We present first principles molecular dynamics simulations of the chloride anion in liquid water performed using gradient-corrected and hybrid density functionals. We show that it is necessary to use hybrid functionals both for the generation of molecular dynamics trajectories and for the calculation of electronic states in order to obtain a qualitatively correct description of the electronic properties of the solution. In particular, it is only with hybrid functionals that the highest occupied molecular orbital of the anion is found above the valence band maximum of water, consistent with photoelectron detachment measurements. Similar results were obtained using many body perturbation theory within the G0W0 approximation.

Aqueous solutions of simple salts such as alkali halides and common acids such as sulfates are of great interest in electrochemistry1,2 and atmospheric chemistry.3 Much progress was made in recent years in the microscopic description of simple aqueous solutions including the determination of the structure of the solvation shells of cations and anions1–8 and of their vibrational properties.9–14 

On the other hand, electronic structure studies are still limited. Experimentally, recent developments in liquid microjet techniques15,16 enabled the measurement of photoelectrons emitted directly from the liquid solutions and constituted a fundamental step forward in understanding the electronic properties of aqueous solutions. From a theoretical standpoint, only few interpretations or predictions15,17–20 of photoelectron spectra appeared in the literature; indeed, the microscopic description of aqueous solutions from first principles remains a challenging task, both from a theoretical and computational standpoint, and especially so in the case of anions, whose electronic structure is in general more complex than that of cations.7,21,22

In this paper, we consider a chloride anion in aqueous solution and report an analysis of its electronic properties based on first principles molecular dynamics (MD) using several levels of theory; these include semi-local (Perdew-Burke-Ernzerhof (PBE)23,24) and hybrid (PBE025) functionals and many body perturbation theory (MBPT)26 within the G0W0 approximation.27,28 To the best of our knowledge, these are the first calculations of the electronic properties of a solvated ion in water using ab initio MD with hybrid functionals and MBPT. By comparing trajectories obtained with the PBE and the PBE0 functionals, we first identified differences between the structural properties provided by the two levels of theory, and we then studied how these differences impact the electronic properties of the solution, e.g., the energy difference between the highest occupied molecular orbital (HOMO) of the anion and the valence band maximum (VBM) of water. In the following, we first briefly describe the methods used in our study, and then discuss our results and compare them with experiments.15 

We performed first principles MD simulations of a chloride anion in water using the gradient-corrected PBE functional23,24 and the PBE0 hybrid functional.25 We used supercells containing one chloride ion and 63 D2O molecules, corresponding to a 0.87 M concentration. The density was fixed at the experimental value of D2O at ambient conditions, and we then removed one water molecule and added one chloride ion. Our calculations were carried out with the Qbox code29 plane wave basis sets and norm-conserving pseudopotentials30,31 with a kinetic energy cutoff of 85 Ry. Simulations were carried out in the NVE ensemble with a time step of 10 a.u., within a Born-Oppenheimer framework. After the solution was equilibrated for ∼5 ps with either the PBE or the PBE0 functional, the PBE trajectories were collected for 20 ps at an average temperature of 388 ± 19 K, and the PBE0 ones were collected for ∼6 ps with an average temperature of 379 ± 18 K. We chose a temperature higher than 300 K since water is overstructured at ambient conditions, when using semi-local32 and some hybrid functionals.33 The electronic contributions to the molecular dipole moment were computed using maximally localized Wannier functions (MLWFs), evaluated at each MD step.34 Electronic structure calculations were carried out at three different levels of theory, using the PBE and PBE0 functionals and within the G0W0 approximation27 with the method developed in Refs. 28 and 35, which allowed us to perform calculations for large supercells for several snapshots. In the following, we denote the results of electronic structure calculations using the PBE (PBE0) functionals and PBE0 (PBE) trajectories as PBE/PBE0 (PBE0/PBE) calculations.

