Pressure plays essential roles in chemistry by altering structures and controlling chemical reactions. The extreme-pressure polarizable continuum model (XP-PCM) is an emerging method with an efficient quantum mechanical description of small- and medium-sized molecules at high pressure (on the order of GPa). However, its application to large molecular systems was previously hampered by a CPU computation bottleneck: the Pauli repulsion potential unique to XP-PCM requires the evaluation of a large number of electric field integrals, resulting in significant computational overhead compared to the gas-phase or standard-pressure polarizable continuum model calculations. Here, we exploit advances in graphical processing units (GPUs) to accelerate the XP-PCM-integral evaluations. This enables high-pressure quantum chemistry simulation of proteins that used to be computationally intractable. We benchmarked the performance using 18 small proteins in aqueous solutions. Using a single GPU, our method evaluates the XP-PCM free energy of a protein with over 500 atoms and 4000 basis functions within half an hour. The time taken by the XP-PCM-integral evaluation is typically 1% of the time taken for a gas-phase density functional theory (DFT) on the same system. The overall XP-PCM calculations require less computational effort than that for their gas-phase counterpart due to the improved convergence of self-consistent field iterations. Therefore, the description of the high-pressure effects with our GPU-accelerated XP-PCM is feasible for any molecule tractable for gas-phase DFT calculation. We have also validated the accuracy of our method on small molecules whose properties under high pressure are known from experiments or previous theoretical studies.

1.
R. J.
Hemley
, “
Effects of high pressure on molecules
,”
Annu. Rev. Phys. Chem.
51
,
763
(
2000
).
2.
O.
Mishima
and
H. E.
Stanley
, “
The relationship between liquid, supercooled and glassy water
,”
Nature
396
,
329
(
1998
).
3.
Y.
Fujii
,
K.
Hase
,
N.
Hamaya
,
Y.
Ohishi
,
A.
Onodera
,
O.
Shimomura
, and
K.
Takemura
, “
Pressure-induced face-centered-cubic phase of monatomic metallic iodine
,”
Phys. Rev. Lett.
58
,
796
(
1987
).
4.
L.
Pauling
,
The Nature of the Chemical Bond
(
Cornell University Press
,
Ithaca, NY
,
1960
), Vol. 260.
5.
S.
Duwal
,
Y.-J.
Ryu
,
M.
Kim
,
C.-S.
Yoo
,
S.
Bang
,
K.
Kim
, and
N. H.
Hur
, “
Transformation of hydrazinium azide to molecular N8 at 40 GPa
,”
J. Chem. Phys.
148
,
134310
(
2018
).
6.
T.
Razzaq
and
C. O.
Kappe
, “
Continuous flow organic synthesis under high-temperature/pressure conditions
,”
Chem.-Asian J.
5
,
1274
(
2010
).
7.
J. V.
Badding
, “
High-pressure synthesis, characterization, and tuning of solid state materials
,”
Annu. Rev. Mater. Sci.
28
,
631
(
1998
).
8.
P.
Stoltze
and
J. K.
Nørskov
, “
Bridging the ‘pressure gap’ between ultrahigh-vacuum surface physics and high-pressure catalysis
,”
Phys. Rev. Lett.
55
,
2502
(
1985
).
9.
M.
Tsuda
and
T. G.
Ebrey
, “
Effect of high pressure on the absorption spectrum and isomeric composition of bacteriorhodopsin
,”
Biophys. J.
30
,
149
(
1980
).
10.
J.
Wang
,
A.
Li
,
S.
Xu
,
B.
Li
,
C.
Song
,
Y.
Geng
,
N.
Chu
,
J.
He
, and
W.
Xu
, “
Tunable luminescence of a novel organic co-crystal based on intermolecular charge transfer under pressure
,”
J. Mater. Chem. C
6
,
8958
(
2018
).
11.
S.
Neumaier
,
M.
Büttner
,
A.
Bachmann
, and
T.
Kiefhaber
, “
Transition state and ground state properties of the helix–coil transition in peptides deduced from high-pressure studies
,”
Proc. Natl. Acad. Sci. U. S. A.
110
,
20988
(
2013
).
12.
H.
Imamura
and
M.
Kato
, “
Effect of pressure on helix-coil transition of an alanine-based peptide: An FTIR study
,”
Mol. Biol. Intell. Unit
75
,
911
(
2009
).
13.
T.
Takekiyo
,
A.
Shimizu
,
M.
