Effective potential methods, obtained by applying a quantum correction to a classical pair potential, are widely used for describing the thermophysical properties of fluids with mild nuclear quantum effects. In case of strong nuclear quantum effects, such as for liquid hydrogen and helium, the accuracy of these quantum corrections deteriorates significantly, but at present no simple alternatives are available. In this work, we solve this issue by developing a new, three-parameter corresponding-states principle that remains applicable in the regions of the phase diagram where quantum effects become significant. The new principle emerges from a mapping procedure, which shows that quantum-corrected pair potentials can be made conformal to their underlying classical pair potential by modifying the latter’s repulsive range. This mapping enables an accurate description of fluids with quantum-corrected interactions based on off-the-shelf methods for classical fluids (e.g., equations of state, classical density functional theory, and entropy scaling) using effective, mapped intermolecular-potential parameters. These effective parameters depend on temperature and molecular mass; simple analytic equations in case of a classical Mie potential with Feynman–Hibbs quantum corrections are presented. Using Mie Feynman–Hibbs force fields from the literature, we show that this procedure provides accurate predictions for the properties of fluids with mild nuclear quantum effects, such as neon or hydrogen at moderate temperatures. Moreover, by adjusting the functional form of the effective intermolecular-potential parameters to experimental data for helium and hydrogen, we are able to apply the corresponding-states principle for optimal quantum-corrected pair potentials that far surpass the accuracy of the Feynman–Hibbs correction.

1.
J. O.
Hirschfelder
,
C. F.
Curtiss
, and
R. B.
Bird
,
Molecular Theory of Gases and Liquids
(
Wiley
,
New York
,
1954
).
2.
L. D.
Landau
and
E. M.
Lifshitz
,
Quantum Mechanics: Non-Relativistic Theory
, 3rd ed. (
Pergamon Press
,
New York
,
1977
).
3.
R. P.
Feynman
,
A. R.
Hibbs
, and
D. F.
Styer
,
Quantum Mechanics and Path Integrals
, amended ed. (
McGraw-Hill
,
New York
,
2005
).
4.
D. A.
McQuarrie
,
Quantum Chemistry
, 2nd ed. (
University Science Books
,
Sausalito, CA
,
2007
).
5.
U. K.
Deiters
and
R. J.
Sadus
, “
An intermolecular potential for hydrogen: Classical molecular simulation of pressure–density–temperature behavior, vapor–liquid equilibria, and critical and triple point properties
,”
J. Chem. Phys.
158
,
194502
(
2023
).
6.
Ø.
Wilhelmsen
,
D.
Berstad
,
A.
Aasen
,
P.
Nekså
, and
G.
Skaugen
, “
Reducing the exergy destruction in the cryogenic heat exchangers of hydrogen liquefaction processes
,”
Int. J. Hydrogen Energy
43
,
5033
5047
(
2018
).
7.
S. Z.
Al Ghafri
,
S.
Munro
,
U.
Cardella
,
T.
Funke
,
W.
Notardonato
,
J. P. M.
Trusler
,
J.
Leachman
,
R.
Span
,
S.
Kamiya
,
G.
Pearce
,
A.
Swanger
,
E. D.
Rodriguez
,
P.
Bajada
,
F.
Jiao
,
K.
Peng
,
A.
Siahvashi
,
M. L.
Johns
, and
E. F.
May
, “
Hydrogen liquefaction: A review of the fundamental physics, engineering practice and future opportunities
,”
Energy Environ. Sci.
15
,
2690
2731
(
2022
).
8.
M.
Hammer
,
G.
Bauer
,
R.
Stierle
,
J.
Gross
, and
O.
Wilhelmsen
, “
Classical density functional theory for interfacial properties of hydrogen, helium, deuterium, neon, and their mixtures
,”
J. Chem. Phys.
158
,
104107
(
2023
).
9.
E.
Wigner
, “
On the quantum correction for thermodynamic equilibrium
,”
Phys. Rev.
40
,
749
759
(
1932
).
10.
J. G.
Kirkwood
, “
Quantum statistics of almost classical assemblies
,”
Phys. Rev.
44
,
31
37
(
1933
).
11.
M.
Neumann
and
M.
Zoppi
, “
Asymptotic expansions and effective potentials for almost classical N-body systems
,”
Phys. Rev. A
40
,
4572
4584
(
1989
).
12.
J. P.
Hansen
and
I. R.
McDonald
,
Theory of Simple Liquids
, 3rd ed. (
Academic Press
,
London
,
2006
).
13.
D.
Frenkel
and
B.
Smit
,
Understanding Molecular Simulation: From Algorithms to Applications
, 2nd ed. (
Academic Press
,
San Diego
,
2002
).
14.
L. M.
Sesé
, “
Feynman-Hibbs quantum effective potentials for Monte Carlo simulations of liquid neon
,”
Mol. Phys.
78
,
1167
1177
(
1993
).
15.
