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van Westen T, Gross J. Double-Hard-Sphere perturbation theory: a perturbation theory that is less dependent on the value of the hard-sphere diameter. Mol Phys 2022. [DOI: 10.1080/00268976.2022.2059410] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
Affiliation(s)
- Thijs van Westen
- Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Stuttgart, Germany
| | - Joachim Gross
- Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Stuttgart, Germany
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van Westen T, Hammer M, Hafskjold B, Aasen A, Gross J, Wilhelmsen Ø. Perturbation theories for fluids with short-ranged attractive forces: A case study of the Lennard-Jones spline fluid. J Chem Phys 2022; 156:104504. [DOI: 10.1063/5.0082690] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
It is generally not straightforward to apply molecular-thermodynamic theories to fluids with short-ranged attractive forces between their constituent molecules (or particles). This especially applies to perturbation theories, which, for short-ranged attractive fluids, typically must be extended to high order or may not converge at all. Here, we show that a recent first-order perturbation theory, the uv-theory, holds promise for describing such fluids. As a case study, we apply the uv-theory to a fluid with pair interactions defined by the Lennard-Jones spline potential, which is a short-ranged version of the LJ potential that is known to provide a challenge for equation-of-state development. The results of the uv-theory are compared to those of third-order Barker–Henderson and fourth-order Weeks–Chandler–Andersen perturbation theories, which are implemented using Monte Carlo simulation results for the respective perturbation terms. Theoretical predictions are compared to an extensive dataset of molecular simulation results from this (and previous) work, including vapor–liquid equilibria, first- and second-order derivative properties, the critical region, and metastable states. The uv-theory proves superior for all properties examined. An especially accurate description of metastable vapor and liquid states is obtained, which might prove valuable for future applications of the equation-of-state model to inhomogeneous phases or nucleation processes. Although the uv-theory is analytic, it accurately describes molecular simulation results for both the critical point and the binodal up to at least 99% of the critical temperature. This suggests that the difficulties typically encountered in describing the vapor–liquid critical region are only to a small extent caused by non-analyticity.
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Affiliation(s)
- Thijs van Westen
- Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Pfaffenwaldring 9, D-70569 Stuttgart, Germany
| | - Morten Hammer
- Department of Gas Technology, SINTEF Energy Research, NO-7465 Trondheim, Norway
| | - Bjørn Hafskjold
- Porelab, Department of Chemistry, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
| | - Ailo Aasen
- Department of Gas Technology, SINTEF Energy Research, NO-7465 Trondheim, Norway
| | - Joachim Gross
- Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Pfaffenwaldring 9, D-70569 Stuttgart, Germany
| | - Øivind Wilhelmsen
- Porelab, Department of Chemistry, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
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van Westen T, Gross J. Accurate thermodynamics of simple fluids and chain fluids based on first-order perturbation theory and second virial coefficients: uv-theory. J Chem Phys 2021; 155:244501. [PMID: 34972377 DOI: 10.1063/5.0073572] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We develop a simplification of our recently proposed uf-theory for describing the thermodynamics of simple fluids and fluids comprising short chain molecules. In its original form, the uf-theory interpolates the Helmholtz energy between a first-order f-expansion and first-order u-expansion as (effective) lower and upper bounds. We here replace the f-bound by a new, tighter (effective) lower bound. The resulting equation of state interpolates between a first-order u-expansion at high densities and another first-order u-expansion that is modified to recover the exact second virial coefficient at low densities. The theory merely requires the Helmholtz energy of the reference fluid, the first-order u-perturbation term, and the total perturbation contribution to the second virial coefficient as input. The revised theory-referred to as uv-theory-is thus simpler than the uf-theory but leads to similar accuracy, as we show for fluids with intermolecular pair interactions governed by a Mie potential. The uv-theory is thereby easier to extend to fluid mixtures and provides more flexibility in extending the model to non-spherical or chain-like molecules. The usefulness of the uv-theory for developing equation-of-state models of non-spherical molecules is here exemplified by developing an equation of state for Lennard-Jones dimers.