Kato
, and
Y.
Taniguchi
, “
Pressure-tuning FT-IR spectroscopic study on the helix–coil transition of Ala-rich oligopeptide in aqueous solution
,”
Biochim. Biophys. Acta, Proteins Proteomics
1750
,
1
(
2005
).
14.
Y.-J.
Kuan
,
S. B.
Charnley
,
H.-C.
Huang
,
W.-L.
Tseng
, and
Z.
Kisiel
, “
Interstellar glycine
,”
Astrophys. J.
593
,
848
(
2003
).
15.
K.
Hadraoui
,
H.
Cottin
,
S. L.
Ivanovski
,
P.
Zapf
,
K.
Altwegg
,
Y.
Benilan
,
N.
Biver
,
V.
Della Corte
,
N.
Fray
,
J.
Lasue
 et al, “
Distributed glycine in comet 67P/Churyumov-Gerasimenko
,”
J. Phys.: Conf. Ser.
630
,
A32
(
2019
).
16.
C. F.
Richardson
and
N. W.
Ashcroft
, “
High temperature superconductivity in metallic hydrogen: Electron-electron enhancements
,”
Phys. Rev. Lett.
78
,
118
(
1997
).
17.
R. P.
Dias
and
I. F.
Silvera
, “
Observation of the Wigner–Huntington transition to metallic hydrogen
,”
Science
355
,
715
(
2017
).
18.
P. M.
Celliers
,
M.
Millot
,
S.
Brygoo
,
R. S.
McWilliams
,
D. E.
Fratanduono
,
J. R.
Rygg
,
A. F.
Goncharov
,
P.
Loubeyre
,
J. H.
Eggert
, and
J. L.
Peterson
, “
Insulator-metal transition in dense fluid deuterium
,”
Science
361
,
677
(
2018
).
19.
J.
Bajorath
, “
Integration of virtual and high-throughput screening
,”
Nat. Rev. Drug Discovery
1
,
882
(
2002
).
20.
J. E.
Jaffe
and
A. C.
Hess
, “
Hartree-Fock study of phase changes in ZnO at high pressure
,”
Phys. Rev. B
48
,
7903
(
1993
).
21.
B.
Moses Abraham
,
J.
Prathap Kumar
, and
G.
Vaitheeswaran
, “
High-pressure studies of hydrogen-bonded energetic material 3,6-dihydrazino-s-tetrazine using DFT
,”
ACS Omega
3
,
9388
(
2018
).
22.
W.
Zhu
,
X.
Zhang
,
T.
Wei
, and
H.
Xiao
, “
DFT studies of pressure effects on structural and vibrational properties of crystalline octahydro-1,3,5,7-tetranitro-1,3,5,7-tetrazocine
,”
Theor. Chem. Acc.
124
,
179
(
2009
).
23.
Q.
Guo
,
Y.
Zhao
,
C.
Jiang
,
W. L.
Mao
, and
Z.
Wang
, “
Phase transformation in Sm2O3 at high pressure: In situ synchrotron X-ray diffraction study and ab initio DFT calculation
,”
Solid State Commun.
145
,
250
(
2008
).
24.
R. B.
Best
,
C.
Miller
, and
J.
Mittal
, “
Role of solvation in pressure-induced helix stabilization
,”
J. Chem. Phys.
141
,
22D522
(
2014
).
25.
D.
Paschek
,
S.
Gnanakaran
, and
A. E.
Garcia
, “
Simulations of the pressure and temperature unfolding of an α-helical peptide
,”
Proc. Natl. Acad. Sci. U. S. A.
102
,
6765
(
2005
).
26.
H. W.
Hatch
,
F. H.
Stillinger
, and
P. G.
Debenedetti
, “
Computational study of the stability of the miniprotein trp-cage, the GB1 β-hairpin, and the AK16 peptide, under negative pressure
,”
J. Phys. Chem. B
118
,
7761
(
2014
).
27.
Y.
Mori
and
H.
Okumura
, “
Molecular dynamics of the structural changes of helical peptides induced by pressure
,”
Mol. Biol. Intell. Unit
82
,
2970
(
2014
).
28.
R.
Cammi
,
V.
Verdolino
,
B.
Mennucci
, and
J.
Tomasi
, “
Towards the elaboration of a QM method to describe molecular solutes under the effect of a very high pressure
,”
Chem. Phys.
344
,
135
(
2008
).
29.
R.
Cammi
,
C.
Cappelli
,
B.