L. M.
Sesé
, “
Study of the Feynman-Hibbs effective potential against the path-integral formalism for Monte Carlo simulations of quantum many-body Lennard-Jones systems
,”
Mol. Phys.
81
,
1297
1312
(
1994
).
16.
F.
Calvo
,
J. P. K.
Doye
, and
D. J.
Wales
, “
Quantum partition functions from classical distributions: Application to rare-gas clusters
,”
J. Chem. Phys.
114
,
7312
7329
(
2001
).
17.
P.
Kowalczyk
,
L.
Brualla
,
P. A.
Gauden
, and
A. P.
Terzyk
, “
Static and thermodynamic properties of low-density supercritical 4He—breakdown of the Feynman–Hibbs approximation
,”
Phys. Chem. Chem. Phys.
11
,
9182
9187
(
2009
).
18.
A.
Aasen
,
M.
Hammer
,
A.
Ervik
,
E.
Müller
, and
O.
Wilhelmsen
, “
Equation of state and force fields for Feynman–Hibbs-corrected mie fluids. I. application to pure helium, neon, hydrogen, and deuterium
,”
J. Chem. Phys.
151
,
064508
(
2019
).
19.
T. T.
Trinh
,
M.
Hammer
,
V.
Sharma
, and
Ø.
Wilhelmsen
, “
Mie–FH: A quantum corrected pair potential in the LAMMPS simulation package for hydrogen mixtures
,”
SoftwareX
26
,
101716
(
2024
).
20.
S.
Kim
,
D.
Henderson
, and
J. A.
Barker
, “
Perturbation theory of fluids and deviations from classical behavior
,”
Can. J. Phys.
47
,
99
102
(
1969
).
21.
V. M.
Trejos
and
A.
Gil-Villegas
, “
Semiclassical approach to model quantum fluids using the statistical associating fluid theory for systems with potentials of variable range
,”
J. Chem. Phys.
136
,
184506
(
2012
).
22.
A.
Aasen
,
M.
Hammer
,
E.
Müller
, and
O.
Wilhelmsen
, “
Equation of state and force fields for Feynman–Hibbs-corrected Mie fluids. II. application to mixtures of helium, neon, hydrogen, and deuterium
,”
J. Chem. Phys.
152
,
074507
(
2020
).
23.
M. E.
Boyd
,
S. Y.
Larsen
, and
J. E.
Kilpatrick
, “
Quantum mechanical second virial coefficient of a Lennard-Jones gas. Helium
,”
J. Chem. Phys.
50
,
4034
4055
(
1969
).
24.
R.
Hellmann
, “
Ab initio potential energy surface for the nitrogen molecule pair and thermophysical properties of nitrogen gas
,”
Mol. Phys.
111
,
387
401
(
2013
).
25.
R.
Subramanian
,
A. J.
Schultz
, and
D. A.
Kofke
, “
Quantum virial coefficients of molecular nitrogen
,”
Mol. Phys.
115
,
869
878
(
2017
).
26.
R.
Hellmann
, “
Eighth-order virial equation of state for methane from accurate two-body and nonadditive three-body intermolecular potentials
,”
J. Phys. Chem. B
126
,
3920
3930
(
2022
).
27.
P.
Ströker
,
R.
Hellmann
, and
K.
Meier
, “
Thermodynamic properties of krypton from Monte Carlo simulations using ab initio potentials
,”
J. Chem. Phys.
157
,
114504
(
2022
).
28.
J.
de Boer
and
A.
Michels
, “
Contribution to the quantum-mechanical theory of the equation of state and the law of corresponding states. Determination of the law of force of helium
,”
Physica
5
,
945
957
(
1938
).
29.
J. D.
van der Waals
, “
Over de continuiteit van den gas- en vloeistoftoestand
,” Ph.D. thesis,
Hogeschool
,
Leiden
,
1873
, [translated by J. S. Rowlinson, On the Continuity of the Gaseous and Liquid States (1988)].
30.
K. S.
Pitzer
, “
Corresponding states for perfect liquids
,”
J. Chem. Phys.
7
,
583
590
(
1939
).
31.
H. C.
Longuet-Higgins
, “
The statistical thermodynamics of multicomponent systems
,”
Proc. R. Soc. London, Ser. A
205
,
247
(
1951
).
32.
E. A.
Guggenheim
, “
The principle of corresponding states
,”
J. Chem. Phys.
13
,
253
261
(
1945
).
33.
N. S.
Ramratan
, “
Simulation and theoretical perspectives of the phase behaviour of solids, liquids and gases using the mie family of intermolecular potentials
,” Ph.D. thesis,
Imperial College of Science, Technology and Medicine
,
2013
.
34.
N. S.
Ramrattan
,
C.
Avendaño
,
E. A.
Müller
, and
A.
Galindo
, “
A corresponding-states framework for the description of the Mie family of intermolecular potentials
,”
Mol. Phys.
113
,
932
947
(
2015
).
35.
