Lattice-switch Monte Carlo and the related diabat methods have emerged as efficient and accurate ways to compute free energy differences between polymorphs. In this work, we introduce a one-to-one mapping from the reference positions and displacements in one molecular crystal to the positions and displacements in another. Two features of the mapping facilitate lattice-switch Monte Carlo and related diabat methods for computing polymorph free energy differences. First, the mapping is unitary so that its Jacobian does not complicate the free energy calculations. Second, the mapping is easily implemented for molecular crystals of arbitrary complexity. We demonstrate the mapping by computing free energy differences between polymorphs of benzene and carbamazepine. Free energy calculations for thermodynamic cycles, each involving three independently computed polymorph free energy differences, all return to the starting free energy with a high degree of precision. The calculations thus provide a force field independent validation of the method and allow us to estimate the precision of the individual free energy differences.
REFERENCES
1.
W. E.
Addison
, The Allotropy of the Elements
(American Elsevier Publishing Company
, 1964
).2.
A.
Putnis
, An Introduction to Mineral Sciences
(Cambridge University Press
, 1992
).3.
J.
Bernstein
, Polymorphism in Molecular Crystals
(Oxford University Press
, 2002
), Vol. 14.4.
W.
Yan
, B.
Chen
, S. M.
Mahurin
, V.
Schwartz
, D. R.
Mullins
, A. R.
Lupini
, S. J.
Pennycook
, S.
Dai
, and S. H.
Overbury
, “Preparation and comparison of supported gold nanocatalysts on anatase, brookite, rutile, and P25 polymorphs of TiO2 for catalytic oxidation of CO
,” J. Phys. Chem. B
109
(21
), 10676
–10685
(2005
).5.
D. M.
Robinson
, Y. B.
Go
, M.
Mui
, G.
Gardner
, Z.
Zhang
, D.
Mastrogiovanni
, E.
Garfunkel
, J.
Li
, M.
Greenblatt
, and G. C.
Dismukes
, “Photochemical water oxidation by crystalline polymorphs of manganese oxides: Structural requirements for catalysis
,” J. Am. Chem. Soc.
135
(9
), 3494
–3501
(2013
).6.
W. L.
Kwong
, C. C.
Lee
, A.
Shchukarev
, E.
Björn
, and J.
Messinger
, “High-performance iron (III) oxide electrocatalyst for water oxidation in strongly acidic media
,” J. Catal.
365
, 29
–35
(2018
).7.
B.
Zheng
, W.
Hua
, Y.
Yue
, and Z.
Gao
, “Dehydrogenation of propane to propene over different polymorphs of gallium oxide
,” J. Catal.
232
(1
), 143
–151
(2005
).8.
X.
Chen
, Y.
Luo
, J.
Zhang
, K.
Jiang
, J. B.
Pendry
, and S.
Zhang
, “Macroscopic invisibility cloaking of visible light
,” Nat. Commun.
2
(1
), 176
(2011
).9.
T.
Wonglakhon
and D.
Zahn
, “Interaction potentials for modelling GaN precipitation and solid state polymorphism
,” J. Phys.: Condens. Matter
32
(20
), 205401
(2020
).10.
K.
Wang
, H.
Zhang
, S.
Chen
, G.
Yang
, J.
Zhang
, W.
Tian
, Z.
Su
, and Y.
Wang
, “Organic polymorphs: One-compound-based crystals with molecular-conformation- and packing-dependent luminescent properties
,” Adv. Mater.
26
(35
), 6168
–6173
(2014
).11.
T.
Beyer
, G. M.
Day
, and S. L.
Price
, “The prediction, morphology, and mechanical properties of the polymorphs of paracetamol
,” J. Am. Chem. Soc.
123
(21
), 5086
–5094
(2001
).12.
C.
Sun
and D. J.
Grant
, “Influence of crystal structure on the tableting properties of sulfamerazine polymorphs
,” Pharm. Res.
18
(3
), 274
–280
(2001
).13.
R.
Censi
and P.
