# Isomorph Invariance of Higher-Order Structural Measures in Four Lennard–Jones Systems

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## Abstract

**:**

## 1. Introduction

## 2. Results and Discussion

#### 2.1. Standard Lennard–Jones System

#### 2.2. Binary Lennard–Jones Mixtures

#### 2.2.1. Wahnström Mixture

#### 2.2.2. Kob–Andersen Mixture

#### 2.2.3. ${\mathrm{NiY}}_{2}$ Mixture

## 3. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Isomorph State-Point Data

$\mathit{\rho}$ | T | R | $\mathit{\gamma}$ |
---|---|---|---|

1.000 | 2.000 | 0.994 | 5.021 |

1.050 | 2.545 | 0.995 | 4.881 |

1.100 | 3.186 | 0.997 | 4.768 |

1.150 | 3.930 | 0.997 | 4.676 |

1.200 | 4.788 | 0.998 | 4.608 |

1.250 | 5.772 | 0.998 | 4.544 |

0.850 | 0.600 | 0.956 | 5.895 |

0.900 | 0.832 | 0.979 | 5.564 |

0.950 | 1.116 | 0.987 | 5.293 |

1.000 | 1.458 | 0.993 | 5.114 |

1.050 | 1.864 | 0.995 | 4.957 |

1.100 | 2.340 | 0.996 | 4.827 |

1.150 | 2.894 | 0.997 | 4.733 |

1.200 | 3.534 | 0.998 | 4.653 |

$\mathit{\rho}$ | T | R | $\mathit{\gamma}$ |
---|---|---|---|

0.75 | 0.646 | 0.982 | 5.050 |

0.80 | 0.893 | 0.990 | 4.911 |

0.85 | 1.200 | 0.994 | 4.785 |

0.90 | 1.572 | 0.996 | 4.681 |

0.95 | 2.018 | 0.997 | 4.596 |

1.00 | 2.549 | 0.998 | 4.525 |

1.50 | 14.738 | 0.999 | 4.205 |

2.00 | 48.610 | 0.999 | 4.110 |

0.65 | 0.493 | 0.936 | 5.600 |

0.70 | 0.748 | 0.977 | 5.364 |

0.75 | 1.076 | 0.988 | 5.106 |

0.80 | 1.489 | 0.993 | 4.912 |

0.85 | 2.000 | 0.995 | 4.772 |

0.90 | 2.620 | 0.997 | 4.659 |

0.95 | 3.364 | 0.998 | 4.573 |

1.00 | 4.248 | 0.998 | 4.505 |

1.50 | 24.564 | 0.999 | 4.197 |

2.00 | 81.016 | 0.999 | 4.105 |

2.50 | 201.615 | 0.999 | 4.066 |

$\mathit{\rho}$ | T | R | $\mathit{\gamma}$ |
---|---|---|---|

1.200 | 0.500 | 0.939 | 5.158 |

1.403 | 1.091 | 0.983 | 4.784 |

1.607 | 2.058 | 0.993 | 4.568 |

1.810 | 3.520 | 0.997 | 4.424 |

2.001 | 5.461 | 0.998 | 4.339 |

1.200 | 0.750 | 0.958 | 5.149 |

1.400 | 1.601 | 0.988 | 4.774 |

1.600 | 2.966 | 0.995 | 4.552 |

1.800 | 5.009 | 0.997 | 4.415 |

2.000 | 7.916 | 0.999 | 4.324 |

1.200 | 1.000 | 0.968 | 5.111 |

1.400 | 2.126 | 0.990 | 4.743 |

1.600 | 3.929 | 0.996 | 4.530 |

1.800 | 6.625 | 0.998 | 4.399 |

2.000 | 10.459 | 0.999 | 4.311 |

1.200 | 1.200 | 0.973 | 5.081 |

1.400 | 2.542 | 0.992 | 4.721 |

1.600 | 4.689 | 0.996 | 4.514 |

1.800 | 7.897 | 0.998 | 4.387 |

2.000 | 12.459 | 0.999 | 4.303 |

$\mathit{\rho}$ | T | R | $\mathit{\gamma}$ |
---|---|---|---|

1.30 | 1.000 | 0.959 | 5.270 |

1.40 | 1.461 | 0.980 | 5.083 |

1.50 | 2.050 | 0.988 | 4.889 |

1.70 | 3.689 | 0.995 | 4.645 |

2.00 | 7.640 | 0.998 | 4.430 |

2.50 | 19.965 | 0.999 | 4.253 |

3.00 | 42.877 | 0.999 | 4.173 |

3.50 | 81.086 | 0.999 | 4.123 |

## References

- Bernal, J.D. The Bakerian Lecture, 1962. The Structure of Liquids. Proc. R. Soc. Lond. Ser. A
**1964**, 280, 299–322. [Google Scholar] - Wong, J.; Angell, C.A. Glass Structure by Spectroscopy; Marcel Dekker: New York, NY, USA, 1976. [Google Scholar]
- Elliott, S.R. Medium-range structural order in covalent amorphous solids. Nature
**1991**, 354, 445–452. [Google Scholar] [CrossRef] - Gutzow, I.; Schmelzer, J. The Vitreous State: Thermodynamics, Structure, Rheology, and Crystallization; Springer: Berlin/Heidelberg, Germany, 1995. [Google Scholar]
- Debenedetti, P.G. Structure, Dynamics and Thermodynamics in Complex Systems: Theoretical Challenges and Opportunities. AICHE J.
