100 Years of the Lennard-Jones Potential Peter Schwerdtfeger∗,† and David Wales∗,‡ †Centre for Theoretical Chemistry and Physics, The New Zealand Institute for Advanced Study, Massey University Auckland, Private Bag 102904, 0745 Auckland, New Zealand ‡Yusuf Hamied Department of Chemistry, Lensfield Road, Cambridge CB2 1EW, UK E-mail: peter.schwerdtfeger@gmail.com; dw34@cam.ac.uk Abstract It is now 100 years since Lennard-Jones pub- lished his first paper introducing the now fa- mous potential that bears his name. It is there- fore timely to reflect on the many achievements, as well as the limitations, of this potential in the theory of atomic and molecular interactions, where applications range from descriptions of intermolecular forces, to clusters and condensed matter. 1 Historical Introduction 100 years ago in April 22, 1924, the Royal Society in London received an article written by a research student at Trinity College at Cambridge, John Edward Jonesa (Figure 1), communicated through the mathematician and geophysicist Prof. Sydney Chapman (FRS) at Cambridge.1 It was the year when Lennard- Jones received his PhD from Cambridge Uni- versity. The paper entitled On the determi- nation of molecular fields.—I. From the vari- ation of the viscosity of a gas with tempera- ture was the first in a series of landmark papers to come,2–5 introducing a general two-body po- tential that separated the interaction between aLennard-Jones added the surname of his wife Kath- leen Mary Lennard to his name when they were married in 1925. Kathleen Mary Lennard was the daughter of a shoe-shop owner, and her brothers had perished in World War I. Under these circumstances, a son-in-law could add the maiden name of their spouse to preserve the family name. Figure 1: John Edward Lennard-Jones (1934), National Portrait Gallery, UK. atoms into a repulsive r−n and attractive part r−m (n > m) (the (n,m)-LJ potential), ϕLJ(r) = An rn − Bm rm (1) in order to improve on the Chapman-Enskog6–8 and Sutherland9 models for the viscosity of a gas. In the first sentence of the introduction Lennard-Jones stated that Until our knowledge of the disposition and motion of the electrons in atoms and molecules is more complete, we cannot hope to make a direct calculation of the nature of the forces called into play during an 1 peter.schwerdtfeger@gmail.com dw34@cam.ac.uk encounter between molecules in a gas.b The Schrödinger equation was introduced soon af- ter in 1925, and published in 1926,10 but at the time was considered far too complicated to ac- curately treat the interactions between atoms by first-principles quantum methods.c In the absence of a more rigorous quantum theoretical treatment one had to rely on empirical forces or interaction potentials, where the parameters (like An and Bm in (1)) can be derived from experimental data. Concerning the theory of intermolecular forces, both Johannes D. van der Waals in 1873 and William Sutherland in 1893 realized that an attractive force is required to correctly describe the static and dynamic behavior of the gaseous state.9,12 Sutherland introduced an at- tractive r−m potential to the hard-sphere model (the so-called Sutherland potential, which we could consider as a (∞,m)-LJ potential). He set m = 2 identical to the van der Waals cohe- sive force to improve on the kinetic gas theory for atomic and molecular collisions.9 The virial expansion introduced by Kamerlingh Onnes for a non-ideal gas followed in 1901.13 Gustav Mie in his 1903 paper entitled About the kinetic theory of one-atomic bodies already separated the two-body interaction potential into a short range repulsive and a long-range attractive form to study liquid-gas properties.14 He applied a repulsive potential of the form in (1) (proposing n = 5). It was, however, Eduard Grüneisen in his 1912 paper on the Theory of the solid state of one-atomic elements based on Mie’s theory who first introduced the potential (1) in its most general form, deriving impor- tant thermodynamic relationships, such as the Grüneisen parameter for the solid state,15 γ = V ( ∂P ∂E ) V ≈ n+m+ 1 6 . (2) Here, E is the