We first discuss the structural properties of the chloride solution obtained at the PBE and PBE0 levels of theory. The position of the first maximum (rmax) and minimum (rmin) of the Cl−O and Cl−D pair correlation functions and the corresponding coordination numbers are reported in Table I, together with experimental values. The latter ones differ depending on the technique used (i.e., neutron diffraction or X-ray measurements) and the type of counterions present in the solution,5,6,36–41 whose effect however appears to be minor. The dependence of experimental data on concentration is instead more pronounced (at least for the Cl−D coordination). At both levels of theory (PBE and PBE0), our results are consistent with experiments and with previous generalized gradient approximation (GGA) calculations, which reported Cl−O solvation shell cutoff radii |${\rm r}_{min}^{{{\rm Cl}}-{\rm O}}$|rmin Cl O varying from 3.76 to 3.90 Å,12,42–45 and coordination numbers between 5.4 and 6.7,42–45 When using PBE0, we obtained a larger |${\rm r}_{max}^{{{\rm Cl}}-{\rm O}}$|rmax Cl O and higher Cl−O coordination number than with PBE, possibly indicating better agreement with experiments. It thus appears that the GGA tends to overestimate the strength of the Cl−O interaction in water, possibly due to an overestimate of the water and Cl polarizability.

Table I.

Structure of the first chloride solvation shell.a

 |${\rm r}_{max}^{{\rm Cl}-{\rm O}}$|rmax Cl O (Å)|${\rm r}_{min}^{{{\rm Cl}}-{\rm O}}$|rmin Cl O (Å)|${\rm N}_{coor}^{{{\rm Cl}}-{\rm O}}$|Ncoor Cl O|${\rm r}_{max}^{{\rm Cl}-D}$|rmax Cl D (Å)|${\rm r}_{min}^{{\rm Cl}-D}$|rmin Cl D (Å)|${\rm N}_{coor}^{{\rm Cl}-D}$|Ncoor Cl D
PBE0 3.15 ± 0.01 3.83 ± 0.04 6.4 ± 0.29 2.21 ± 0.01 2.90 ± 0.06 5.2 ± 0.31 
PBE 3.11 ± 0.01 3.79 ± 0.17 6.1 ± 0.87 2.15 ± 0.03 2.84 ± 0.03 4.9 ± 0.05 
Expt.b 3.05–3.25 ± 0.20c … 6.0–6.5 ± 0.5c 2.18–2.29 ± 0.16d … 5.3–5.8 ± 0.5d 
  3.14–3.34 ± 0.20d         4.4 ± 0.3e 
 |${\rm r}_{max}^{{\rm Cl}-{\rm O}}$|rmax Cl O (Å)|${\rm r}_{min}^{{{\rm Cl}}-{\rm O}}$|rmin Cl O (Å)|${\rm N}_{coor}^{{{\rm Cl}}-{\rm O}}$|Ncoor Cl O|${\rm r}_{max}^{{\rm Cl}-D}$|rmax Cl D (Å)|${\rm r}_{min}^{{\rm Cl}-D}$|rmin Cl D (Å)|${\rm N}_{coor}^{{\rm Cl}-D}$|Ncoor Cl D
PBE0 3.15 ± 0.01 3.83 ± 0.04 6.4 ± 0.29 2.21 ± 0.01 2.90 ± 0.06 5.2 ± 0.31 
PBE 3.11 ± 0.01 3.79 ± 0.17 6.1 ± 0.87 2.15 ± 0.03 2.84 ± 0.03 4.9 ± 0.05 
Expt.b 3.05–3.25 ± 0.20c … 6.0–6.5 ± 0.5c 2.18–2.29 ± 0.16d … 5.3–5.8 ± 0.5d 
  3.14–3.34 ± 0.20d         4.4 ± 0.3e 
a

|${\rm r}_{max}^{{{\rm Cl}}-{\rm O}}$|rmax Cl O, |${\rm r}_{min}^{{{\rm Cl}}-{\rm O}}$|rmin Cl O, and |${\rm N}_{coor}^{{{\rm Cl}}-{\rm O}}$|Ncoor Cl O are the positions of the first maximum and minimum of the Cl−O pair, correlation function, and the coordination number corresponding to |${\rm r}_{min}^{{{\rm Cl}}-{\rm O}}$|rmin Cl O, respectively; |${\rm r}_{max}^{{\rm Cl}-D}$|rmax Cl D, |${\rm r}_{min}^{{\rm Cl}-D}$|rmin Cl D, and |${\rm N}_{coor}^{{\rm Cl}-D}$|Ncoor Cl D are the same quantities for the Cl−O pair correlation.

