In this study, we investigate the nuclear quantum effects (NQEs) on the acidity constant (pKA) of liquid water isotopologs under the ambient condition by path integral molecular dynamics (PIMD) simulations. We compared simulations using a fully explicit solvent model with a classical polarizable force field, density functional tight binding, and ab initio density functional theory, which correspond to empirical, semiempirical, and ab initio PIMD simulations, respectively. The centroid variable with respect to the proton coordination number of a water molecule was restrained to compute the gradient of the free energy, which measures the reversible work of the proton abstraction for the quantum mechanical system. The free energy curve obtained by thermodynamic integration was used to compute the pKA value based on probabilistic determination. This technique not only reproduces the pKA value of liquid D2O experimentally measured (14.86) but also allows for a theoretical prediction of the pKA values of liquid T2O and aqueous HDO and HTO, which are unknown due to their scarcity. It is also shown that the NQEs on the free energy curve can result in a downshift of 4.5 ± 0.9 pKA units in the case of liquid water, which indicates that the NQEs plays an indispensable role in the absolute determination of pKA. The results of this study can help inform further extensions into the calculation of the acidity constants of isotope substituted species with high accuracy.

1.
K. S.
Alongi
and
G. C.
Shields
, “
Theoretical calculations of acid dissociation constants: A review article
,” in
Annual Reports in Computational Chemistry
, edited by
R. A.
Wheeler
(
Elsevier
,
2010
), Chap. 8, Vol. 6, pp.
113
138
.
2.
E.
Alexov
,
E. L.
Mehler
,
N.
Baker
,
A.
M. Baptista
,
Y.
Huang
,
F.
Milletti
,
J.
Erik Nielsen
,
D.
Farrell
,
T.
Carstensen
,
M. H. M.
Olsson
,
J. K.
Shen
,
J.
Warwicker
,
S.
Williams
, and
J. M.
Word
, “
Progress in the prediction of pKa values in proteins
,”
Proteins
79
,
3260
3275
(
2011
).
3.
J.
Ho
and
M. L.
Coote
, “
A universal approach for continuum solvent pK a calculations: Are we there yet?
,”
Theor. Chem. Acc.
125
,
3
(
2010
).
4.
J.
Tomasi
,
B.
Mennucci
, and
R.
Cammi
, “
Quantum mechanical continuum solvation models
,”
Chem. Rev.
105
,
2999
3094
(
2005
).
5.
S.
Miertuš
,
E.
Scrocco
, and
J.
Tomasi
, “
Electrostatic interaction of a solute with a continuum. A direct utilization of AB initio molecular potentials for the prevision of solvent effects
,”
Chem. Phys.
55
,
117
129
(
1981
).
6.
J. B.
Foresman
,
T. A.
Keith
,
K. B.
Wiberg
,
J.
Snoonian
, and
M. J.
Frisch
, “
Solvent effects. 5. Influence of cavity shape, truncation of electrostatics, and electron correlation on ab initio reaction field calculations
,”
J. Phys. Chem.
100
,
16098
16104
(
1996
).
7.
J. L.
Pascual-ahuir
,
E.
Silla
, and
I.
Tuñon
, “
GEPOL: An improved description of molecular surfaces. III. A new algorithm for the computation of a solvent-excluding surface
,”
J. Comput. Chem.
15
(
10
),
1127
1138
(
1994
).
8.
S.
Miertuš
and
J.
Tomasi
, “
Approximate evaluations of the electrostatic free energy and internal energy changes in solution processes
,”
Chem. Phys.
65
,
239
245
(
1982
).
9.
M.
Cossi
,
N.
Rega
,
G.
Scalmani
, and
V.
Barone
, “
Energies, structures, and electronic properties of molecules in solution with the C-PCM solvation model
,”
J. Comput. Chem.
24
(
6
),
669
681
(
2003
).
10.
A.
Klamt
and
G.
Schüürmann
, “
COSMO: A new approach to dielectric screening in solvents with explicit expressions for the screening energy and its gradient
,”
J. Chem. Soc., Perkin Trans. 2
2
,
799
805
(
1993
).
