the error does not change significantly as you go to the larger

cluster sizes.

Incremental RDFs

Here we provide incremental RDFs for the other functionals studied.

26

2 3 4r [Å]

0

0.5

1

1.5

2

2.5

3

g (r

)

(a) Sodium (Na+)

2 2.5 3 3.5 4 4.5 5r [Å]

0

0.5

1

1.5

2

g (r

)(b) Potassium (K+)

Figure 6: Incremental RDFs of the sodium and potassium ions in water with the revPBE-D3functional.

2 2.5 3 3.5 4 4.5 5r [Å]

0

0.5

1

1.5

2

2.5

g (r

)

(a) Sodium (Na+)

Figure 7: Incremental RDFs of the potassium ion in water with the RPA functional.

27

Graphical TOC Entry

0 2 4 6 8r [Å]

0

2

4

6

8

g (r

)

revPBE-D3SCANExp. peak position

28

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