Di Martino
, “Polymorph impact on the bioavailability and stability of poorly soluble drugs
,” Molecules
20
(10
), 18759
–18776
(2015
).14.
C. A.
Lipinski
, F.
Lombardo
, B. W.
Dominy
, and P. J.
Feeney
, “Experimental and computational approaches to estimate solubility and permeability in drug discovery and development settings
,” Adv. Drug Delivery Rev.
23
(1
), 3
–25
(1997
).15.
J.
Aaltonen
, M.
Alleso
, S.
Mirza
, V.
Koradia
, K.
Gordon
, and J.
Rantanen
, “Solid form screening—A review
,” Eur. J. Pharm. Biopharm.
71
(1
), 23
–37
(2009
).16.
G. R.
Desiraju
, “Crystal engineering: A brief overview
,” J. Chem. Sci.
122
(5
), 667
–675
(2010
).17.
S.
Price
, “The computational prediction of pharmaceutical crystal structures and polymorphism
,” Adv. Drug Delivery Rev.
56
(3
), 301
–319
(2004
).18.
S. L.
Price
, D. E.
Braun
, and S. M.
Reutzel-Edens
, “Can computed crystal energy landscapes help understand pharmaceutical solids?
,” Chem. Commun.
52
(44
), 7065
–7077
(2016
).19.
J.
Kendrick
, F. J. J.
Leusen
, M. A.
Neumann
, and J.
van de Streek
, “Progress in crystal structure prediction
,” Chem. Eur. J.
17
(38
), 10736
–10744
(2011
).20.
G. M.
Day
, “Current approaches to predicting molecular organic crystal structures
,” Crystallogr. Rev.
17
(1
), 3
–52
(2011
).21.
C. C.
Pantelides
, C. S.
Adjiman
, and A. V.
Kazantsev
, “General computational algorithms for ab initio crystal structure prediction for organic molecules
, in Prediction and Calculation of Crystal Structures
(Springer
, 2014
), pp. 25
–58
.22.
G. J. O.
Beran
, “Modeling polymorphic molecular crystals with electronic structure theory
,” Chem. Rev.
116
(9
), 5567
–5613
(2016
).23.
T.-Q.
Yu
and M. E.
Tuckerman
, “Temperature-accelerated method for exploring polymorphism in molecular crystals based on free energy
,” Phys. Rev. Lett.
107
(1
), 015701
(2011
).24.
B.
Peters
, “Competing nucleation pathways in a mixture of oppositely charged colloids: Out-of-equilibrium nucleation revisited
,” J. Chem. Phys.
131
(24
), 244103
(2009
).25.
N.
Duff
, Y. R.
Dahal
, J. D.
Schmit
, and B.
Peters
, “Salting out the polar polymorph: Analysis by alchemical solvent transformation
,” J. Chem. Phys.
140
(1
), 014501
(2014
).26.
M.
Salvalaglio
, C.
Perego
, F.
Giberti
, M.
Mazzotti
, and M.
Parrinello
, “Molecular-dynamics simulations of urea nucleation from aqueous solution
,” Proc. Natl. Acad. Sci. U. S. A.
112
(1
), E6
–E14
(2015
).27.
G. C.
Sosso
, J.
Chen
, S. J.
Cox
, M.
Fitzner
, P.
Pedevilla
, A.
Zen
, and A.
Michaelides
, “Crystal nucleation in liquids: Open questions and future challenges in molecular dynamics simulations
,” Chem. Rev.
116
(12
), 7078
–7116
(2016
).28.
M. N.
Joswiak
, B.
Peters
, and M. F.
Doherty
, “In silico crystal growth rate prediction for NaCl from aqueous solution
,” Cryst. Growth Des.
18
(10
), 6302
–6306
(2018
).29.
D.
Han
, T.
Karmakar
, Z.
Bjelobrk
, J.
Gong
, and M.
Parrinello
, “Solvent-mediated morphology selection of the active pharmaceutical ingredient isoniazid: Experimental and simulation studies
,” Chem. Eng. Sci.