b

Experimental values of alkali and alkaline earth metal chloride solutions.

c

X-ray measurements.5,39,40

d

Neutron diffraction experiments on solutions at low concentration (<10 M).6,36,38,41

e

Neutron diffraction experiments on a LiCl solution at a high concentration of 14.9 M.37 

The major difference between the PBE0 and PBE descriptions of the structural properties of the solution was observed in the Cl−O angular correlations. The distributions of the tilt angle Φ between the Cl−O distance, and the dipole moment of water molecules within the chloride first solvation shell are shown in Fig. 1. A main peak centered at ∼55° and a shoulder around at ∼100° are present in both distributions, with an additional shoulder at ∼35° in the PBE case. The peak at ∼55° (a) indicates that in the first chloride solvation shell the dipole moment of the water molecules is mainly oriented towards the interior of the shell, with a smaller contribution (b) coming from molecules with dipoles oriented outwards. However, there are important differences between the PBE and PBE0 results: the main peak at ∼55° is less prominent in the PBE data, and a sensibly larger number of molecules adopt configuration (c), leading to an overall broader distribution of tilt angles than in the PBE0 case. Such orientation differences are robust against the choice of solvation shell radii, i.e., the results are unchanged irrespective of whether one chooses |${\rm r}_{min}^{{{\rm Cl}}-{\rm O}}$|rmin Cl O computed at the PBE or PBE0 level of theory (see Fig. 1).

FIG. 1.

Distributions of the tilt angle Φ between the Cl−O distance and the dipole moment of water molecules within the first Cl−O solvation shell, obtained with the PBE0 (black) and PBE (red) functionals. Insets (a)–(c) show three orientations of water molecules within the first chloride solvation shell. Solid and dashed lines correspond to results obtained with two different cutoff radii for the Cl−O distance within the first solvation shell.

FIG. 1.

Distributions of the tilt angle Φ between the Cl−O distance and the dipole moment of water molecules within the first Cl−O solvation shell, obtained with the PBE0 (black) and PBE (red) functionals. Insets (a)–(c) show three orientations of water molecules within the first chloride solvation shell. Solid and dashed lines correspond to results obtained with two different cutoff radii for the Cl−O distance within the first solvation shell.

Close modal

We now turn to the discussion of the electronic structure of the solvated chloride anion, and we show how the subtle structural differences between the PBE and PBE0 results observed in our simulations influence the electronic properties of the system. The computed band gap of the solution is 6.55 ± 0.18 eV and 4.18 ± 0.18 eV, in our PBE0/PBE0 and PBE/PBE calculations, respectively. The values were obtained as averages over 8 and 15 configurations extracted from the respective trajectories, and are consistent with previous calculations using smaller cells with 32 water molecules.33 (As discussed below, in our calculations the difference between the band gap of the Cl solution and of pure water is at most 0.3 eV.) Although a substantial improvement on the PBE result, the PBE0 gap underestimates the value of the quasi particle gap obtained by photoemission experiments (8.7 ± 0.5 eV46), and by GW calculations (8.1 ± 0.2 eV47 and 8.7 eV48,49). We note that within statistical errors, we obtained the same value of the gap (6.58 ± 0.21 eV) in PBE0/PBE calculations. This result indicates that the value of the electronic gap of the pristine liquid is not sensitive to the details of the atomic structure and the differences in hydrogen bonding configurations found between PBE and PBE0 samples.33 