Mennucci
, and
J.
Tomasi
, “
Calculation and analysis of the harmonic vibrational frequencies in molecules at extreme pressure: Methodology and diborane as a test case
,”
J. Chem. Phys.
137
,
154112
(
2012
).
30.
R.
Cammi
,
B.
Chen
, and
M.
Rahm
, “
Analytical calculation of pressure for confined atomic and molecular systems using the extreme-pressure polarizable continuum model
,”
J. Comput. Chem.
39
,
2243
(
2018
).
31.
R.
Cammi
, “
Quantum chemistry at the high pressures: The extreme pressure polarizable continuum model (XP-PCM)
,” in
Frontiers of Quantum Chemistry
(
Springer
,
2018
), pp.
273
287
.
32.
M.
Rahm
,
M.
Ångqvist
,
J. M.
Rahm
,
P.
Erhart
, and
R.
Cammi
, “
Non-bonded radii of the atoms under compression
,”
ChemPhysChem
21
,
2441
(
2020
).
33.
M.
Rahm
,
R.
Cammi
,
N. W.
Ashcroft
, and
R.
Hoffmann
, “
Squeezing all elements in the periodic table: Electron configuration and electronegativity of the atoms under compression
,”
J. Am. Chem. Soc.
141
,
10253
(
2019
).
34.
R.
Cammi
, “
A new extension of the polarizable continuum model: Toward a quantum chemical description of chemical reactions at extreme high pressure
,”
J. Comput. Chem.
36
,
2246
(
2015
).
35.
M.
Pagliai
,
R.
Cammi
,
G.
Cardini
, and
V.
Schettino
, “
XP-PCM calculations of high pressure structural and vibrational properties of P4S3
,”
J. Phys. Chem. A
120
,
5136
(
2016
).
36.
C.
Caratelli
,
R.
Cammi
,
R.
Chelli
,
M.
Pagliai
,
G.
Cardini
, and
V.
Schettino
, “
Insights on the realgar crystal under pressure from XP-PCM and periodic model calculations
,”
J. Phys. Chem. A
121
,
8825
(
2017
).
37.
M.
Pagliai
,
G.
Cardini
, and
R.
Cammi
, “
Vibrational frequencies of fullerenes C60 and C70 under pressure studied with a quantum chemical model including spatial confinement effects
,”
J. Phys. Chem. A
118
,
5098
(
2014
).
38.
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
(
1993
).
39.
V.
Barone
and
M.
Cossi
, “
Quantum calculation of molecular energies and energy gradients in solution by a conductor solvent model
,”
J. Phys. Chem. A
102
,
1995
(
1998
).
40.
U. J.
Kapasi
,
S.
Rixner
,
W. J.
Dally
,
B.
Khailany
,
J. H.
Ahn
,
P.
Mattson
, and
J. D.
Owens
, “
Programmable stream processors
,”
Computer
36
,
54
(
2003
).
41.
A.
Asadchev
,
V.
Allada
,
J.
Felder
,
B. M.
Bode
,
M. S.
Gordon
, and
T. L.
Windus
, “
Uncontracted Rys quadrature implementation of up to G functions on graphical processing units
,”
J. Chem. Theory Comput.
6
,
696
(
2010
).
42.
K.
Yasuda
, “
Two-electron integral evaluation on the graphics processor unit
,”
J. Comput. Chem.
29
,
334
(
2008
).
43.
I. S.
Ufimtsev
and
T. J.
Martínez
, “
Quantum chemistry on graphical processing units. 1. Strategies for two-electron integral evaluation
,”
J. Chem. Theory Comput.
4
,
222
(
2008
).
44.
F.
Liu
,
N.
Luehr
,
H. J.
Kulik
, and
T. J.
Martínez
, “
Quantum chemistry for solvated molecules on graphical processing units using polarizable continuum models
,”
J. Chem. Theory Comput.
11
,
3131
(
2015
).
45.
I. S.
Ufimtsev
and
T. J.
Martinez
, “
Quantum chemistry on graphical processing units. 2. Direct self-consistent-field implementation
,”
J. Chem. Theory Comput.
5
,
1004
(
2009
).
46.
X.
Andrade
and
A.
Aspuru-Guzik
, “
Real-space density functional theory on graphical processing units: Computational approach and comparison to Gaussian basis set methods
,”
J. Chem. Theory Comput.
9
,
4360
(
2013
).
47.
L.
Vogt
,
R.