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Affiliation(s)
- Thijs van Westen
- Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Pfaffenwaldring 9, D-70569 Stuttgart, Germany
| | - Joachim Gross
- Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Pfaffenwaldring 9, D-70569 Stuttgart, Germany
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Convergence of Coupling-Parameter Expansion-Based Solutions to Ornstein–Zernike Equation in Liquid State Theory. CONDENSED MATTER 2021. [DOI: 10.3390/condmat6030029] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
The objective of this paper is to investigate the convergence of coupling-parameter expansion-based solutions to the Ornstein–Zernike equation in liquid state theory. The analytically solved Baxter’s adhesive hard sphere model is analyzed first by using coupling-parameter expansion. It was found that the expansion provides accurate approximations to solutions—including the liquid-vapor phase diagram—in most parts of the phase plane. However, it fails to converge in the region where the model has only complex solutions. Similar analysis and results are obtained using analytical solutions within the mean spherical approximation for the hardcore Yukawa potential. However, numerical results indicate that the expansion converges in all regions in this model. Next, the convergence of the expansion is analyzed for the Lennard-Jones potential by using an accurate density-dependent bridge function in the closure relation. Numerical results are presented which show convergence of correlation functions, compressibility versus density profiles, etc., in the single as well as two-phase regions. Computed liquid-vapor phase diagrams, using two independent schemes employing the converged profiles, compare excellently with simulation data. The results obtained for the generalized Lennard-Jones potential, with varying repulsive exponent, also compare well with the simulation data. Solution-spaces and the bifurcation of the solutions of the Ornstein–Zernike equation that are relevant to coupling-parameter expansion are also briefly discussed. All of these results taken together establish the coupling-parameter expansion as a practical tool for studying single component fluid phases modeled via general pair-potentials.
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van Westen T, Gross J. Accurate first-order perturbation theory for fluids: uf-theory. J Chem Phys 2021; 154:041102. [PMID: 33514104 DOI: 10.1063/5.0031545] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022] Open
Abstract
We propose a new first-order perturbation theory that provides a near-quantitative description of the thermodynamics of simple fluids. The theory is based on the ansatz that the Helmholtz free energy is bounded below by a first-order Mayer-f expansion. Together with the rigorous upper bound provided by a first-order u-expansion, this brackets the actual free energy between an upper and (effective) lower bound that can both be calculated based on first-order perturbation theory. This is of great practical use. Here, the two bounds are combined into an interpolation scheme for the free energy. The scheme exploits the fact that a first-order Mayer-f perturbation theory is exact in the low-density limit, whereas the accuracy of a first-order u-expansion grows when density increases. This allows an interpolation between the lower "f"-bound at low densities and the upper "u" bound at higher liquid-like densities. The resulting theory is particularly well behaved. Using a density-dependent interpolating function of only two adjustable parameters, we obtain a very accurate representation of the full fluid-phase behavior of a Lennard-Jones fluid. The interpolating function is transferable to other intermolecular potential types, which is here shown for the Mie m-6 family of fluids. The extension to mixtures is simple and accurate without requiring any dependence of the interpolating function on the composition of the mixture.
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Affiliation(s)
- Thijs van Westen
- Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Pfaffenwaldring 9, D-70569 Stuttgart, Germany
| | - Joachim Gross
- Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Pfaffenwaldring 9, D-70569 Stuttgart, Germany
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Abstract
Although the freezing of liquids and melting of crystals are fundamental for many areas of the sciences, even simple properties like the temperature–pressure relation along the melting line cannot be predicted today. Here we present a theory in which properties of the coexisting crystal and liquid phases at a single thermodynamic state point provide the basis for calculating the pressure, density and entropy of fusion as functions of temperature along the melting line, as well as the variation along this line of the reduced crystalline vibrational mean-square displacement (the Lindemann ratio), and the liquid's diffusion constant and viscosity. The framework developed, which applies for the sizable class of systems characterized by hidden scale invariance, is validated by computer simulations of the standard 12-6 Lennard-Jones system. Melting is a classic first-order phase transition, but an accurate thermodynamic description is still lacking. Here, Pedersen et al. develop a theory, validated by simulations of the Lennard-Jones system, for the melting thermodynamics applicable to all systems characterized by hidden scale invariance.