On the contrary, we found that the electronic structure of the solvated anion is sensitive to the structural properties of the solution: in particular, the position of the HOMO of the anion with respect to the VBM of water is qualitatively different when using PBE and PBE0 trajectories. It is only when the PBE0 functional is adopted to describe both the trajectories and the electronic structure of the solvated Cl that we obtain a HOMO level of the anion on average above the VBM of the liquid. Within PBE, irrespective of the trajectories used, the HOMO of the anion is below the VBM of water. In PBE0/PBE simulations, the Cl HOMO moves slightly upward but not enough to be above the water VBM. These findings are illustrated in Fig. 2, where we report the five topmost energy levels of the solutions, obtained from PBE0/PBE0 and PBE/PBE0 simulations (left panels) and PBE0/PBE and PBE/PBE simulations (right panels). The energy levels were classified as belonging to water (black circles) or the anion (red squares) based on the localization of the square moduli of the wavefunctions. In general the single particle states are more delocalized at the PBE level than at the PBE0 level of theory, as illustrated in Fig. S1 of the supplementary material.57 This result is consistent with other reports on Cl/H2O clusters21 and shows that the PBE0 functional at least partially corrects the delocalization error of the semi-local PBE description.

FIG. 2.

(Left panel) Top five valence bands of a 0.87 M solution of Cl in water calculated with the PBE0 and PBE functionals on an ∼6 ps PBE0 trajectory. (Right panel) Top five valence bands computed with the PBE0 and PBE functionals on a 20 ps PBE trajectory. Red squares and black circles represent states assigned to the anion and to water, respectively. The assignment was made based on the localization of the corresponding eigenfunctions.

FIG. 2.

(Left panel) Top five valence bands of a 0.87 M solution of Cl in water calculated with the PBE0 and PBE functionals on an ∼6 ps PBE0 trajectory. (Right panel) Top five valence bands computed with the PBE0 and PBE functionals on a 20 ps PBE trajectory. Red squares and black circles represent states assigned to the anion and to water, respectively. The assignment was made based on the localization of the corresponding eigenfunctions.

Close modal

We also carried out MBPT calculations, within the G0W0 approximation, for snapshots extracted from PBE0 trajectories, starting from PBE wavefunctions. We used the techniques proposed in Refs. 28 and 35 with 1000 eigenpotentials for the spectral decomposition of the dielectric matrix and the same kinetic energy cutoff as that adopted for the solution of the Kohn-Sham equations (85 Ry). Table SI of the supplementary material57 reports the energy difference between the Cl HOMO and the VBM of water for eight snapshots. Figure 3 reports the top five uppermost valence band eigenvalues as computed within G0W0. Although the use of MBPT significantly improves on the PBE results, it does not suffice to bring the Cl HOMO significantly above the water valence band.

FIG. 3.

Top five valence bands of a 0.87 M solution of Cl in water calculated using G0W0 corrections of PBE eigenvalues for snapshots extracted from an ∼6 ps PBE0 trajectory. Red squares and black circles represent states assigned to the anion and to water, respectively. The assignment was made based on the localization of the corresponding eigenfunctions.

FIG. 3.

Top five valence bands of a 0.87 M solution of Cl in water calculated using G0W0 corrections of PBE eigenvalues for snapshots extracted from an ∼6 ps PBE0 trajectory. Red squares and black circles represent states assigned to the anion and to water, respectively. The assignment was made based on the localization of the corresponding eigenfunctions.