**2005**, 51, 2391–2395. [Google Scholar] [CrossRef] - Cheng, Y.Q.; Ma, E. Atomic-level structure and structure—Property relationship in metallic glasses. Prog. Mater. Sci.
**2011**, 56, 379–473. [Google Scholar] [CrossRef] - Coslovich, D. Locally preferred structures and many-body static correlations in viscous liquids. Phys. Rev. E
**2011**, 83, 051505. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Banerjee, A.; Sengupta, S.; Sastry, S.; Bhattacharyya, S.M. Role of Structure and Entropy in Determining Differences in Dynamics for Glass Formers with Different Interaction Potentials. Phys. Rev. Lett.
**2014**, 113, 225701. [Google Scholar] [CrossRef] [Green Version] - Böhmer, R.; Gainaru, C.; Richert, R. Structure and dynamics of monohydroxy alcohols—Milestones towards their microscopic understanding, 100 years after Debye. Phys. Rep.
**2014**, 545, 125–195. [Google Scholar] [CrossRef] - Royall, C.P.; Williams, S.R. The role of local structure in dynamical arrest. Phys. Rep.
**2015**, 560, 1–75. [Google Scholar] [CrossRef] [Green Version] - Cubuk, E.D.; Ivancic, R.J.S.; Schoenholz, S.S.; Strickland, D.J.; Basu, A.; Davidson, Z.S.; Fontaine, J.; Hor, J.L.; Huang, Y.R.; Jiang, Y.; et al. Structure-property relationships from universal signatures of plasticity in disordered solids. Science
**2017**, 358, 1033–1037. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Gangopadhyay, A.; Kelton, K. Recent progress in understanding high temperature dynamical properties and fragility in metallic liquids, and their connection with atomic structure. J. Mater. Res.
**2017**, 32, 2638–2657. [Google Scholar] [CrossRef] - Wei, D.; Yang, J.; Jiang, M.Q.; Dai, L.H.; Wang, Y.J.; Dyre, J.C.; Douglass, I.; Harrowell, P. Assessing the utility of structure in amorphous materials. J. Chem. Phys.
**2019**, 150, 114502. [Google Scholar] [CrossRef] [Green Version] - Dyre, J.C. The Glass Transition and Elastic Models of Glass-Forming Liquids. Rev. Mod. Phys.
**2006**, 78, 953–972. [Google Scholar] [CrossRef] [Green Version] - Dyre, J.C. Master-equation approach to the glass transition. Phys. Rev. Lett.
**1987**, 58, 792–795. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Schrøder, T.B.; Dyre, J.C. Simplicity of condensed matter at its core: Generic definition of a Roskilde-simple system. J. Chem. Phys.
**2014**, 141, 204502. [Google Scholar] [CrossRef] [Green Version] - Dyre, J.C. Simple liquids’ quasiuniversality and the hard-sphere paradigm. J. Phys. Condens. Matter
**2016**, 28, 323001. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dyre, J.C. Perspective: Excess-entropy scaling. J. Chem. Phys.
**2018**, 149, 210901. [Google Scholar] [CrossRef] [Green Version] - Dyre, J.C. Isomorph theory beyond thermal equilibrium. J. Chem. Phys.
**2020**, 153, 134502. [Google Scholar] [CrossRef] - Gnan, N.; Schrøder, T.B.; Pedersen, U.R.; Bailey, N.P.; Dyre, J.C. Pressure-energy correlations in liquids. IV. “Isomorphs” in liquid phase diagrams. J. Chem. Phys.
**2009**, 131, 234504. [Google Scholar] [CrossRef] [PubMed] - Ingebrigtsen, T.S.; Schrøder, T.B.; Dyre, J.C. Isomorphs in Model Molecular Liquids. J. Phys. Chem. B
**2012**, 116, 1018–1034. [Google Scholar] [CrossRef] - Dyre, J.C. Hidden scale envariance in condensed matter. J. Phys. Chem. B
**2014**, 118, 10007–10024. [Google Scholar] [CrossRef] - Hummel, F.; Kresse, G.; Dyre, J.C.; Pedersen, U.R. Hidden scale invariance of metals. Phys. Rev. B
**2015**, 92, 174116. [Google Scholar] [CrossRef] [Green Version] - Costigliola, L.; Pedersen, U.R.; Heyes, D.; Schrøder, T.B.; Dyre, J.C. Communication: Simple liquids’ high-density viscosity. J. Chem. Phys.