internal energy, P is the pressure, and V the volume. The LJ potential is therefore bIn his 1924 paper Lennard-Jones introduced the force f(r) = −dϕ(r)/dr = λnr −n − λmr−m and termed the coefficients λn and λm force constants.1 cThe first molecule treated using the Schrödinger equation was H2 in 1927 by Walter H. Heitler and Fritz W. London.11 also called the Mie-Grüneisen potential within the solid-state and materials physics commu- nity. Grüneisen pointed out that one requires the conditions n > m and n > 3 for a solid, the latter to assure convergence of the sum over all interactions for an inverse power potential.15 Two other papers should be mentioned for his- torical completeness. The first by Franz E. Si- mon and Clara von Simson in 1924,16 who sub- mitted their paper a few weeks after Lennard- Jones in 1924,d entitled The crystal structure of argon. Simon and Simson used the LJ potential with exponents of n = 15 and m = 9 estimated from the Grüneisen parameter.17,18 The other is a paper by Adolf Kratzer in 192019 using n = 2 and m = 1 (the Kratzer potential), for which one can obtain analytical solutions for the ra- dial Schrödinger equation.20 Coming back to Lennard-Jones, he tried to adjust the exponents (n,m) in (1) to experi- mental gas phase or solid state data, i.e. he was experimenting with different values of n and m to obtain a best fit.1,2 In his first paper he found that n = 131 3 and m = 2 for (1) gave good re- sults for gaseous argon at various temperatures. We notice that the famous attractive long-range r−6 behavior for the dispersion force was intro- duced later in 1930 by London,21,22 and recog- nised by Lennard-Jones23 soon after.e It has been noted by London that the Drude model24 of oscillating dipoles between two atoms pro- vides a straightforward way to introduce the dispersion force.22,25,26 In his second paper, Lennard-Jones consid- ered the virial coefficients of a gas.2 He now realised that the condition m > 3 is required to ensure the convergence of the integral required for the first viral coefficient B in line with Grüneisen’s earlier finding. He also found a se- ries expansion for the second viral coefficient B(T ) in terms of Γ-functions for the (n,m)-LJ potential. Lennard-Jones experimented with different exponents for (1), but settled for n = 131 3 and m = 4 to give good agreement with dSimon and Simson gave reference to the Grüneisen paper,15 while Lennard-Jones did not. eWhile the r−6 attractive potential has a physical meaning through the dispersion force, the r−n repulsive potential is purely empirical 2 gaseous argon. Applications for the first virial coefficient and viscosity for gaseous helium and neon and some ionic crystals were considered soon after for different n values setting m = 4.4 In Lennard-Jones’ third paper of 1924,3 lat- tice sums were introduced to obtain bulk prop- erties for cubic crystals. This work is particu- larly interesting (and perhaps less well known to the chemistry community) as it opened up a whole new field in mathematical lattice the- ory.27–29 Together with the mathematician Al- bert E. Ingham at Cambridge they realized that summing over all interactions in a cubic lat- tice using inverse power potentials of the form r−n leads to a special case of the Epstein zeta function.30 Ingham finished his PhD on Epstein zeta functions in 1922 in Cambridge and spe- cialised in infinite sums. He was in close contact with Godfrey H. Hardy and John E. Littlewood in Cambridge, well known for their work in number theory and their famous relation with Srinivasa Ramanujan (the man who knew infin- ity31). Ingham must have come across all these activities during his PhD. In 1924/25, Lennard- Jones and Ingham were able to obtain useful series expressions for the lattice sums and ob- tained values to 4-5 significant digits behind the decimal point.3,5 They pointed out that m > 3 in (1) needs to be chosen to avoid divergences in the lattice sum, similar to the earlier investi- gated virial coefficient (thus the Kratzer poten- tial cannot be used in solid-state calculations without a long-range cut-off). They