Supplementary Material for
An Analysis of Hydrated Proton Diffusion in Ab Initio Molecular Dynamics
Ying-Lung Steve Tse,†, § Chris Knight,‡, § Gregory A. Voth†,a)
†
Department of Chemistry, James Franck Institute, and Computation Institute, University of Chicago,
Chicago, Illinois 60637, United States
‡
Leadership Computing Facility, Argonne National Laboratory, Argonne, Illinois 60439, USA, United
States
§
Y.-L. S. T. and C.K. contributed equally
The calculation of the hn(τ) function
What is important in this investigation are the criteria used to associate an excess proton with a particular
water molecule. Typically, distance criteria, such as that employed here, are utilized and ultimately there
can be some degree of ambiguity in the assignment, particularly for transitionary proton transfer
complexes (e.g. compressed water wires) involving multiple excess protons, as spatially delocalized
charged defects are effectively mapped onto single, localized points in space.
In the calculation of the hn(τ) function, for each time interval τ, the excess proton has to be first assigned
to a water molecule. If the excess proton is associated the most with a particular water molecule during
the interval, then it is assigned to that water molecule. If the excess proton is associated with two or more
water molecules for equivalent times, then the excess proton is assigned to the water molecule that was
assigned during the previous time interval. If none of these candidate water molecules are the previously
assigned water, then the excess proton is assigned to the first water molecule encountered during the
current time interval. The value of n corresponds to the number of distinct water molecules counted
during n consecutive time intervals of length τ. For example, if the excess proton starts on water A and
after a time τ the proton remains with water A, then n takes the value of one and h1(τ) is incremented. The
h2(τ) function would be incremented if instead the proton hopped to a different water molecule after time t
and subsequently hopped back to A or stayed on B after a second time τ passed.
S1
Table S1: The estimated probability for the fraction of time that the void is filled is in the F state (See
Fig. 6 in the main text for the definition of the F state) at temperature 300 K. The estimated conditional
probability for  is in F in its current state given the last state was in F is also shown in this table.
∈ F
Setup
BLYP-D3-TZV2P-2
BLYP-1
BLYP-2
BLYP-D3-DZVP-1
BLYP-D3-DZVP-2
HCTH-1
HCTH-2
Burst
0.555
0.610
0.658
0.588
0.647
0.264
0.406
Rest
0.311
0.384
0.572
0.161
0.262
0.221
0.246
∈ F given the
previous ∈ F
Burst
0.579
0.595
0.559
0.414
0.545
0.213
0.404
Rest
0.347
0.245
0.385
0.105
0.242
0.115
0.124
M06-L AIMD Simulation
Preliminary results are shown for AIMD simulations with the M06-L functional to indicate a possible
candidate for a density functional approximation with improved water-water properties. The
computational setup for the M06-L simulation in CP2K is identical to those discussed in the manuscript
with the exception of the use of a 500 Ry plane wave cutoff. The M06-L functional was accessed using
the Libxc library for these simulations.1 The Goedecker-Teter-Hutter pseudopotential parameters for the
oxygen and hydrogen atoms were computed using the Atom module within CP2K and reported in Table
S2. The last configuration from the BLYP–D3–TZV2P–2 simulations was used as the initial
configuration for this simulation. The system was initially equilibrated in the constant NVT ensemble with
a target temperature of 300 K for 10.5 ps and subsequently simulated in the constant NVE ensemble for
36.5 ps. The average temperature during the NVE simulation was 289.8 K. The limited statistics and
lowered temperature relative to all other reported simulations in this work may contribute to some of the
observed discrepancies, however, the results are encouraging compared to other functional
approximations and work is currently in progress to examine converge properties at temperatures in the
300-330 K range. The inclusion of empirical dispersion corrections may further improve agreement with
experiment as observed with other functionals and reduce some of the discrepancies observed in this
single, short simulation.
Table S2: GTH pseudopotential parameters for the M06-L density functional for hydrogen and oxygen.
The format is that same as that used in the original GTH reference.