**2018**, 148, 081101. [Google Scholar] [CrossRef] - Rosenfeld, Y. Relation between the transport coefficients and the internal entropy of simple systems. Phys. Rev. A
**1977**, 15, 2545–2549. [Google Scholar] [CrossRef] - Allen, M.P.; Tildesley, D.J. Computer Simulation of Liquids; Oxford Science Publications: Oxford, UK, 1987. [Google Scholar]
- Hansen, J.P.; McDonald, I.R. Theory of Simple Liquids: With Applications to Soft Matter, 4th ed.; Academic Press: New York, NY, USA, 2013. [Google Scholar]
- Bailey, N.P.; Pedersen, U.R.; Gnan, N.; Schrøder, T.B.; Dyre, J.C. Pressure-energy correlations in liquids. I. Results from computer simulations. J. Chem. Phys.
**2008**, 129, 184507. [Google Scholar] [CrossRef] [PubMed] - Ingebrigtsen, T.S.; Schrøder, T.B.; Dyre, J.C. What is a simple liquid? Phys. Rev. X
**2012**, 2, 011011. [Google Scholar] [CrossRef] [Green Version] - Schrøder, T.B.; Gnan, N.; Pedersen, U.R.; Bailey, N.P.; Dyre, J.C. Pressure-energy correlations in liquids. V. Isomorphs in generalized Lennard–Jones systems. J. Chem. Phys.
**2011**, 134, 164505. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Albrechtsen, D.E.; Olsen, A.E.; Pedersen, U.R.; Schrøder, T.B.; Dyre, J.C. Isomorph Invariance of the Structure and Dynamics of Classical Crystals. Phys. Rev. B
**2014**, 90, 094106. [Google Scholar] [CrossRef] [Green Version] - Costigliola, L.; Schrøder, T.B.; Dyre, J.C. Freezing and melting line invariants of the Lennard–Jones system. Phys. Chem. Chem. Phys.
**2016**, 18, 14678–14690. [Google Scholar] [CrossRef] [Green Version] - Bacher, A.K.; Schrøder, T.B.; Dyre, J.C. The EXP pair-potential system. II. Fluid phase isomorphs. J. Chem. Phys.
**2018**, 149, 114502. [Google Scholar] [CrossRef] - Friedeheim, L.; Dyre, J.C.; Bailey, N.P. Hidden scale invariance at high pressures in gold and five other face-centered-cubic metal crystals. Phys. Rev. E
**2019**, 99, 022142. [Google Scholar] [CrossRef] [Green Version] - Pedersen, U.R.; Bacher, A.K.; Schrøder, T.B.; Dyre, J.C. The EXP pair-potential system. III. Thermodynamic phase diagram. J. Chem. Phys.
**2019**, 150, 174501. [Google Scholar] [CrossRef] [Green Version] - Tolias, P.; Castello, F.L. Isomorph-based empirically modified hypernetted-chain approach for strongly coupled Yukawa one-component plasmas. Phys. Plasmas
**2019**, 26, 043703. [Google Scholar] [CrossRef] [Green Version] - Saw, S.; Dyre, J.C. Structure of the Lennard–Jones liquid estimated from a single simulation. Phys. Rev. E
**2021**, 103, 012110. [Google Scholar] [CrossRef] [PubMed] - Ingebrigtsen, T.S.; Tanaka, H. Effect of size polydispersity on the nature of Lennard–Jones liquids. J. Phys. Chem. B
**2015**, 119, 11052–11062. [Google Scholar] [CrossRef] [Green Version] - Ingebrigtsen, T.S.; Tanaka, H. Effect of energy polydispersity on the nature of Lennard–Jones liquids. J. Phys. Chem. B
**2016**, 120, 7704–7713. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Malins, A.; Eggers, J.; Royall, C.P. Investigating Isomorphs with the Topological Cluster Classification. J. Chem. Phys.