E. A.
Mason
and
W. E.
Rice
, “
The intermolecular potentials of helium and hydrogen
,”
J. Chem. Phys.
22
,
522
535
(
1954
).
36.
M. G.
Dondi
,
U.
Valbusa
, and
G.
Scoles
, “
Energy dependence of the differential collision cross section for hydrogen at thermal energies
,”
Chem. Phys. Lett.
17
,
137
141
(
1972
).
37.
T.
van Westen
and
J.
Gross
, “
Accurate thermodynamics of simple fluids and chain fluids based on first-order perturbation theory and second virial coefficients: Uv-theory
,”
J. Chem. Phys.
155
,
244501
(
2021
).
38.
E.
Helfand
and
S. A.
Rice
, “
Principle of corresponding states for transport properties
,”
J. Chem. Phys.
32
,
1642
1644
(
1960
).
39.
J. W.
Leachman
,
R. T.
Jacobsen
,
S. G.
Penoncello
, and
E. W.
Lemmon
, “
Fundamental equations of state for parahydrogen, normal hydrogen, and orthohydrogen
,”
J. Phys. Chem. Ref. Data
38
,
721
748
(
2009
).
40.
A.
Mulero
,
I.
Cachadiña
, and
M. I.
Parra
, “
Recommended correlations for the surface tension of common fluids
,”
J. Phys. Chem. Ref. Data
41
,
043105
(
2012
).
41.
C. D.
Muzny
,
M. L.
Huber
, and
A. F.
Kazakov
, “
Correlation for the viscosity of normal hydrogen obtained from symbolic regression
,”
J. Chem. Eng. Data
58
,
969
979
(
2013
).
42.
E. W.
Lemmon
,
I. H.
Bell
,
M. L.
Huber
, and
M. O.
McLinden
, “
Thermophysical properties of fluid systems
,” in
NIST Chemistry WebBook, NIST Standard Reference Database Number 69
, edited by
P.
Linstrom
and
W.
Mallard
(
National Institute of Standards and Technology
,
Gaithersburg, MD
,
2024
).
43.
P.
Rehner
,
G.
Bauer
, and
J.
Gross
, “
Feos: An open-source framework for equations of state and classical density functional theory
,”
Ind. Eng. Chem. Res.
62
,
5347
5357
(
2023
).
44.
D. O.
Ortiz-Vega
, “
A new wide range equation of state for helium-4
,” Ph.D. thesis,
Texas A&M University
,
2013
.
45.
V. G.
Jervell
and
O.
Wilhelmsen
, “
Revised Enskog theory for Mie fluids: Prediction of diffusion coefficients, thermal diffusion coefficients, viscosities, and thermal conductivities
,”
J. Chem. Phys.
158
,
224101
(
2023
).
46.
Y.
Rosenfeld
, “
Relation between the transport coefficients and the internal entropy of simple systems
,”
Phys. Rev. A
15
,
2545
2549
(
1977
).
47.
Y.
Rosenfeld
, “
A quasi-universal scaling law for atomic transport in simple fluids
,”
J. Phys.: Condens. Matter
11
,
5415
(
1999
).
48.
D.
Saric
,
I. H.
Bell
,
G.
Guevara-Carrion
, and
J.
Vrabec
, “
Influence of repulsion on entropy scaling and density scaling of monatomic fluids
,”
J. Chem. Phys.
160
,
104503
(
2024
).
49.
I. H.
Bell
,
R.
Messerly
,
M.
Thol
,
L.
Costigliola
, and
J. C.
Dyre
, “
Modified entropy scaling of the transport properties of the Lennard-Jones fluid
,”
J. Phys. Chem. B
123
,
6345
6363
(
2019
).
50.
T.
Lafitte
,
A.
Apostolakou
,
C.
Avendãno
,
A.
Galindo
,
C. S.
Adjiman
,
E. A.
Müller
, and
G.
Jackson
, “
Accurate statistical associating fluid theory for chain molecules formed from Mie segments
,”
J. Chem. Phys.
139
,
154504
(
2013
).
51.
P.
Rehner
and
G.
Bauer
, “
Application of generalized (hyper-) dual numbers in equation of state modeling
,”
Front. Chem. Eng.
3
,
758090
(
2021
).
52.
P. J.
Walker
,
T.
Zhao
,
A. J.
Haslam
, and
G.
Jackson
, “
Ab initio development of generalized Lennard-Jones (Mie) force fields for predictions of thermodynamic properties in advanced molecular-based SAFT equations of state
,”
J. Chem. Phys.
156
,
154106
(
2022
).
53.
G. A.
Voth
, “
Calculation of equilibrium averages with Feynman-Hibbs effective classical potentials and similar variational approximations
,”
Phys. Rev. A
44
,
5302
5305
(
1991
).
54.
J.
Cao
and
G. A.
Voth
, “
The formulation of quantum statistical mechanics based on the Feynman path centroid density. I. Equilibrium properties
,”
J. Chem. Phys.
100
,
5093
5105
(
1994
).
You do not currently have access to this content.