204
, 320
–328
(2019
).30.
M.
Salvalaglio
, T.
Vetter
, M.
Mazzotti
, and M.
Parrinello
, “Controlling and predicting crystal shapes: The case of urea
,” Angew. Chem., Int. Ed.
52
(50
), 13369
–13372
(2013
).31.
A. J.
Cruz-Cabeza
, S. M.
Reutzel-Edens
, and J.
Bernstein
, “Facts and fictions about polymorphism
,” Chem. Soc. Rev.
44
(23
), 8619
–8635
(2015
).32.
J.
Nyman
and G. M.
Day
, “Static and lattice vibrational energy differences between polymorphs
,” CrystEngComm
17
(28
), 5154
–5165
(2015
).33.
S. R.
Whittleton
, A.
Otero-de-la-Roza
, and E. R.
Johnson
, “Exchange-hole dipole dispersion model for accurate energy ranking in molecular crystal structure prediction
,” J. Chem. Theory Comput.
13
(2
), 441
–450
(2017
).34.
J. G.
Brandenburg
, T.
Maas
, and S.
Grimme
, “Benchmarking DFT and semiempirical methods on structures and lattice energies for ten ice polymorphs
,” J. Chem. Phys.
142
(12
), 124104
(2015
).35.
K.
Hongo
, M. A.
Watson
, R. S.
Sánchez-Carrera
, T.
Iitaka
, and A.
Aspuru-Guzik
, “Failure of conventional density functionals for the prediction of molecular crystal polymorphism: A quantum Monte Carlo study
,” J. Phys. Chem. Lett.
1
(12
), 1789
–1794
(2010
).36.
J.
Yang
, W.
Hu
, D.
Usvyat
, D.
Matthews
, M.
Schütz
, and G. K.-L.
Chan
, “Ab initio determination of the crystalline benzene lattice energy to sub-kilojoule/mole accuracy
,” Science
345
(6197
), 640
–643
(2014
).37.
G. J. O.
Beran
, “A new era for ab initio molecular crystal lattice energy prediction
,” Angew. Chem., Int. Ed.
54
(2
), 396
–398
(2015
).38.
J.
Hoja
, H.-Y.
Ko
, M. A.
Neumann
, R.
Car
, R. A.
DiStasio
, and A.
Tkatchenko
, “Reliable and practical computational description of molecular crystal polymorphs
,” Sci. Adv.
5
(1
), eaau3338
(2019
).39.
A. M.
Reilly
and A.
Tkatchenko
, “Role of dispersion interactions in the polymorphism and entropic stabilization of the aspirin crystal
,” Phys. Rev. Lett.
113
(5
), 055701
(2014
).40.
J. G.
Brandenburg
, J.
Potticary
, H. A.
Sparkes
, S. L.
Price
, and S. R.
Hall
, “Thermal expansion of carbamazepine: Systematic crystallographic measurements challenge quantum chemical calculations
,” J. Phys. Chem. Lett.
8
(17
), 4319
–4324
(2017
).41.
J. L.
McKinley
and G. J. O.
Beran
, “Identifying pragmatic quasi-harmonic electronic structure approaches for modeling molecular crystal thermal expansion
,” Faraday Discuss.
211
, 181
–207
(2018
).42.
N. S.
Abraham
and M. R.
Shirts
, “Statistical mechanical approximations to more efficiently determine polymorph free energy differences for small organic molecules
,” J. Chem. Theory Comput.
16
(10
), 6503
–6512
(2020
).43.
D.
Frenkel
and B.
Smit
, Understanding Molecular Simulation: From Algorithms to Applications
(Elsevier
, 2001
).44.
I. J.
Nessler
, J. M.
Litman
, and M. J.
Schnieders
, “Toward polarizable AMOEBA thermodynamics at fixed charge efficiency using a dual force field approach: Application to organic crystals
,” Phys. Chem. Chem. Phys.
18
(44
), 30313
–30322
(2016
).45.