**2013**, 139, 234505. [Google Scholar] [CrossRef] [Green Version] - Kob, W.; Andersen, H.C. Testing mode-coupling theory for a supercooled binary Lennard–Jones mixture I: The van Hove correlation function. Phys. Rev. E
**1995**, 51, 4626–4641. [Google Scholar] [CrossRef] [Green Version] - Lennard–Jones, J.E. On the determination of molecular fields. I. From the variation of the viscosity of a gas with temperature. Proc. R. Soc. Lond. A
**1924**, 106, 441–462. [Google Scholar] - Bailey, N.P.; Ingebrigtsen, T.S.; Hansen, J.S.; Veldhorst, A.A.; Bøhling, L.; Lemarchand, C.A.; Olsen, A.E.; Bacher, A.K.; Costigliola, L.; Pedersen, U.R.; et al. RUMD: A general purpose molecular dynamics package optimized to utilize GPU hardware down to a few thousand particles. Scipost Phys.
**2017**, 3, 038. [Google Scholar] [CrossRef] [Green Version] - Rycroft, C.H.; Grest, G.S.; Landry, J.; Bazant, M.Z. Analysis of granular flow in a pebble-bed nuclear reactor. Phys. Rev. E
**2006**, 74, 021306. [Google Scholar] [CrossRef] [Green Version] - Rycroft, C.H. Voro++: A three-dimensional Voronoi cell library in C++. Chaos
**2009**, 19, 041111. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wahnström, G. Molecular-dynamics study of a supercooled two-component Lennard–Jones system. Phys. Rev. A
**1991**, 44, 3752. [Google Scholar] [CrossRef] [PubMed] - Bøhling, L.; Ingebrigtsen, T.S.; Grzybowski, A.; Paluch, M.; Dyre, J.C.; Schrøder, T.B. Scaling of viscous dynamics in simple liquids: Theory, simulation and experiment. New J. Phys.
**2012**, 14, 113035. [Google Scholar] [CrossRef] - Ingebrigtsen, T.S.; Bøhling, L.; Schrøder, T.B.; Dyre, J.C. Thermodynamics of Condensed Matter with Strong Pressure-Energy Correlations. J. Chem. Phys.
**2012**, 136, 061102. [Google Scholar] [CrossRef] - Costigliola, L.; Heyes, D.M.; Schrøder, T.B.; Dyre, J.C. Revisiting the Stokes–Einstein relation without a hydrodynamic diameter. J. Chem. Phys.
**2019**, 150, 021101. [Google Scholar] [CrossRef] [Green Version] - Frank, F.C.; Kasper, J.S. Complex alloy structures regarded as sphere packings. I. Definitions and basic principles. Acta Crystallograph.
**1958**, 11, 184–190. [Google Scholar] [CrossRef] - Frank, F.C. Supercooling of Liquids. Proc. R. Soc. Lond. A
**1952**, 215, 43–46. [Google Scholar] - Pedersen, U.R.; Schrøder, T.B.; Dyre, J.C.; Harrowell, P. Geometry of slow structural fluctuations in a supercooled binary alloy. Phys. Rev. Lett.
**2010**, 104, 105701. [Google Scholar] [CrossRef] [Green Version] - Pedersen, U.R.; Douglass, I.; Harrowell, P. How a supercooled liquid borrows structure from the crystal. J. Chem. Phys.
**2021**, 154, 054503. [Google Scholar] [CrossRef] - Rahman, A. Liquid Structure and Self-Diffusion. J. Chem. Phys.
**1966**, 45, 2585–2592. [Google Scholar] [CrossRef] - Tanemura, M.; Hiwatari, Y.; Matsuda, H.; Ogawa, T.; Ogita, N.; Ueda, A. Geometrical Analysis of Crystallization of the Soft-Core Model. Prog. Theor. Phys.
**1977**, 58, 1079–1095. [Google Scholar] [CrossRef] - Steinhardt, P.J.; Nelson, D.R.; Ronchetti, M. Bond-orientational order in liquids and glasses. Phys. Rev. B
**1983**, 28, 784–805. [Google Scholar] [CrossRef] - Valle, R.G.D.; Gazzillo, D.; Frattini, R.; Pastore, G. Microstructural analysis of simulated Ni
_{33}Y_{67}glass. Phys. Rev. B**1994**, 49, 12625–12632. [Google Scholar] [CrossRef] - Coslovich, D.; Pastore, G. Understanding fragility in supercooled Lennard–Jones mixtures. II. Potential energy surface. J. Chem. Phys.
**2007**, 127, 124505. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Toxvaerd, S.; Dyre, J.C. Communication: Shifted forces in molecular dynamics. J. Chem. Phys.
**2011**, 134, 081102. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**(