Close modal

The results on the electronic structure of the anion obtained at the PBE0 and G0W0 levels of theory are in qualitative agreement with experiment, however, the energy difference between the Cl HOMO and the VBM of water (∼0.23 eV within PBE0 and ∼0.08 eV within G0W0) is an underestimate of the experimental value. The measured electron binding energies of a 3 M chloride solution and water are 9.6 ± 0.0715 and 11.16 ± 0.04 eV,50 respectively, and the experimental values of the threshold energy of photoelectron emission spectra are 8.81 and 10.06 eV51 for a 2 M chloride solution and liquid water, respectively. This indicates that the HOMO of Cl is located about 1.25−1.56 eV above the VBM of water. We note that the comparison between our calculations and the experiments of Ref. 15 is not straightforward. The measured photoemission spectra15 of water and those of (NaCl)aq. show a considerable overlap of the bands of the solution and of pristine water below 10 eV, and the onset of the (NaCl)aq. spectrum is not easy to detect. A weak shoulder is present between 8.8 and 9.6 eV (see Fig. 1 of Ref. 15). Most importantly, as pointed out in Ref. 15, the spectra of the solution may have contributions from both bulk and surface anions. The structural and electronic properties of the latter differ from those of the bulk anions, as Cl anions are known to preferably reside at the surface.52–55 Hence, the signal detected experimentally may have an overall significant component originating from surface anions. Finally, we note that the concentration of the solution simulated here (∼0.87 M) is lower than those studied experimentally (2 M51,56 and 3 M15) and the concentration difference may also be partially responsible for the discrepancy with experiments.

In summary, we presented a study of the electronic properties of the solvated chloride anion using first principles MD and both semi-local (PBE) and hybrid (PBE0) functionals. We showed that it is only when using both PBE0 trajectories and the PBE0 functional to compute electronic properties that the HOMO of the anion is found above the VBM of water. This finding is consistent with photoelectron detachment measurements, although our results underestimate the energy difference between the HOMO of the chloride ion and the VBM of water found experimentally. Work is in progress to investigate the contribution of surface anions to the signals detected in photodetachment experiments, and to improve on MBPT calculations of aqueous solutions.

We thank M. Sprik for useful discussions. This work was supported by DOE/BES (Grant No. DE-SC0008938) and DOE/CMCSN (Grant No. DE-SC0005180). This research used resources of the Argonne Leadership Computing Facility at Argonne National Laboratory, which is supported by the Office of Science of the U.S. DOE under Contract No. DE-AC02-06CH11357. Part of this work was performed under the auspices of the U.S. DOE by LLNL under Contract No. DE-AC52-07NA27344. T.A.P. acknowledges support from the Lawrence Scholar Program.