E.
Schneider
, L.
Vogt
, and M. E.
Tuckerman
, “Exploring polymorphism of benzene and naphthalene with free energy based enhanced molecular dynamics
,” Acta Crystallogr., Sect. B: Struct. Sci., Cryst. Eng. Mater.
72
(4
), 542
–550
(2016
).46.
E. C.
Dybeck
, N. S.
Abraham
, N. P.
Schieber
, and M. R.
Shirts
, “Capturing entropic contributions to temperature-mediated polymorphic transformations through molecular modeling
,” Cryst. Growth Des.
17
(4
), 1775
–1787
(2017
).47.
D.
Frenkel
and A. J. C.
Ladd
, “New Monte Carlo method to compute the free energy of arbitrary solids. Application to the FCC and HCP phases of hard spheres
,” J. Chem. Phys.
81
(7
), 3188
–3193
(1984
).48.
C.
Vega
and E. G.
Noya
, “Revisiting the Frenkel–Ladd method to compute the free energy of solids: The Einstein molecule approach
,” J. Chem. Phys.
127
(15
), 154113
(2007
).49.
M.
Yang
, E.
Dybeck
, G.
Sun
, C.
Peng
, B.
Samas
, V. M.
Burger
, Q.
Zeng
, Y.
Jin
, M. A.
Bellucci
, Y.
Liu
et al., “Prediction of the relative free energies of drug polymorphs above zero Kelvin
,” Cryst. Growth Des.
20
(8
), 5211
–5224
(2020
).50.
R.
Martoňák
, D.
Donadio
, A. R.
Oganov
, and M.
Parrinello
, “Crystal structure transformations in SiO2 from classical and ab initio metadynamics
,” Nat. Mater.
5
(8
), 623
–626
(2006
).51.
P.
Raiteri
, R.
Martoňák
, and M.
Parrinello
, “Exploring polymorphism: The case of benzene
,” Angew. Chem., Int. Ed.
44
(24
), 3769
–3773
(2005
).52.
A. D.
Bruce
, N. B.
Wilding
, and G. J.
Ackland
, “Free energy of crystalline solids: A lattice-switch Monte Carlo method
,” Phys. Rev. Lett.
79
(16
), 3002
(1997
).53.
A. D.
Bruce
, A. N.
Jackson
, G. J.
Ackland
, and N. B.
Wilding
, “Lattice-switch Monte Carlo method
,” Phys. Rev. E
61
(1
), 906
(2000
).54.
K.
Kamat
and B.
Peters
, “Diabat interpolation for polymorph free-energy differences
,” J. Phys. Chem. Lett.
8
(3
), 655
–660
(2017
).55.
K.
Kamat
and B.
Peters
, “Gibbs free-energy differences between polymorphs via a diabat approach
,” J. Chem. Phys.
149
(21
), 214106
(2018
).56.
R. W.
Zwanzig
, “High-temperature equation of state by a perturbation method. I. Nonpolar gases
,” J. Chem. Phys.
22
(8
), 1420
–1426
(1954
).57.
C. H.
Bennett
, “Efficient estimation of free energy differences from Monte Carlo data
,” J. Comput. Phys.
22
(2
), 245
–268
(1976
).58.
A. N.
Jackson
, A. D.
Bruce
, and G. J.
Ackland
, “Lattice-switch Monte Carlo method: Application to soft potentials
,” Phys. Rev. E
65
(3
), 036710
(2002
).59.
T. L.
Underwood
and G. J.
Ackland
, “Monteswitch: A package for evaluating solid–solid free energy differences via lattice-switch Monte Carlo
,” Comput. Phys. Commun.
215
, 204
–222
(2017
).60.
S.
Bridgwater
and D.
Quigley
, “Lattice-switching Monte Carlo method for crystals of flexible molecules
,” Phys. Rev. E
90
(6
), 063313
(2014
).61.
S.
Bridgwater
, “Accurate free energy methods for model organic solids
,” Ph.D. thesis, University of Warwick
, 2014
.62.