also con- sidered ionic crystals, but did not make the step to use alternating series for the Coulomb in- teraction between fractional charges as Erwin Madelung did in 1918.32 Lennard-Jones published many more papers on atomic and molecular interactions33–35 to obtain bulk properties, including investigations into melting and boiling points of bulk sys- tems. A more complete list of Lennard-Jones’ papers together with a nice historical essay on his life and work has been given by Nevill Fran- cis Mott in 1955.36 It was during his tenure in Cambridge as Plummer Professor of Theo- retical Chemistry that Lennard-Jones founded the Cambridge University Mathematical Labo- ratory in 1937. This facility laid foundations for Figure 2: The 1951 Shelter Island Conference on Quantum Mechanics in Valence Theory. From left to right. Standing: Klaus Rüdenberg, Theodore H. Berlin, Michael P. Barnett, Bryce Crawford, Duncan A. Mac Innes, Henry Margenau, Kenneth Pitzer, George E. Kimball, C. W. Ufford, Robert S. Mulliken, John H. Van Vleck, Per-Olov Löwdin, John E. Lennard-Jones, Henry Eyring, John C. Slater, Charles A. Coulson, Joseph O. Hirschfelder, George W. Wheland, Harrison Shull, Leslie E. Sut- ton, Robert Parr. Seated: Joseph Edward Mayer, William Moffitt, Clemens C. J. Roothaan, Masao Kotani. the future of computer science in Cambridge,37 and was recognised by the formation of a virtual Lennard-Jones Centre to focus interdisciplinary research in molecular modelling.38 In 1951, the giants in quantum chemistry met close to New York at the Shelter Island Confer- ence on Quantum Mechanics in Valence The- ory organized by Robert S. Mulliken and spon- sored by the National Academy of Sciences. A whole section of the conference was ded- icated to intermolecular and non-bonded in- teratomic forces with prominent speakers like Hirschfelder, Coulson, Mulliken and Van Vleck. However, Lennard-Jones decided to give a talk on The Spatial Correlation of Electrons in Molecules in another session on Chemical Va- lence Concepts. A summary of the conference has been provided by Parr and Crawford,39 which makes for interesting reading. A histor- ical photograph with many well known players in the field at the time is shown in Figure 2. Without any doubt, the Lennard-Jones (LJ) potential (1) is the most famous and widely ap- 3 plied interatomic potential for exploring impor- tant interactions in atomic and molecular sys- tems and the bulk. If we consider the boundary conditions that the minimum for a diatomic is at the equilibrium distance re with a potential depth of ϵ we obtain for the coefficients An = mϵrne n−m and Bm = nϵrme n−m (3) with the condition that n > m. The often used van der Waals radius σ where ϕ(σ) = 0 is then given by σ = (m n ) 1 n−m re (4) The LJ potentials for different (n,m) values Figure 3: Lennard-Jones potential for different (n,m) values (n > m). Reduced units are used (re = 1 and ϵ = 1). For n,m → ∞ the sticky hard-sphere limit and for n → ∞ the Sutherland potential9 is obtained. The two potentials with n = 240 are close to those limits. are depicted in Figure 3, also showing the rela- tion to the sticky hard-sphere and Sutherland model. The advantage is that the LJ potential is simple, computationally efficient, and many properties of extended systems (e.g. the melt- ing point, cohesive energy, bulk modulus etc.) can be obtained to computer precision. The parameters An and Bm are often chosen differ- ently from eq.(3), and adjusted to experimental data in order to achieve better agreement. In this case, the (more empirical) LJ potential in- corporates implicitly to a certain extent impor- tant many-body effects, and better agreement with experiment is therefore achieved. The LJ potential is ideal to test various al- gorithms used in atomistic molecular dynamics (MD) and Monte Carlo (MC) simulations40,41 to gain deeper insight into the theory of in- teratomic interactions in extended systems. Of course, there are many more accurate interac- tion potentials around, most notably the Morse potential42 and the Aziz potential,43 required for example to obtain accurate vibrational- rotational spectra of diatomic molecules. Re- views of different interaction potentials in use are given in refs. 44,45. In the following, we concentrate on some important selected appli- cations for the LJ potential and provide a brief overview of the field. 