**a**) The two LJ isomorphs studied. State points on the isomorphs were found by integrating Equation (4) numerically. (

**b**) Mean-square displacement (MSD) for the state points of isomorph 1 plotted in reduced units, demonstrating good isomorph invariance of the dynamics. (

**c**) The same for the state points of isomorph 2.

**Figure 2.**Voronoi-structure variation along the two isomorphs and along the $\rho =1.00$ isochore of the single-component LJ system. Top panels: Fraction of local environments as a function of temperature for four common occurrences of the number of edges, vertices, and faces (marked on top of each figure). There is some variation along the isomorphs, but it is much smaller than along the isochore. The deviations decrease with increasing temperature and the consequent increase of R (Appendix A). Bottom panels: Fraction of the two most common Voronoi polyhedra of the conventional indexing system. Again we find approximate isomorph invariance.

**Figure 3.**A particle mean-square displacement in reduced units of three different sets of simulations of the Wahnström binary LJ mixture [46]. (

**a**) gives MSD data along the $\rho =0.85$ isochore (where density is given in AA particle units). (

**b**) is for the isomorph with reference state point $(\rho ,T)=(0.85,1.2)$. (

**c**) is for the isomorph with reference state point $(\rho ,T)=(0.85,2.0)$.

**Figure 4.**Different higher-order structure measures along an isomorph and along the isochore of Figure 3a, plotted as functions of the temperature for the Wahnström binary LJ mixture. The solid lines represent values along the $\rho =0.85$ isochore, while the points give values along the isomorph generated from the reference state point $(\rho ,T)=(0.85,1.2)$ (Figure 3b). (

**a**) shows results for the occurrence of Frank–Kasper bonds (black, denoted by “n”) and small particles in icosahedral local order (blue, denoted by “${\mathrm{Ico}}_{\mathrm{A}}$”). (

**b**) shows the average number of four-, five-, and six-sided faces of the Voronoi polyhedra. (

**c**) shows the Shannon entropy of the cell types of the Voronoi tessellation (Equation (6)). (

**d**) shows the relative frequency of occurrence of the five most common Voronoi cell types. Overall, there is good isomorph invariance.

**Figure 5.**As in Figure 4, but for the isomorph with reference state point $(\rho ,T)=(0.85,2.0)$, which is above the freezing line. The isochore is given by $\rho =0.85$. The structures are isomorph invariant to a good approximation, but vary along the isochore. Note that compared to Figure 4 there are more low-probability Voronoi structures, with no dominant structural motif.

**Figure 6.**Kob–Andersen (KA) binary LJ system characteristics. (

**a**) Four isomorphs. (

**b**) Reduced A particle MSD at four state points of the $\rho =1.20$ isochore. (

**c**) Reduced MSD along the lowest-temperature isomorph (reference state point $(\rho ,T)=(1.20,0.50)$) for the A and B particles, respectively, confirming that the state points are isomorphic by collapsing the reduced-unit mean-square displacements as a function of the reduced time. Note that the B particles are considerably faster than the A particles.