N.
Metropolis
, A. W.
Rosenbluth
, M. N.
Rosenbluth
, A. H.
Teller
, and E.
Teller
, “Equation of state calculations by fast computing machines
,” J. Chem. Phys.
21
(6
), 1087
–1092
(1953
).63.
G. R.
Smith
and A. D.
Bruce
, “A study of the multi-canonical Monte Carlo method
,” J. Phys. Math. Gen.
28
(23
), 6623
(1995
).64.
M.
Fitzgerald
, R. R.
Picard
, and R. N.
Silver
, “Canonical transition probabilities for adaptive metropolis simulation
,” Europhys. Lett.
46
(3
), 282
(1999
).65.
J. K.
Hwang
and A.
Warshel
, “Microscopic examination of free-energy relationships for electron transfer in polar solvents
,” J. Am. Chem. Soc.
109
(3
), 715
–720
(1987
).66.
J.
Blumberger
, I.
Tavernelli
, M. L.
Klein
, and M.
Sprik
, “Diabatic free energy curves and coordination fluctuations for the aqueous Ag+/Ag2+ redox couple: A biased Born–Oppenheimer molecular dynamics investigation
,” J. Chem. Phys.
124
(6
), 064507
(2006
).67.
M.
Tachiya
, “Relation between the electron-transfer rate and the free energy change of reaction
,” J. Phys. Chem.
93
(20
), 7050
–7052
(1989
).68.
D. W.
Small
, D. V.
Matyushov
, and G. A.
Voth
, “The theory of electron transfer reactions: What may Be missing?
,” J. Am. Chem. Soc.
125
(24
), 7470
–7478
(2003
).69.
Y.
Tateyama
, J.
Blumberger
, M.
Sprik
, and I.
Tavernelli
, “Density-functional molecular-dynamics study of the redox reactions of two anionic, aqueous transition-metal complexes
,” J. Chem. Phys.
122
(23
), 234505
(2005
).70.
M. M.
Thiéry
and J. M.
Léger
, “High pressure solid phases of benzene. I. Raman and X-ray studies of C6H6 at 294 K up to 25 GPa
,” J. Chem. Phys.
89
(7
), 4255
–4271
(1988
).71.
T.
Shoda
, K.
Yamahara
, K.
Okazaki
, and D. E.
Williams
, “Molecular packing analysis of benzene crystals. Part 2. Prediction of experimental crystal structure polymorphs at low and high pressure
,” J. Mol. Struct.: THEOCHEM
333
(3
), 267
–274
(1995
).72.
F.
Cansell
, D.
Fabre
, and J. P.
Petitet
, “Phase transitions and chemical transformations of benzene up to 550 °C and 30 GPa
,” J. Chem. Phys.
99
(10
), 7300
–7304
(1993
).73.
W. L.
Jorgensen
and J.
Tirado-Rives
, “The OPLS (optimized potentials for liquid simulations) potential functions for proteins, energy minimizations for crystals of cyclic peptides and crambin
,” J. Am. Chem. Soc.
110
(6
), 1657
–1666
(1988
).74.
L. S.
Dodda
, I.
Cabeza de Vaca
, J.
Tirado-Rives
, and W. L.
Jorgensen
, “LigParGen web server: An automatic OPLS-AA parameter generator for organic ligands
,” Nucleic Acids Res.
45
(W1
), W331
–W336
(2017
).75.
A.
Budzianowski
and A.
Katrusiak
, “Pressure-frozen benzene I revisited
,” Acta Crystallogr., Sect. B: Struct. Sci.
62
(1
), 94
–101
(2006
).76.
G. J.
Piermarini
, A. D.
Mighell
, C. E.
Weir
, and S.
Block
, “Crystal structure of benzene II at 25 kilobars
,” Science
165
(3899
), 1250
–1255
(1969
).77.
C. F.
Macrae
, I. J.
Bruno
, J. A.
Chisholm
, P. R.
Edgington
, P.