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Kelly BD, Smith WR, Henderson D. Analytical representation of the density derivative of the Percus–Yevick hard-sphere radial distribution function. Mol Phys 2016. [DOI: 10.1080/00268976.2016.1164908] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
Affiliation(s)
- Braden D. Kelly
- Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada
| | - William R. Smith
- Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada
- Faculty of Science, University of Ontario Institute of Technology, Oshawa, Ontario, Canada
| | - Douglas Henderson
- Department of Chemistry and Biochemistry, Brigham Young University, Provo, UT, USA
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van Westen T, Vlugt TJH, Gross J. On the vapor-liquid equilibrium of attractive chain fluids with variable degree of molecular flexibility. J Chem Phys 2015; 142:224504. [DOI: 10.1063/1.4922264] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Thijs van Westen
- Process and Energy Laboratory, Delft University of Technology, Leeghwaterstraat 39, 2628 CB Delft, The Netherlands
| | - Thijs J. H. Vlugt
- Process and Energy Laboratory, Delft University of Technology, Leeghwaterstraat 39, 2628 CB Delft, The Netherlands
| | - Joachim Gross
- Institut für Thermodynamik und Thermische Verfahrenstechnik, Universität Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany
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Henderson D. Understanding liquids: reflections about Melbourne, 1966–67. Mol Phys 2011. [DOI: 10.1080/00268976.2010.521203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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Pedersen UR, Schrøder TB, Dyre JC. Repulsive reference potential reproducing the dynamics of a liquid with attractions. PHYSICAL REVIEW LETTERS 2010; 105:157801. [PMID: 21230939 DOI: 10.1103/physrevlett.105.157801] [Citation(s) in RCA: 55] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2010] [Indexed: 05/30/2023]
Abstract
A well-known result of liquid state theory is that the structure of dense fluids is mainly determined by their repulsive forces. The Weeks-Chandler-Andersen potential, which cuts intermolecular potentials at their minima, is therefore often used as a reference. However, this cannot reproduce the viscous dynamics of the Kob-Andersen binary Lennard-Jones liquid [Berthier and Tarjus, Phys. Rev. Lett. 103, 170601 (2009)]. This paper shows that repulsive inverse-power-law potentials provide a reference for this liquid that reproduces its structure, dynamics, and isochoric heat capacity.
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Affiliation(s)
- Ulf R Pedersen
- Department of Chemistry, University of California, Berkeley, California 94720-1460, USA.
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CUMMINGS PETERT, JACKSON GEORGE, ROWLINSON JOHNS. Keith E. Gubbins: A celebration of statistical mechanics. Mol Phys 2009. [DOI: 10.1080/00268970210142666] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/14/2022]
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Andersen HC, Chandler D, Weeks JD. Roles of Repulsive and Attractive Forces in Liquids : The Equilibrium Theory of Classical Fluids. ADVANCES IN CHEMICAL PHYSICS 2007. [DOI: 10.1002/9780470142530.ch2] [Citation(s) in RCA: 212] [Impact Index Per Article: 12.5] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 12/23/2022]
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Vogelsang R, Hoheisel C. Molecular dynamics studies on potential energy fluctuations of particles in Lennard-Jones systems. Mol Phys 2006. [DOI: 10.1080/00268978400102121] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Ram J, Singh Y. First quantum correction to the radial distribution function and to the thermodynamic properties for a fluid of hard spheres. Mol Phys 2006. [DOI: 10.1080/00268977500102191] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
Affiliation(s)
- J. Ram
- a Department of Physics , Banaras Hindu University , Varanasi , 221005 , India
| | - Y. Singh
- a Department of Physics , Banaras Hindu University , Varanasi , 221005 , India
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Ben-Amotz D, Stell G. Reformulation of Weeks−Chandler−Andersen Perturbation Theory Directly in Terms of a Hard-Sphere Reference System. J Phys Chem B 2004. [DOI: 10.1021/jp037810s] [Citation(s) in RCA: 67] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Dor Ben-Amotz
- Department of Chemistry, Purdue University, West Lafayette, Indiana 47907
| | - George Stell
- State University of New York, Stony Brook, New York 11794