2 The Lennard-Jones Po- tential in Gas-Phase Sim- ulations Present-day understanding of the states of mat- ter draws heavily on computer simulations of gases, liquids and the solid state using realistic interaction potentials.26 For these applications, it is important to accurately know the equation- of-state (EOS), f(P, Vm, T ) = 0 originally in- troduced for gases, where T is the absolute temperature, P the pressure, and Vm = V/n the molar volume.46 The EOS can be obtained from either experimental data or computational simulations and subsequently fitted, for exam- ple, to some of the analytical forms available for f(P, Vm, T ) for the different phases.47 How- ever, state-of-the-art MD or MC simulations using sophisticated quantum chemical methods are computationally too demanding, because a large number of atoms or molecules are re- quired to obtain accurate results close to the infinite particle limit. One therefore relies on reliable many-body interaction potentials for atomic and molecular interactions or on force- field methods parameterized for specific sys- tems. Hence one has to be aware of the ac- curacy of such approaches, e.g. how to deal with the many-body effects beyond a two-body approximation, which are, for example, respon- sible for obtaining the correct structure in clus- 4 ters or moleculesf.48–51 The most widely used EOS for a real gas is the virial form (or van der Waals virial equation),52 PVm = RTZ = RT ∑ j=1 Bj(T )V −j+1 m , (5) which, for many-body potentials, follows di- rectly from the Ursell–Mayer cluster expan- sion of the system’s grand-canonical partition function.26,53 Here R = NAkB is the ideal gas constant and Bj(T ) are the (temperature- dependent) jth virial coefficients with B1 = 1. For an ideal gas, the compressibility factor Z becomes Z = 1 (hard-sphere model with ϕ(r) = 0 for r > rs and ϕ(r) = ∞ for r ≤ rs). Despite the straightforward analytical defini- tion for the virial coefficients, there are difficul- ties concerning the convergence of the series ex- pansion54–56 for small molar volumes (low tem- peratures or high pressures, or close to the liq- uid phase), and the complexity of calculating the higher-order virial coefficients.57,58 Never- theless, most gas phase simulations employ the (12,6)-Lennard-Jones potential, as basic infor- mation on f(P, Vm, T ) can already be obtained from such a simple model. The dominant second virial coefficient B2(T ) for atomic interactions has a very simple form,59 B2(T ) = 2π ∫ ∞ 0 dr r2 ( e−ϕ(r)/kBT − 1 ) , (6) with the two-body potential ϕ(r) such as in (1), and the property that at the Boyle tempera- ture TB, where a real gas behaves ideally, we have B2(TB) = 0. There are closed-form ex- pressions for B2 in terms of standard functions for a LJ potential.2,26,60,61 For example, for the (12,6)-LJ potential one obtains the simple series expansion,2,26 B2(T ∗) = −2π 3 NAr 3 e ∞∑ i=0 2i 4i! Γ ( 2i− 1 4 ) T ∗ 2i+1 4 (7) with the temperature T ∗ = kBT/ϵ in reduced fFor example, Jahn-Teller distortions require higher than two-body forces. units. For the hard-sphere model we simply get a temperature-independent term of B2 = 2πσ3/3, where the van der Waals radius is de- fined in (4). An excellent discussion on the sec- ond virial coefficients for LJ systems is provided by Heyes et al,62 and a historical overview and relevant data are provided by Stephan and Deit- ers.63 The 2nd and 3rd virial coefficients as a function of the temperature are shown in Fig- ure 4. Figure 4: Virial coefficients B2(T ) and B3(T ) as a function of the temperature for the (12,6)-LJ po- tential (in reduced units). The Boyle temperature with T ∗ B = 3.417927982 is indicated by a dashed line. The data are taken from ref. 63. In 1924, Lennard-Jones used his potential and obtained B2(T ) for various choices of exponents (n,m) and a series of different temperatures.2 By comparison with experimental data he was able to fit the constants An and Bm in eq.