McCabe
, E.
Pidcock
, L.
Rodriguez-Monge
, R.
Taylor
, J.
van de Streek
, and P. A.
Wood
, “Mercury CSD 2.0—New features for the visualization and investigation of crystal structures
,” J. Appl. Crystallogr.
41
(2
), 466
–470
(2008
).78.
S. J.
Leon
, I.
Bica
, and T.
Hohn
, Linear Algebra with Applications
(Pearson Prentice Hall Upper
, Saddle River, NJ, USA
, 2006
).79.
S.
Kumar
, J. M.
Rosenberg
, D.
Bouzida
, R. H.
Swendsen
, and P. A.
Kollman
, “The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method
,” J. Comput. Chem.
13
(8
), 1011
–1021
(1992
).80.
B.
Roux
, “The calculation of the potential of mean force using computer simulations
,” Comput. Phys. Commun.
91
(1–3
), 275
–282
(1995
).81.
A.
Grossfield
, “WHAM: The weighted histogram analysis method
,” Version
2
(9
), 06
(2012
).82.
F. H.
Allen
, “The Cambridge structural Database: A quarter of a million crystal structures and rising
,” Acta Crystallogr., Sect. B: Struct. Sci.
58
(3
), 380
–388
(2002
).83.
C. R.
Groom
and F. H.
Allen
, “The Cambridge structural database in retrospect and prospect
,” Angew. Chem., Int. Ed.
53
(3
), 662
–671
(2014
).84.
L.
D’Ascenzo
and P.
Auffinger
, “A comprehensive classification and nomenclature of carboxyl–carboxyl(Ate) supramolecular motifs and related catemers: Implications for biomolecular systems
,” Acta Crystallogr. Sect. B Struct. Sci. Cryst. Eng. Mater.
71
(Pt 2
), 164
–175
(2015
).85.
T.
Beyer
and S. L.
Price
, “Dimer or catemer? Low-energy crystal packings for small carboxylic acids
,” J. Phys. Chem. B
104
(12
), 2647
–2655
(2000
).86.
J.-B.
Arlin
, L. S.
Price
, S. L.
Price
, and A. J.
Florence
, “A strategy for producing predicted polymorphs: Catemeric carbamazepine form V
,” Chem. Commun.
47
(25
), 7074
–7076
(2011
).87.
A. L.
Grzesiak
, M.
Lang
, K.
Kim
, and A. J.
Matzger
, “Comparison of the four anhydrous polymorphs of carbamazepine and the crystal structure of form I
,” J. Pharm. Sci.
92
(11
), 2260
–2271
(2003
).88.
A. J.
Florence
, A.
Johnston
, S. L.
Price
, H.
Nowell
, A. R.
Kennedy
, and N.
Shankland
, “An automated parallel crystallisation search for predicted crystal structures and packing motifs of carbamazepine
,” J. Pharm. Sci.
95
(9
), 1918
–1930
(2006
).89.
K.
Park
, J. M. B.
Evans
, and A. S.
Myerson
, “Determination of solubility of polymorphs using differential scanning calorimetry
,” Cryst. Growth Des.
3
(6
), 991
–995
(2003
).90.
J.
Wang
, R. M.
Wolf
, J. W.
Caldwell
, P. A.
Kollman
, and D. A.
Case
, “Development and testing of a general amber force field
,” J. Comput. Chem.
25
(9
), 1157
–1174
(2004
).91.
J.
Wang
, W.
Wang
, P. A.
Kollman
, and D. A.
Case
, “Automatic atom type and bond type perception in molecular mechanical calculations
,” J. Mol. Graphics Modell.
25
(2
), 247
–260
(2006
).92.
J.
Nyman
and G. M.
Day
, “Modelling temperature-dependent properties of polymorphic organic molecular crystals
,” Phys. Chem. Chem. Phys.
18
(45
), 31132
–31143
(2016
).93.
S. L.
Price
, M.
Leslie
, G. W. A.
Welch
, M.
Habgood
, L. S.