1.
Q.
Mi
,
A.
Zhanaidarova
,
B. S.
Brunschwig
,
H. B.
Gray
, and
N. S.
Lewis
,
Energy Environ. Sci.
5
,
5694
(
2012
).
2.
J. C.
Hill
and
K.-S.
Choi
,
J. Phys. Chem. C
116
,
7612
(
2012
).
3.
J. P. D.
Abbatt
,
S.
Benz
,
D. J.
Cziczo
,
Z.
Kanji
,
U.
Lohmann
, and
O.
Möhler
,
Science
313
,
1770
(
2006
).
4.
A. W.
Omta
,
M. F.
Kropman
,
S.
Woutersen
, and
H. J.
Bakker
,
Science
301
,
347
(
2003
).
5.
L. X.
Dang
,
G. K.
Schenter
,
V.-A.
Glezakou
, and
J. L.
Fulton
,
J. Phys. Chem. B
110
,
23644
(
2006
).
6.
R.
Mancinelli
,
A.
Botti
,
F.
Bruni
,
M. A.
Ricci
, and
A. K.
Soper
,
J. Phys. Chem. B
111
,
13570
(
2007
).
7.
H. J.
Kulik
,
N.
Marzari
,
A. A.
Correa
,
D.
Prendergast
,
E.
Schwegler
, and
G.
Galli
,
J. Phys. Chem. B
114
,
9594
(
2010
).
8.
A.
Bankura
,
V.
Carnevale
, and
M. L.
Klein
,
J. Chem. Phys.
138
,
014501
(
2013
).
9.
P. B.
Petersen
and
R. J.
Saykally
,
J. Phys. Chem. B
110
,
14060
(
2006
).
10.
S.
Park
and
M. D.
Fayer
,
Proc. Natl. Acad. Sci. U.S.A.
104
,
16731
(
2007
).
11.
J. D.
Smith
,
R. J.
Saykally
, and
P. L.
Geissler
,
J. Am. Chem. Soc.
129
,
13847
(
2007
).
12.
B. S.
Mallik
,
A.
Semparithi
, and
A.
Chandra
,
J. Chem. Phys.
129
,
194512
(
2008
).
13.
A. M.
Jubb
and
H. C.
Allen
,
J. Phys. Chem. C
116
,
13161
(
2012
).
14.
J. L.
Skinner
,
P. A.
Pieniazek
, and
S. M.
Gruenbaum
,
Acc. Chem. Res.
45
,
93
(
2012
).
15.
B.
Winter
,
R.
Weber
,
I. V.
Hertel
,
M.
Faubel
,
P.
Jungwirth
,
E. C.
Brown
, and
S. E.
Bradforth
,
J. Am. Chem. Soc.
127
,
7203
(
2005
).
16.
R.
Seidel
,
S.
Thürmer
, and
B.
Winter
,
J. Phys. Chem. Lett.
2
,
633
(
2011
).
17.
C.
Adriaanse
,
M.
Sulpizi
,
J.
VandeVondele
, and
M.
Sprik
,
J. Am. Chem. Soc.
131
,
6046
(
2009
).
18.
C.
Adriaanse
,
J.
Cheng
,
V.
Chau
,
M.
Sulpiz
,
J.
VandeVondele
, and
M.
Sprik
,
J. Phys. Chem. Lett.
3
,
3411
(
2012
).
19.
D.
Ghosh
,
A.
Roy
,
R.
Seidel
,
B.
Winter
,
S.
Bradforth
, and
A. I.
Krylov
,
J. Phys. Chem. B
116
,
7269
(
2012
).
20.
E.
Pluhařová
,
M.
Ončák
,
R.
Seidel
,
C.
Schroeder
,
W.
Schroeder
,
B.
Winter
,
S. E.
Bradforth
,
P.
Jungwirth
, and
P.
Slavíček
,
J. Phys. Chem. B
116
,
13254
(
2012
).
21.
A. J.
Cohen
,
P.
Mori-Sánchez
, and
W. T.
Yang
,
Science
321
,
792
(
2008
).
22.
A. J.
Cohen
,
P.
Mori-Sánchez
, and
W.
Yang
,
Chem. Rev.
112
,
289
(
2012
).
23.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
77
,
3865
(
1996
).
24.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
78
,
1396
(
1997
).
25.
C.
Adamo
and
V.
Barone
,
J. Chem. Phys.
110
,
6158
(
1999
).
26.
Y.
Ping
,
D.
Rocca
, and
G.
Galli
,
Chem. Soc. Rev.
42
,
2437
(
2013
).
28.
H.-V.
Nguyen
,
T. A.
Pham
,
D.
Rocca
, and
G.
Galli
,
Phys. Rev. B
85
,
081101
(
2012
).
30.
D. R.
Hamann
,
Phys. Rev. B
40
,
2980
(
1989
).
31.
L.
Kleinman