Olivares-Amaya
,
S.
Kermes
,
Y.
Shao
,
C.
Amador-Bedolla
, and
A.
Aspuru-Guzik
, “
Accelerating resolution-of-the-identity second-order Møller-Plesset quantum chemistry calculations with graphical processing units
,”
J. Phys. Chem. A
112
,
2049
(
2008
).
48.
A. E.
DePrince
 III
and
J. R.
Hammond
, “
Coupled cluster theory on graphics processing units I. The coupled cluster doubles method
,”
J. Chem. Theory Comput.
7
,
1287
(
2011
).
49.
E. G.
Hohenstein
,
N.
Luehr
,
I. S.
Ufimtsev
, and
T. J.
Martínez
, “
An atomic orbital-based formulation of the complete active space self-consistent field method on graphical processing units
,”
J. Chem. Phys.
142
,
224103
(
2015
).
50.
C.
Song
and
T. J.
Martínez
, “
Reduced scaling CASPT2 using supporting subspaces and tensor hyper-contraction
,”
J. Chem. Phys.
149
,
044108
(
2018
).
51.
T. N.
Truong
and
E. V.
Stefanovich
, “
A new method for incorporating solvent effect into the classical, ab initio molecular orbital and density functional theory frameworks for arbitrary shape cavity
,”
Chem. Phys. Lett.
240
,
253
(
1995
).
52.
B.
Mennucci
,
E.
Cancès
, and
J.
Tomasi
, “
Evaluation of solvent effects in isotropic and anisotropic dielectrics and in ionic solutions with a unified integral equation method: Theoretical bases, computational implementation, and numerical applications
,”
J. Phys. Chem. B
101
,
10506
(
1997
).
53.
E.
Cancès
,
B.
Mennucci
, and
J.
Tomasi
, “
A new integral equation formalism for the polarizable continuum model: Theoretical background and applications to isotropic and anisotropic dielectrics
,”
J. Chem. Phys.
107
,
3032
(
1997
).
54.
J.
Tomasi
,
B.
Mennucci
, and
E.
Cancès
, “
The IEF version of the PCM solvation method: An overview of a new method addressed to study molecular solutes at the QM ab initio level
,”
J. Mol. Struct.: THEOCHEM
464
,
211
(
1999
).
55.
D. M.
York
and
M.
Karplus
, “
A smooth solvation potential based on the conductor-like screening model
,”
J. Phys. Chem. A
103
,
11060
(
1999
).
56.
A. W.
Lange
and
J. M.
Herbert
, “
A smooth, nonsingular, and faithful discretization scheme for polarizable continuum models: The switching/Gaussian approach
,”
J. Chem. Phys.
133
,
244111
(
2010
).
57.
F.
Liu
,
D. M.
Sanchez
,
H. J.
Kulik
, and
T. J.
Martínez
, “
Exploiting graphical processing units to enable quantum chemistry calculation of large solvated molecules with conductor-like polarizable continuum models
,”
Int. J. Quantum Chem.
119
,
e25760
(
2019
).
58.
E.
Cancès
,
Y.
Maday
, and
B.
Stamm
, “
Domain decomposition for implicit solvation models
,”
J. Chem. Phys.
139
,
054111
(
2013
).
59.
B.
Stamm
,
E.
Cancès
,
F.
Lipparini
, and
Y.
Maday
, “
A new discretization for the polarizable continuum model within the domain decomposition paradigm
,”
J. Chem. Phys.
144
,
054101
(
2016
).
60.
F.
Lipparini
,
B.
Stamm
,
E.
Cancès
,
Y.
Maday
, and
B.
Mennucci
, “
Fast domain decomposition algorithm for continuum solvation models: Energy and first derivatives
,”
J. Chem. Theory Comput.
9
,
3637
(
2013
).
61.
J.
Tomasi
,
B.
Mennucci
, and
R.
Cammi
, “
Quantum mechanical continuum solvation models
,”
Chem. Rev.
105
,
2999
(
2005
).
62.
J.
Tomasi
and
M.
Persico
, “
Molecular interactions in solution: An overview of methods based on continuous distributions of the solvent
,”
Chem. Rev.
94
,
2027
(
1994
).
63.
C.
Amovilli
and
B.
Mennucci
, “
Self-consistent-field calculation of Pauli repulsion and dispersion contributions to the solvation free energy in the polarizable continuum model
,”
J. Phys. Chem. B
101
,
1051
(
1997
).
64.