Price
, P. G.
Karamertzanis
, and G. M.
Day
, “Modelling organic crystal structures using distributed multipole and polarizability-based model intermolecular potentials
,” Phys. Chem. Chem. Phys.
12
(30
), 8478
–8490
(2010
).94.
S. J.
Clark
, M. D.
Segall
, C. J.
Pickard
, P. J.
Hasnip
, M. I.
Probert
, K.
Refson
, and M. C.
Payne
, “First principles methods using CASTEP
,” Z. Kristallogr.-Cryst. Mater.
220
(5–6
), 567
–570
(2005
).95.
A.
Tkatchenko
and M.
Scheffler
, “Accurate molecular van der Waals interactions from ground-state electron density and free-atom reference data
,” Phys. Rev. Lett.
102
(7
), 073005
(2009
).96.
A.
Ambrosetti
, A. M.
Reilly
, R. A.
DiStasio
, Jr., and A.
Tkatchenko
, “Long-range correlation energy calculated from coupled atomic response functions
,” J. Chem. Phys.
140
(18
), 18A508
(2014
).97.
R. J.
Behme
and D.
Brooke
, “Heat of fusion measurement of a low melting polymorph of carbamazepine that undergoes multiple-phase changes during differential scanning calorimetry analysis
,” J. Pharm. Sci.
80
(10
), 986
–990
(1991
).98.
E. C.
Dybeck
, D. P.
McMahon
, G. M.
Day
, and M. R.
Shirts
, “Exploring the multi-minima behavior of small molecule crystal polymorphs at finite temperature
,” Cryst. Growth Des.
19
(10
), 5568
–5580
(2019
).99.
N. F.
Francia
, L. S.
Price
, J.
Nyman
, S. L.
Price
, and M.
Salvalaglio
, “Systematic finite-temperature reduction of crystal energy landscapes
,” Cryst. Growth Des.
20
(10
), 6847
–6862
(2020
).100.
S. L.
Price
, “Is zeroth order crystal structure prediction (CSP_0) coming to maturity? What should we aim for in an ideal crystal structure prediction code?
,” Faraday Discuss.
211
, 9
–30
(2018
).101.
L. R.
Pestana
, N.
Mardirossian
, M.
Head-Gordon
, and T.
Head-Gordon
, “Ab initio molecular dynamics simulations of liquid water using high quality meta-GGA functionals
,” Chem. Sci.
8
(5
), 3554
–3565
(2017
).102.
T.
Cheng
, H.
Xiao
, and W. A.
Goddard
, “Full atomistic reaction mechanism with kinetics for CO reduction on Cu(100) from ab initio molecular dynamics free-energy calculations at 298 K
,” Proc. Natl. Acad. Sci. U. S. A.
114
(8
), 1795
–1800
(2017
).103.
M.
Ceriotti
, J.
More
, and D. E.
Manolopoulos
, “A Python interface for ab initio path integral molecular dynamics simulations
,” Comput. Phys. Commun.
185
(3
), 1019
–1026
(2014
).104.
M.
Parrinello
and A.
Rahman
, “Study of an F center in molten KCl
,” J. Chem. Phys.
80
(2
), 860
–867
(1984
).105.
O.
Marsalek
and T. E.
Markland
, “Ab initio molecular dynamics with nuclear quantum effects at classical cost: Ring polymer contraction for density functional theory
,” J. Chem. Phys.
144
(5
), 054112
(2016
).106.
V.
Kapil
, J.
VandeVondele
, and M.
Ceriotti
, “Accurate molecular dynamics and nuclear quantum effects at low cost by multiple steps in real and imaginary time: Using density functional theory to accelerate wavefunction methods
,” J. Chem. Phys.
144
(5
), 054111
(2016
).107.
T. E.
Markland
and M.
Ceriotti
, “Nuclear quantum effects enter the mainstream
,” Nat. Rev. Chem.
2
(3
), 0109
(2018
).© 2020 Author(s).
2020
Author(s)
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