11.
A.
Klamt
, “
Conductor-like screening model for real solvents: A new approach to the quantitative calculation of solvation phenomena
,”
J. Phys. Chem.
99
,
2224
2235
(
1995
).
12.
A. V.
Marenich
,
C. J.
Cramer
, and
D. G.
Truhlar
, “
Universal solvation model based on solute electron density and on a continuum model of the solvent defined by the bulk dielectric constant and atomic surface tensions
,”
J. Phys. Chem. B
113
,
6378
6396
(
2009
).
13.
I.
Soteras
,
C.
Curutchet
,
A.
Bidon-Chanal
,
M.
Orozco
, and
F. J.
Luque
, “
Extension of the MST model to the IEF formalism: HF and B3lyp parametrizations
,”
J. Mol. Struct.: THEOCHEM
727
,
29
40
(
2005
).
14.
N. L.
Doltsinis
and
M.
Sprik
, “
Theoretical pKa estimates for solvated P(OH)5 from coordination constrained Car–Parrinello molecular dynamics
,”
Phys. Chem. Chem. Phys.
5
,
2612
2618
(
2003
).
15.
M.
Schilling
and
S.
Luber
, “
Determination of pKa values via ab initio molecular dynamics and its application to transition metal-based water oxidation catalysts
,”
Inorganics
7
,
73
(
2019
).
16.
Y.-L.
Chen
,
N. L.
Doltsinis
,
R. C.
Hider
, and
D. J.
Barlow
, “
Prediction of absolute hydroxyl pKa values for 3-hydroxypyridin-4-ones
,”
J. Phys. Chem. Lett.
3
,
2980
2985
(
2012
).
17.
J. E.
Davies
,
N. L.
Doltsinis
,
A. J.
Kirby
,
C. D.
Roussev
, and
M.
Sprik
, “
Estimating pKa values for pentaoxyphosphoranes
,”
J. Am. Chem. Soc.
124
,
6594
6599
(
2002
).
18.
N.
Sandmann
,
J.
Bachmann
,
A.
Hepp
,
N. L.
Doltsinis
, and
J.
Müller
, “
Copper(II)-mediated base pairing involving the artificial nucleobase 3h-imidazo[4,5-f]quinolin-5-ol
,”
Dalton Trans.
48
,
10505
10515
(
2019
).
19.
A. K.
Tummanapelli
and
S.
Vasudevan
, “
Dissociation constants of weak acids from ab initio molecular dynamics using metadynamics: Influence of the inductive effect and hydrogen bonding on pKa values
,”
J. Phys. Chem. B
118
,
13651
13657
(
2014
).
20.
A. K.
Tummanapelli
and
S.
Vasudevan
, “
Ab initio MD simulations of the brønsted acidity of glutathione in aqueous solutions: Predicting pKa shifts of the cysteine residue
,”
J. Phys. Chem. B
119
,
15353
15358
(
2015
).
21.
A. K.
Tummanapelli
and
S.
Vasudevan
, “
Ab initio molecular dynamics simulations of amino acids in aqueous solutions: Estimating pKa values from metadynamics sampling
,”
J. Phys. Chem. B
119
,
12249
12255
(
2015
).
22.
C. D.
Daub
and
L.
Halonen
, “
Ab initio molecular dynamics simulations of the influence of lithium bromide salt on the deprotonation of formic acid in aqueous solution
,”
J. Phys. Chem. B
123
,
6823
6829
(
2019
).
23.
L.
Xu
and
M. L.
Coote
, “
Methods to improve the calculations of solvation model density solvation free energies and associated aqueous pKa values: Comparison between choosing an optimal theoretical level, solute cavity scaling, and using explicit solvent molecules
,”
J. Phys. Chem. A
123
,
7430
7438
(
2019
).
24.
C. C. R.
Sutton
,
G. V.
Franks
, and
G.
da Silva
, “
First principles pKa calculations on carboxylic acids using the SMD solvation model: Effect of thermodynamic cycle, model chemistry, and explicit solvent molecules
,”
J. Phys. Chem. B
116
,
11999
12006
(
2012
).