**Figure 7.**Two Voronoi structures of the KA system probed along the reference state point $(\rho ,T)$ = $(1.20,0.50)$ isomorph and along several isochores, plotted as a function of the temperature. (

**a**) shows results for the $<0,2,8,\ast >$ Voronoi structure around either an A or a B particle, and (

**b**) shows analogous results for the $<0,3,6,\ast >$ Voronoi structure. While there is significant variation along the isochores, both structures are isomorph invariant to a good approximation.

**Figure 8.**Normalized time-autocorrelation function of ${\overline{Q}}_{6}(t)$ plotted as a function of the reduced time. (

**a**) Results along the $\rho =1.20$ isochore. (

**b**) Results along the $T=1.20$ isotherm. (

**c**) Results along the reference-state-point $(\rho ,T)=(1.20,1.20)$ isomorph. (

**d**) Results along the reference-state-point $(\rho ,T)=(1.20,0.75)$ isomorph. (

**e**) Results along the reference-state-point $(\rho ,T)=(1.20,0.50)$ isomorph. There is good isomorph invariance.

**Figure 9.**Snapshot of the ${\mathrm{NiY}}_{2}$ binary LJ system at $(\rho ,T)=(1.3,1.0)$, the reference state point for the isomorph studied. The A particles representing the Yttrium atoms are blue and the B particles representing the Nickel atoms are red. We see that the system is homogeneous.

**Figure 10.**(

**a**–

**d**): RDF of central particles A or B counting all surrounding particles independent of their identity, monitored along the isomorph, plotted in both LJ units (

**left**) and reduced units (

**right**). There is good isomorph invariance of the reduced RDFs, although the first peak of the B particle RDF is visibly not isomorph invariant.

**Figure 11.**A particle (Y atom) MSD of the ${\mathrm{NiY}}_{2}$ binary LJ mixture along the isomorph in LJ and reduced units (

**a**,

**b**), and similarly along the $\rho =1.30$ isochore (

**c**,

**d**). Only along the isomorph is the MSD invariant in reduced units.

**Figure 12.**Temperature dependence of the occurrence of eight of the most abundant Voronoi structures at the reference state point $(\rho ,T)=(1.30,1.00)$, plotted for the ${\mathrm{NiY}}_{2}$ binary mixture as follows: along the isomorph as a function of the logarithm of the temperature (

**a**,

**b**), along the $\rho =1.3$ isochore as a function of the logarithm of the temperature (

**c**,

**d**). In (

**a**,

**c**) the central particle is of type B, in (

**b**,

**d**) it is an A particle. The Voronoi structures are isomorph invariant to a good approximation, but vary significantly along the isochore.

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**MDPI and ACS Style**

Rahman, M.; Carter, B.M.G.D.; Saw, S.; Douglass, I.M.; Costigliola, L.; Ingebrigtsen, T.S.; Schrøder, T.B.; Pedersen, U.R.; Dyre, J.C.
Isomorph Invariance of Higher-Order Structural Measures in Four Lennard–Jones Systems. *Molecules* **2021**, *26*, 1746.
https://doi.org/10.3390/molecules26061746

**AMA Style**

Rahman M, Carter BMGD, Saw S, Douglass IM, Costigliola L, Ingebrigtsen TS, Schrøder TB, Pedersen UR, Dyre JC.
Isomorph Invariance of Higher-Order Structural Measures in Four Lennard–Jones Systems. *Molecules*. 2021; 26(6):1746.
https://doi.org/10.3390/molecules26061746

**Chicago/Turabian Style**

Rahman, Mahajabin, Benjamin M. G. D. Carter, Shibu Saw, Ian M. Douglass, Lorenzo Costigliola, Trond S. Ingebrigtsen, Thomas B. Schrøder, Ulf R. Pedersen, and Jeppe C. Dyre.
2021. "Isomorph Invariance of Higher-Order Structural Measures in Four Lennard–Jones Systems" *Molecules* 26, no. 6: 1746.
https://doi.org/10.3390/molecules26061746