(1). He also determined the viscosity for gaseous ar- gon. In 1938, Buckingham studied the EOSs for helium, neon and argon using (n, 6)-LJ po- tentials, as well Slater repulsive potentials.64 At the time, these calculations were inconclu- sive concerning the best choice of the poten- tial form. Since then, experimental data and theoretical calculations have become far more accurate and the LJ potential has been shown to be a reasonable first approximation for most weakly interacting gases, especially the rare gas elements. A comprehensive review on the per- formance of the LJ potential for the rare gases is provided by Rutkai et al.65 Belov discusses the different approximations for the compress- 5 ibility factor Z using the (12,6)-LJ potential in- cluding higher-order contributions in the virial expansion.66 For the critical parameters of a LJ gas see Majumdar and RamaRao.67 It is clear that any improvement on EOS simulations re- lies on more realistic interaction potentials in- cluding higher than two-body terms, such as the Axilrod-Teller-Muto potential.68,69 Concerning the EOS for mixtures of gases, as early as in 1927 Lennard-Jones and Cook con- sidered few atomic and molecular gas mixtures where experimental data were available.70 Us- ing a (n, 6)-LJ potential they found generally good agreement with experimental data with the LJ coefficients fitted to experimental gas phase data. A critical analysis of the perfor- mance of LJ systems for gas mixtures is pro- vided by Rutkai et al.65 Figure 5: Argon isotherms P (V, T ) at tempera- tures T=140, 148, 150.7, 153, 155, 157, 160, 170, 180, 190, 200, 250, and 295 K as computed from various 2+3-body potentials: ELJ-EAT, LJ-AT, HFD-EAT, MTT-EAT, and MTT-JHBV (in red, cyan, green, blue, and black, respectively); Empty black symbols are experimental and saturated va- por pressures;71,72 dashed black linear splines inter- polating the experimental data points are to guide the eye. Figure and text are taken from ref. 57 © American Institute of Physics. The performance of the LJ potential is best illustrated for the case of gaseous argon, which (unlike helium) still behaves in a semi-classical way. If we use the LJ Boyle (reduced) tem- perature63 of T ∗ B = 3.417927982 and the most accurate value for the dissociation energy73 of Ar2 with ϵ=1.1885 kJ mol−1, we obtain for ar- gon TB = 488.57K compared to the experimen- tal value of 407.76K.71,72 This is a rather large discrepancy, which has a significant influence on the EOS isotherms. Figure 5 shows some more recent calcula- tions by Wiebke et al.57 for argon using a variety of different two-body potentials (i.e. LJ with the parameters given by Jäger,74 an extended LJ version (ELJ) using an inverse power expansion of the internuclear distance, the Aziz potential (HFD),43 and a modified Tang–Toennies potential MTT74), plus three different three-body potentials (i.e. an Axilrod- Teller-Muto (AT), a three-body ATM potential with (MAT) and without (AT) correction for the short range,75 and a more accurate three- body Jäger-Hellmann-Bich-Vogel (JHVB) po- tential76), and a virial expansion up to fourth order. The LJ-AT model predicts pressures that are too small, and condensation happens at Tc = 174.26K compared to the experimental value of 150.68K.71,72 All the other potentials are in rather good agreement with experimen- tal values, except for those close to the liquid phase, which will be discussed further below. 