25.
S.
Zhang
, “
A reliable and efficient first principles-based method for predicting pKa values. III. Adding explicit water molecules: Can the theoretical slope be reproduced and pKa values predicted more accurately?
,”
J. Comput. Chem.
33
(
5
),
517
526
(
2012
).
26.
B.
Thapa
and
H. B.
Schlegel
, “
Calculations of pKa’s and redox potentials of nucleobases with explicit waters and polarizable continuum solvation
,”
J. Phys. Chem. A
119
,
5134
5144
(
2015
).
27.
B.
Thapa
and
H. B.
Schlegel
, “
Density functional theory calculation of pKa’s of thiols in aqueous solution using explicit water molecules and the polarizable continuum model
,”
J. Phys. Chem. A
120
,
5726
5735
(
2016
).
28.
N. L.
Haworth
,
Q.
Wang
, and
M. L.
Coote
, “
Modeling flexible molecules in solution: A pKa case study
,”
J. Phys. Chem. A
121
,
5217
5225
(
2017
).
29.
E. A.
Carter
,
G.
Ciccotti
,
J. T.
Hynes
, and
R.
Kapral
, “
Constrained reaction coordinate dynamics for the simulation of rare events
,”
Chem. Phys. Lett.
156
,
472
477
(
1989
).
30.
M.
Sprik
and
G.
Ciccotti
, “
Free energy from constrained molecular dynamics
,”
J. Chem. Phys.
109
,
7737
7744
(
1998
).
31.
A. V.
Bandura
and
S. N.
Lvov
, “
The ionization constant of water over wide ranges of temperature and density
,”
J. Phys. Chem. Ref. Data
35
,
15
30
(
2005
).
32.
N.
Mora-Diez
,
Y.
Egorova
,
H.
Plommer
, and
P. R.
Tremaine
, “
Theoretical study of deuterium isotope effects on acid–base equilibria under ambient and hydrothermal conditions
,”
RSC Adv.
5
,
9097
9109
(
2015
).
33.
D. W.
Shoesmith
and
W.
Lee
, “
The ionization constant of heavy water (D2o) in the temperature range 298 to 523 K
,”
Can. J. Chem.
54
,
3553
3558
(
1976
).
34.
M.
Sprik
, “
Computation of the pK of liquid water using coordination constraints
,”
Chem. Phys.
258
,
139
150
(
2000
).
35.
E.
Perlt
,
M. v.
Domaros
,
B.
Kirchner
,
R.
Ludwig
, and
F.
Weinhold
, “
Predicting the ionic product of water
,”
Sci. Rep.
7
,
1
10
(
2017
).
36.
M.
Štrajbl
,
G.
Hong
, and
A.
Warshel
, “
Ab initio QM/MM simulation with proper sampling: ‘First principle’ calculations of the free energy of the autodissociation of water in aqueous solution
,”
J. Phys. Chem. B
106
,
13333
13343
(
2002
).
37.
R.
Wang
,
V.
Carnevale
,
M. L.
Klein
, and
E.
Borguet
, “
First-principles calculation of water pKa using the newly developed SCAN functional
,”
J. Phys. Chem. Lett.
11
,
54
59
(
2020
).
38.
M.
Moqadam
,
A.
Lervik
,
E.
Riccardi
,
V.
Venkatraman
,
B. K.
Alsberg
, and
T. S.
van Erp
, “
Local initiation conditions for water autoionization
,”
Proc. Natl. Acad. Sci. U. S. A.
115
,
E4569
E4576
(
2018
).
39.
P. L.
Geissler
,
C.
Dellago
,
D.
Chandler
,
J.
Hutter
, and
M.
Parrinello
, “
Autoionization in liquid water
,”
Science
291
,
2121
2124
(
2001
).
40.
B. L.
Trout
and
M.
Parrinello
, “
Analysis of the dissociation of H2o in water using first-principles molecular dynamics
,”
J. Phys. Chem. B
103
,
7340
7345
(
1999
).
41.
M.