and
D. M.
Bylander
,
Phys. Rev. Lett.
48
,
1425
(
1982
).
32.
J. C.
Grossman
,
E.
Schwegler
,
E. W.
Draeger
,
F.
Gygi
, and
G.
Galli
,
J. Chem. Phys.
120
,
300
(
2004
).
33.
C.
Zhang
,
D.
Donadio
,
F.
Gygi
, and
G.
Galli
,
J. Chem. Theory Comput.
7
,
1443
(
2011
).
34.
F.
Gygi
,
J. L.
Fattebert
, and
E.
Schwegler
,
Comput. Phys. Commun.
155
,
1
(
2003
).
35.
T. A.
Pham
,
H.-V.
Nguyen
,
D.
Rocca
, and
G.
Galli
,
Phys. Rev. B
87
,
155148
(
2013
).
36.
S.
Cummings
,
J. E.
Enderby
,
G. W.
Neilson
,
J. R.
Newsome
,
R. A.
Howe
,
W. S.
Howells
, and
A. K.
Soper
,
Nature (London)
287
,
714
(
1980
).
37.
A. P.
Copestake
,
G. W.
Neilson
, and
J. E.
Enderby
,
J. Phys. C
18
,
4211
(
1985
).
38.
M.
Yamagami
,
H.
Wakita
, and
T.
Yamaguchi
,
J. Chem. Phys.
103
,
8174
(
1995
).
39.
T.
Megyes
,
I.
Bakó
,
S.
Bálint
,
T.
Grósz
, and
T.
Radnai
,
J. Mol. Liq.
129
,
63
(
2006
).
40.
A.
Tongraar
,
J.
T-Thienprasert
,
S.
Rujirawat
, and
S.
Limpijumnong
,
Phys. Chem. Chem. Phys.
12
,
10876
(
2010
).
41.
F.
Bruni
,
S.
Imberti
,
R.
Mancinelli
, and
M. A.
Ricci
,
J. Chem. Phys.
136
,
064520
(
2012
).
42.
T.
Ikeda
,
M.
Hirata
, and
T.
Kimura
,
J. Chem. Phys.
119
,
12386
(
2003
).
43.
J. M.
Heuft
and
E. J.
Meijer
,
J. Chem. Phys.
119
,
11788
(
2003
).
44.
R.
Scipioni
,
D. A.
Schmidt
, and
M.
Boero
,
J. Chem. Phys.
130
,
024502
(
2009
).
45.
N.
Galamba
,
R. A.
Mata
, and
B. J. C.
Cabral
,
J. Phys. Chem. A
113
,
14684
(
2009
).
46.
A.
Bernas
,
C.
Ferradini
, and
J.-P.
Jay-Gerin
,
Chem. Phys.
222
,
151
(
1997
).
47.
T. A.
Pham
 et al, “
Electronic structure of liquid water using many body perturbation theory
” (unpublished); GW calculations were carried out using 64 water molecule samples and the method of Ref. 28.
48.
V.
Garbuio
,
M.
Cascella
,
L.
Reining
,
R. D.
Sole
, and
O.
Pulci
,
Phys. Rev. Lett.
97
,
137402
(
2006
).
49.
D.
Lu
,
F.
Gygi
, and
G.
Galli
,
Phys. Rev. Lett.
100
,
147601
(
2008
).
50.
B.
Winter
,
R.
Weber
,
W.
Widdra
,
M.
Dittmar
,
M.
Faubel
, and
I. V.
Hertel
,
J. Phys. Chem. A
108
,
2625
(
2004
).
51.
P.
Delahay
,
Acc. Chem. Res.
15
,
40
(
1982
).
52.
S.
Ghosal
,
J. C.
Hemminger
,
H.
Bluhm
,
B. S.
Mun
,
E. L. D.
Hebenstreit
,
G.
Ketteler
,
D. F.
Ogletree
,
F. G.
Requejo
, and
M.
Salmeron
,
Science
307
,
563
(
2005
).
53.
D. J.
Tobias
and
J. C.
Hemminger
,
Science
319
,
1197
(
2008
).
54.
C.
Calemana
,
J. S.
Hubb
,
P. J.
van Maaren
, and
D.
van der Spoel
,
Proc. Natl. Acad. Sci. U.S.A.
108
,
6838
(
2011
).
55.
M. D.
Baer
and
C. J.
Mundy
,
J. Phys. Chem. Lett.
2
,
1088
(
2011
).
56.
I.
Watanabe
,
J. B.
Flanagan
, and
P.
Delahay
,
J. Chem. Phys.
73
,
2057
(
1980
).
57.
See supplementary material at http://dx.doi.org/10.1063/1.4804621 for Figure S1 and Table SI.

Supplementary Material