H
0.20
O
0.24084771
0.22187778












S2
hn(): Normalized counts per 50 ps
1000
BLYP-D3-TZV2P-1
BLYP-D3-TZV2P-2
BLYP-1
BLYP-2
HCTH-1
HCTH-2
n=1
n=2
100
n=3
10
1
0.0
0.1
0.2
0.3
0.4
0.5
(ps)
Figure S1: hn(τ) is plotted for different AIMD simulations equilibrated at 300 K as a function of τ. hn(τ) is
normalized to 50 ps to account for the different simulation times in different setups.
S3
Figure S2: In the left column, the distance of hydronium oxygen (O*) relative to its initial position is
plotted as a function of time for (a) BLYP–2, (b) HCTH–2, (c) BLYP–D3–DZVP–1, and (d) BLYP–D3–
DZVP–2. The right column shows the corresponding O*–Hw RDF from each of the “burst” and “rest”
segments defined in the left column.
S4
Figure S3: The joint probability density distributions between d and a single  is shown to illustrate how
to determine the direction of a blue or green arrow in the top left figure. In the lower right picture, label
“2y1” indicates the 2y oxygen that is closer to O*. The normalization of the distribution is
50
50
 P( x , y
n 1 n 1
n
m
)xy  1 , where xn and ym are the centers of the bins on the x and y axes, and Δx =
range(x)/50 and Δy = range(y)/50 are the bin sizes. See also the main text for more explanation.
S5
Figure S4: The radial distribution function g(r) between water oxygens (Ow) for all the setups at two
different temperatures at 300 and 330 K. The “Expt 1” and “Expt 2” data come from Skinner and
Benmore2 and Soper and Benmore,3 respectively.
Figure S5: The radial distribution function g(r) between water hydrogens (Hw) for all the setups at two
different temperatures at 300 and 330 K. The “Expt 2” data comes from Skinner and Benmore.2
S6
Figure S6: The radial distribution function g(r) between water oxygen (Ow) and water hydrogen (Hw) for
all the setups at two different temperatures at 300 and 330 K. The “Expt 2” data comes from Skinner and
Benmore.2
Figure S7: The radial distribution function g(r) between hydronium oxygen (O*) and Ow for all the setups
at two different temperatures at 300 and 330 K. The “Expt 3” data is from Botti, Ricci, and Soper.4
S7
Figure S8: The radial distribution function g(r) between hydronium oxygen (O*) and water hydrogen (Hw)
for all the setups at two different temperatures at 300 and 330 K. The “Expt 3” data is from Botti, Ricci,
and Soper.4
Figure S9: The radial distribution function g(r) between hydronium hydrogen (H*) and water hydrogen
(Hw) for all the setups at two different temperatures at 300 and 330 K. The “Expt 3” data is from Botti,
Ricci, and Soper.4
S8
Figure S10: The radial distribution function g(r) between hydronium hydrogen (H*) and water oxygen
(Ow) for all the setups at two different temperatures at 300 and 330 K. The “Expt 3” data is from Botti,
Ricci, and Soper.4
Figure S11: The water mean-squared displacement (MSD) as a function of time at 300 K.
S9
Figure S12: The water mean-squared displacement (MSD) as a function of time at 330 K.
Figure S13: The proton mean-squared displacement (MSD) as a function of time at 300 K.
S10
Figure S14: The proton mean-squared displacement (MSD) as a function of time at 330 K.
Figure S15: Free energy profile as a function of the proton transfer coordinate that is (as typically)
defined as the positive difference of the distances between the excess proton and its closest water oxygen
neighbor and its second closest water oxygen neighbor.
S11
Supplemental References:
[1]
M. A. L. Marques, M. J. T. Oliveira, and T. Burnus, Comput. Phys. Commun. 183, 2272 (2012).
[2]
L. B. Skinner, C. C. Huang, D. Schlesinger, L. G. M. Pettersson, A. Nilsson, and C. J. Benmore, J.
Chem. Phys. 138, 074506 (2013).
[3]
A. K. Soper, and C. J. Benmore, Phys. Rev. Lett. 101, 065502 (2008).
[4]
A. Botti, F. Bruni, S. Imberti, M. A. Ricci, and A. K. Soper, J. Chem. Phys. 121, 7840 (2004).
a)
Electronic mail: gavoth@uchicago.edu
S12