Shiga
, “
Path integral simulations
,” in
Reference Module in Chemistry, Molecular Sciences and Chemical Engineering
(
Elsevier
,
2018
).
42.
M.
Parrinello
and
A.
Rahman
, “
Study of an F center in molten KCl
,”
J. Chem. Phys.
80
,
860
867
(
1984
).
43.
R. W.
Hall
and
B. J.
Berne
, “
Nonergodicity in path integral molecular dynamics
,”
J. Chem. Phys.
81
,
3641
3643
(
1984
).
44.
M.
Ceriotti
,
M.
Parrinello
,
T. E.
Markland
, and
D. E.
Manolopoulos
, “
Efficient stochastic thermostatting of path integral molecular dynamics
,”
J. Chem. Phys.
133
,
124104
(
2010
).
45.
M. E.
Tuckerman
,
B. J.
Berne
,
G. J.
Martyna
, and
M. L.
Klein
, “
Efficient molecular dynamics and hybrid Monte Carlo algorithms for path integrals
,”
J. Chem. Phys.
99
,
2796
2808
(
1993
).
46.
M.
Ceriotti
,
W.
Fang
,
P. G.
Kusalik
,
R. H.
McKenzie
,
A.
Michaelides
,
M. A.
Morales
, and
T. E.
Markland
, “
Nuclear quantum effects in water and aqueous systems: Experiment, theory, and current challenges
,”
Chem. Rev.
116
,
7529
7550
(
2016
).
47.
D.
Marx
,
M. E.
Tuckerman
,
J.
Hutter
, and
M.
Parrinello
, “
The nature of the hydrated excess proton in water
,”
Nature
397
,
601
604
(
1999
).
48.
G.
Cassone
, “
Nuclear quantum effects largely influence molecular dissociation and proton transfer in liquid water under an electric field
,”
J. Phys. Chem. Lett.
11
,
8983
8988
(
2020
).
49.
D.
Chandler
,
Introduction to Modern Statistical Mechanics
(
Oxford University Press
,
Oxford
,
1987
).
50.
M.
Shiga
, PIMD,
2020
.
51.
S.
Nosé
, “
A unified formulation of the constant temperature molecular dynamics methods
,”
J. Chem. Phys.
81
,
511
519
(
1984
).
52.
W. G.
Hoover
, “
Canonical dynamics: Equilibrium phase-space distributions
,”
Phys. Rev. A
31
,
1695
1697
(
1985
).
53.
G. J.
Martyna
,
M. L.
Klein
, and
M.
Tuckerman
, “
Nosé–Hoover chains: The canonical ensemble via continuous dynamics
,”
J. Chem. Phys.
97
,
2635
2643
(
1992
).
54.
G.
Kresse
and
J.
Hafner
, “
Ab initio molecular dynamics for liquid metals
,”
Phys. Rev. B
47
,
558
561
(
1993
).
55.
G.
Kresse
and
J.
Hafner
, “
Ab initio molecular-dynamics simulation of the liquid-metal–amorphous-semiconductor transition in germanium
,”
Phys. Rev. B
49
,
14251
14269
(
1994
).
56.
G.
Kresse
and
J.
Furthmüller
, “
Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set
,”
Comput. Mater. Sci.
6
,
15
50
(
1996
).
57.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
, “
Generalized gradient approximation made simple
,”
Phys. Rev. Lett.
77
,
3865
3868
(
1996
).
58.
S.
Grimme
,
J.
Antony
,
S.
Ehrlich
, and
H.
Krieg
, “
A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu
,”
J. Chem. Phys.
132
,
154104
(
2010
).
59.
B.
Hourahine
,
B.
Aradi
,
V.
Blum
,
F.
Bonafé
,
A.
Buccheri
,
C.
Camacho
,
C.
Cevallos
,
M. Y.
Deshaye
,
T.
Dumitrică
,
A.
Dominguez
,
S.
Ehlert
,
M.
Elstner
,
T.
van der Heide
,
J.
Hermann
,
S.
Irle
,
J. J.
Kranz
,
C.
Köhler
,
T.