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LÍSAL MARTIN, SMITH WILLIAMR, BUREŠ MICHAL, VACEK VÁCLAV, NAVRÁTIL JIŘÍ. REMC computer simulations of the thermodynamic properties of argon and air plasmas. Mol Phys 2002. [DOI: 10.1080/00268970210130227] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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Lı́sal M, Smith WR, Nezbeda I. Computer simulation of the thermodynamic properties of high-temperature chemically-reacting plasmas. J Chem Phys 2000. [DOI: 10.1063/1.1289245] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Tang Y, Lu BC. First‐order radial distribution functions based on the mean spherical approximation for square‐well, Lennard‐Jones, and Kihara fluids. J Chem Phys 1994. [DOI: 10.1063/1.466449] [Citation(s) in RCA: 41] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Tang Y, Lu BC. A new solution of the Ornstein–Zernike equation from the perturbation theory. J Chem Phys 1993. [DOI: 10.1063/1.465465] [Citation(s) in RCA: 97] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Dufty JW, Mo KC, Gubbins KE. Models for self‐diffusion in the square well fluid. J Chem Phys 1991. [DOI: 10.1063/1.459783] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Abascal JLF, Martín C, Lombardero M, Vazquez J, Bañon A, Santamaria J. MC test of perturbation theories based on site–site potentials. J Chem Phys 1985. [DOI: 10.1063/1.448288] [Citation(s) in RCA: 15] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Wozniak S, Linder B, Zawodny R. Role of molecular symmetry and molecular interactions in the Faraday effect of fluids. ACTA ACUST UNITED AC 1983. [DOI: 10.1051/jphys:01983004403040300] [Citation(s) in RCA: 17] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022]
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Mier y Terán L, del Río F. Orthonormal series expansion solution of the Percus–Yevick equation. J Chem Phys 1980. [DOI: 10.1063/1.439221] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
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Lee LL. Correlation functions of classical fluids. V. The perturbation theory for soft spheres. J Chem Phys 1977. [DOI: 10.1063/1.433730] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Ram J, Singh Y. Perturbation theory for the radial distribution function in the presence of three‐body forces. J Chem Phys 1977. [DOI: 10.1063/1.434000] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Scarfe KD, McLaughlin IL, Collings AF. The transport coefficients for a fluid of square‐well rough spheres: Comparison with methane. J Chem Phys 1976. [DOI: 10.1063/1.433536] [Citation(s) in RCA: 25] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Madden WG, Fitts DD. Integral-equation perturbation theory for the radial distribution function of simple fluids. Mol Phys 1975. [DOI: 10.1080/00268977500102361] [Citation(s) in RCA: 16] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Mo KC, Gubbins KE. A new perturbation expansion for fluids of nonspherical molecules. J Chem Phys 1975. [DOI: 10.1063/1.431513] [Citation(s) in RCA: 43] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Singh Y. Perturbation theory for fluids of non-spherical molecules in the presence of three-body forces. I. Mol Phys 1975. [DOI: 10.1080/00268977500100111] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Sandler SI. The use of a non-spherical reference potential in statistical mechanical perturbation theory. Mol Phys 1974. [DOI: 10.1080/00268977400102521] [Citation(s) in RCA: 51] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Madden WG, Fitts DD. Perturbation theory for the radial distribution function for simple polar fluids. Chem Phys Lett 1974. [DOI: 10.1016/0009-2614(74)80382-9] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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Madden WG, Fitts DD. A new perturbation technique for the radial distribution function of simple fluids. Mol Phys 1974. [DOI: 10.1080/00268977400102411] [Citation(s) in RCA: 11] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Singh Y, Ram J. On the quantum corrections to the radial distribution function and to the thermodynamic properties of fluids. II. Mol Phys 1974. [DOI: 10.1080/00268977400101631] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Mo K, Gubbins K. Perturbation theory for molecular fluids using a nonspherical reference potential. Chem Phys Lett 1974. [DOI: 10.1016/0009-2614(74)80465-3] [Citation(s) in RCA: 36] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Mo K, Gubbins K, Jacucci G, McDonald IR. The radial distribution function in fluid mixtures: Conformal solution theory and molecular dynamics results. Mol Phys 1974. [DOI: 10.1080/00268977400101041] [Citation(s) in RCA: 44] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Wang S, Egelstaff P, Gubbins K. Monte Carlo study of perturbation theory for the radial distribution function. Mol Phys 1973. [DOI: 10.1080/00268977300100401] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Gubbins K, Gray C. Perturbation theory for the angular pair correlation function in molecular fluids. Mol Phys 1972. [DOI: 10.1080/00268977200100171] [Citation(s) in RCA: 103] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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