A.
Szabo
and
N. S.
Ostlund
,
Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory
(
Courier Corporation
,
2012
).
65.
A.
Bondi
, “
van der Waals volumes and radii
,”
J. Phys. Chem.
68
,
441
(
1964
).
66.
M.
Mantina
,
A. C.
Chamberlin
,
R.
Valero
,
C. J.
Cramer
, and
D. G.
Truhlar
, “
Consistent van der Waals radii for the whole main group
,”
J. Phys. Chem. A
113
,
5806
(
2009
).
67.
V. I.
Lebedev
, “
Quadratures on a sphere
,”
USSR Comput. Math. Math. Phys.
16
,
10
(
1976
).
68.
R.
Fukuda
,
M.
Ehara
, and
R.
Cammi
, “
Modeling molecular systems at extreme pressure by an extension of the polarizable continuum model (PCM) based on the symmetry-adapted cluster-configuration interaction (SAC–CI) method: Confined electronic excited states of furan as a test case
,”
J. Chem. Theory Comput.
11
,
2063
(
2015
).
69.
J. L.
Whitten
, “
Coulombic potential energy integrals and approximations
,”
J. Chem. Phys.
58
,
4496
(
1973
).
70.
L. E.
McMurchie
and
E. R.
Davidson
, “
One- and two-electron integrals over Cartesian Gaussian functions
,”
J. Comput. Phys.
26
,
218
(
1978
).
71.
S. F.
Boys
and
A. C.
Egerton
, “
Electronic wave functions—I. A general method of calculation for the stationary states of any molecular system
,”
Proc. R. Soc. London, Ser. A
200
,
542
(
1950
).
72.
See http://www.petachem.com/ for Petachem; accessed 8 April 2021.
73.
S.
Seritan
,
C.
Bannwarth
,
B. S.
Fales
,
E. G.
Hohenstein
,
C. M.
Isborn
,
S. I. L.
Kokkila-Schumacher
,
X.
Li
,
F.
Liu
,
N.
Luehr
,
J. W.
Snyder
, Jr.
,
C.
Song
,
A. V.
Titov
,
I. S.
Ufimtsev
,
L.-P.
Wang
, and
T. J.
Martínez
, “
TeraChem: A graphical processing unit-accelerated electronic structure package for large-scale ab initio molecular dynamics
,”
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
11
,
e1494
(
2021
).
74.
H. J.
Kulik
,
N.
Luehr
,
I. S.
Ufimtsev
, and
T. J.
Martinez
, “
Ab initio quantum chemistry for protein structures
,”
J. Phys. Chem. B
116
,
12501
(
2012
).
75.
P. J.
Stephens
,
F. J.
Devlin
,
C. F.
Chabalowski
, and
M. J.
Frisch
, “
Ab initio calculation of vibrational absorption and circular dichroism spectra using density functional force fields
,”
J. Phys. Chem. A
98
,
11623
(
1994
).
76.
A. D.
Becke
, “
Density-functional thermochemistry. III. The role of exact exchange
,”
J. Chem. Phys.
98
,
5648
(
1993
).
77.
C.
Lee
,
W.
Yang
, and
R. G.
Parr
, “
Development of the Colle-Salvetti correlation-energy formula into a functional of the electron density
,”
Phys. Rev. B
37
,
785
(
1988
).
78.
P. C.
Hariharan
and
J. A.
Pople
, “
Influence of polarization functions on molecular-orbital hydrogenation energies
,”
Theor. Chim. Acta
28
,
213
(
1973
).
79.
D. A.
Young
,
H.
Cynn
,
P.
Söderlind
, and
A.
Landa
, “
Zero-kelvin compression isotherms of the elements 1 ≤ Z ≤ 92 to 100 GPa
,”
J. Phys. Chem. Ref. Data
45
,
043101
(
2016
).
80.
J.
Towns
,
T.
Cockerill
,
M.
Dahan
,
I.
Foster
,
K.
Gaither
,
A.
Grimshaw
,
V.
Hazlewood
,
S.
Lathrop
,
D.
Lifka
, and
G. D.
Peterson
, “
XSEDE: Accelerating scientific discovery
,”
Comput. Sci. Eng.
16
,
62
(
2014
).
81.
A.
Asadchev
and
M. S.
Gordon
, “
New multithreaded hybrid CPU/GPU approach to Hartree–Fock
,”
J. Chem. Theory Comput.
8
,
4166
(
2012
).

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