Kowalczyk
,
T.
Kubař
,
I. S.
Lee
,
V.
Lutsker
,
R. J.
Maurer
,
S. K.
Min
,
I.
Mitchell
,
C.
Negre
,
T. A.
Niehaus
,
A. M. N.
Niklasson
,
A. J.
Page
,
A.
Pecchia
,
G.
Penazzi
,
M. P.
Persson
,
J.
Řezáč
,
C. G.
Sánchez
,
M.
Sternberg
,
M.
Stöhr
,
F.
Stuckenberg
,
A.
Tkatchenko
,
V. W.-z.
Yu
, and
T.
Frauenheim
, “
DFTB+, a software package for efficient approximate density functional theory based atomistic simulations
,”
J. Chem. Phys.
152
,
124101
(
2020
).
60.
M.
Elstner
,
D.
Porezag
,
G.
Jungnickel
,
J.
Elsner
,
M.
Haugk
,
T.
Frauenheim
,
S.
Suhai
, and
G.
Seifert
, “
Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties
,”
Phys. Rev. B
58
,
7260
7268
(
1998
).
61.
M.
Gaus
,
A.
Goez
, and
M.
Elstner
, “
Parametrization and benchmark of DFTB3 for organic molecules
,”
J. Chem. Theory Comput.
9
,
338
354
(
2013
).
62.
L.
Ojamäe
,
I.
Shavitt
, and
S. J.
Singer
, “
Potential models for simulations of the solvated proton in water
,”
J. Chem. Phys.
109
,
5547
5564
(
1998
).
63.
J.
Sala
,
E.
Guàrdia
, and
M.
Masia
, “
The polarizable point dipoles method with electrostatic damping: Implementation on a model system
,”
J. Chem. Phys.
133
,
234101
(
2010
).
64.
N.
Michaud-Agrawal
,
E. J.
Denning
,
T. B.
Woolf
, and
O.
Beckstein
, “
MDAnalysis: A toolkit for the analysis of molecular dynamics simulations
,”
J. Comput. Chem.
32
(
10
),
2319
2327
(
2011
).
65.
R. J.
Gowers
,
M.
Linke
,
J.
Barnoud
,
T. J. E.
Reddy
,
M. N.
Melo
,
S. L.
Seyler
,
J.
Domański
,
D. L.
Dotson
,
S.
Buchoux
,
I. M.
Kenney
, and
O.
Beckstein
, “
MDAnalysis: A Python package for the rapid analysis of molecular dynamics simulations
,” in
Proceedings of the 15th Python in Science Conference
(
SciPy
,
2016
), pp.
98
105
.
66.
H.
Flyvbjerg
and
H. G.
Petersen
, “
Error estimates on averages of correlated data
,”
J. Chem. Phys.
91
,
461
466
(
1989
).
67.
W.
Humphrey
,
A.
Dalke
, and
K.
Schulten
, “
VMD: Visual molecular dynamics
,”
J. Mol. Graphics
14
,
33
38
(
1996
).
68.
M.
Shiga
and
H.
Fujisaki
, “
A quantum generalization of intrinsic reaction coordinate using path integral centroid coordinates
,”
J. Chem. Phys.
136
,
184103
(
2012
).
69.
M.
Machida
,
K.
Kato
, and
M.
Shiga
, “
Nuclear quantum effects of light and heavy water studied by all-electron first principles path integral simulations
,”
J. Chem. Phys.
148
,
102324
(
2017
).
70.
T. P.
Silverstein
and
S. T.
Heller
, “
pKa values in the undergraduate curriculum: What is the real pKa of water?
,”
J. Chem. Educ.
94
,
690
695
(
2017
).
71.
E. C.
Meister
,
M.
Willeke
,
W.
Angst
,
A.
Togni
, and
P.
Walde
, “
Confusing quantitative descriptions of brønsted-lowry acid-base equilibria in chemistry textbooks – A critical review and clarifications for chemical educators
,”
Helv. Chim. Acta
97
(
1
),
1
31
(
2014
).

Supplementary Material

You do not currently have access to this content.