3 The Lennard-Jones Po- tential in Solid-State Sim- ulations 3.1 Lattice sums If we sum over all the interactions between the atoms in a (Bravais) lattice using a (n,m)- LJ potential we obtain the following expression for the cohesive energy for a monoatomic lat- tice,3,77,78 Ecoh = nmϵ 2(n−m) [ Ln n (re R )n − Lm m (re R )m ] (8) where the so-called lattice sums Ln can be de- fined such that the distance R reflects the near- 6 est neighbor distance in the crystal. The lat- tice sums Ln contain expressions for quadratic forms or functions and depend on various cell parameters, i.e. Ln(xi) where xi are the lattice constants (and Wyckoff positions in the case of multi-lattices).79,80 Lattice sums for inverse power potentials are related to the more general Epstein zeta function,30,81–83 Z (c;B; u⃗, v⃗) = ∑ z⃗∈ZN ′ e2πiu⃗·Bz⃗ |Bz⃗ + v⃗|−c (9) with c ∈ C, u⃗, v⃗ ∈ RN , where N is the dimen- sion, and B is a N × N real positive definite matrix and v⃗ a shift vector required for multi- lattices. The prime on the sum indicates that we avoid division by zero. The Epstein zeta function can be seen as an extension of the Rie- mann and Hurwitz zeta functions. In princi- ple, the Epstein zeta function can be evaluated directly, but this calculation is difficult,84 and the function is therefore not in the repertoire of standard mathematical software packages. For lattices in dimension N = 3, the generator matrix B contains the three lattice vectors b⃗i of the Bravais lattice, and G = B⊤B = (⃗bi · b⃗j) is the corresponding symmetric and positive defi- nite Gram matrix. v⃗ is the shift-vector used for the lattice points at specific Wyckoff positions, otherwise we have for Bravais lattices v⃗ = 0⃗ (for multi-lattices one obtains a linear combi- nation of Epstein zetas functions for the lat- tice sums). For (8) we simply have the relation Ln = Z (n;B, 0⃗, 0⃗). This is best illustrated by a few examples. We consider a simple cubic (sc) lattice with lattice vectors b⃗⊤1 = (1, 0, 0), b⃗⊤2 = (0, 1, 0), b⃗⊤3 = (0, 0, 1). This geometry results in B = I, where I is the unit matrix. We set u⃗ = v⃗ = 0⃗ and obtain the following lattice sum (n > 3) from (9), Lsc n = ∑ i,j,k∈Z ′ ( i2 + j2 + k2 )−n 2 (10) and we have limn→∞ Lsc n = 6, which is the number of nearest neighbors (kissing number or coordination number) for the simple cu- bic structure. These lattice sum expressions can become quite lengthy for more compli- cated lattices.29,80,85 Unfortunately, this lattice sum (and most others) is slowly convergent for small values of the exponent n. This problem was already noticed by Lennard-Jones in 1924,3 e.g. for n = 4 he obtained Lsc 4 = 16.54 compared to the correct value of 16.5323159597617 to double precision accuracy.80 To transform such lattice sums into fast converging series (or for very few cases into analytical formulas) is one of the major research areas in lattice theory.29,80 There are various techniques for achieving fast convergence by representing the series in terms of infinite series containing Bessel functions, and the reader is referred to refs. 80,81. A table of lattice sum values for the simple cubic (sc), body-centered cubic (bcc) face-centered cubic (fcc) lattices and for the hexagonal closed pack- ing (hcp) can be found in ref. 80, and a Fortran program to compute these lattice sums is freely available.86 An interesting modification of the lattice sum (10) comes from choosing u⃗ = (1, 1, 1) in the exponential of the Epstein zeta function. This choice results in the following alternating series, Mn 2 = ∑ i,j,k∈Z ′ (−1)i+j+k ( i2 + j2 + k2 )−n 2 (11) which for n = 1 is known as the Madelung con- stant32 used for describing the ionic Coulomb interactions (n = 1) for NaCl-type crystals. This infinite series is well known to be condi- tionally convergent, which makes direct evalu- ation difficult.87 Instead, one can use different series representations that are fast converging88 such as89–93 M 1 2 = −12π ∑ i,j∈N sech2 [π 2 √ (2i− 1)2 + (2j − 1)2 ] (12) In fact, the Madelung constant has been eval- uated to 60 significant digits, i.e. M 1 2 = −1.747564594633182....94 A fast converging ex- pansion in terms of Bessel functions in any di- mension has been provided recently.95 Fast converging series in terms of Bessel func- tion expansions have also been derived for the lattice sums of the sc, bcc and fcc lattices,29,80 and more recently for the hcp structure.85 Once 7 obtained, they can be used to calculate impor- tant bulk properties, such as the lattice con- stant, cohesive energy, pressure, bulk modulus, Einstein frequency and the Grüneisen parame- ter for (n,m)-LJ potentials.78 For example, for the cubic lattices (sc, bcc, fcc and hcp with c/a = √ 8/3) we obtain the simple relation for the cohesive energy of a (n,m)-LJ potential in terms of powers of the lattice sums Ln at the optimized lattice parameter,96 Ecoh LJ = ϵf(n,m) = ϵ 2 (Lm) n n−m (Ln) m n−m (13) Using this formula for the fcc lattice one obtains f fcc(12, 6) = 8.610200157 close (but below!) the hcp value with fhcp(12, 6) = 8.611069732.97 Furthermore, one can derive minimum energy paths for solid-to-solid phase transitions, such as the martensitic bcc→fcc Bain transforma- tion.96,98,99 However, most studies so far use di- rect summation techniques (employing, for ex- ample, the Ewald sum decomposition100–102 or estimating the remainder of the lattice sum) in MD simulations to obtain (P, T ) phase di- agrams for the most widely used (12,6)-LJ po- tential103 as shown in Figure 6. Interestingly, Figure 6: (P, T ) phase diagram for a LJ solid us- ing a (12,6)-LJ potential (from ref. 103). Reduced units are used for the pressure and temperature. © American Institute of Physics the (P, T ) phase diagram shows that the low temperature and pressure phase for the (12,6)- LJ potential is hcp, contrary to experimental X- ray measurements for neon or argon.104,105 This result arises because f fcc < fhcp in equation (13) for the (12,6)-LJ potential (see discussion below). Only for small values m < 5.705, can the fcc phase become more stable than the hcp phase, see Figure 7.78 Figure 7: Relative difference in cohesive energies ∆fcc hcp(n,m) = 1 − Ehcp LJ /Efcc LJ between the fcc and hcp phase at the optimized nearest neighbor dis- tances as a function of the the exponents (n,m) of the LJ potential. From ref. 78 © American Chem- ical Society In 1924, Lennard-Jones considered the fcc phase of solid argon in his lattice sum calcu- lations using a (131 3 , 4)-LJ potential (for the lattice sum Lfcc 4 Lennard-Jones already applied a fast convergent Bessel function expansion).3 He obtained a solid-state bond distance of Re=3.89Å , compared to the experimental X- ray value of 3.756Å.104 A more accurate calcu- lation using this potential and the correct coeffi- cients obtained from (3) (using for diatomic ar- gon the equilibrium distance re = 3.7624 Å and dissociation energy ϵ=1.1885 kJ mol−1)73 gives a significantly shorter distance nearest neigh- bor distance for the solid of Re=3.475 Å. The large difference compared to Lennard-Jones’ re- sult comes from the fact that his coefficients An and Bm could not be accurately obtained from gas phase data2 at that time, and the fact that the r−4 potential is far too attrac- tive. A (12,6)-LJ potential gives a more re- alistic nearest neighbor distance of Re=3.654 Å. In 1931, Lennard-Jones obtained a nearest neighbor distance of Re=3.74 Å using a (12,6) 8 LJ potential from an improved set of coeffi- cients An and Bm, much closer to the experi- mental value.23 For comparison, a more realistic diatomic potential energy curve of Patkowski and Szalewicz73 gives a nearest neighbor dis- tance of Re=3.686 Å for solid argon. Only if one includes higher than two-body interactions and zero-point vibrational effects does the an- swer approach the experimental bond distance to high accuracy.106 Accurate cohesive energies were not available in the early part of the last century, otherwise Lennard-Jones would have realized that an r−4 potential leads to severe over-binding. Using the (131 3 , 4)-LJ potential we get Ecoh= 20.69 kJ mol−1 for fcc argon, compared to 10.23 kJ mol−1 for a (12,6)-LJ potential and 7.72 kJ mol−1 experimentally. Again, a more realistic two-body potential73 gives 9.20 kJ mol−1. Most of the difference from the experimental value comes from neglect of three-body terms and zero-point vibrational contributions. If all these contributions are taken into account, the result becomes very close to the experimental value, within a few 10−3 kJ mol−1 accuracy.106 Concerning the difference between the two close-packed structures fcc and hcp for argon, a (12,6)-LJ potential prefers hcp over fcc by a tiny amount of ∆Ecoh = Efcc coh − Ehcp coh = 8.70 × 10−4 kJ mol−1.79,107 Again, a more realistic two-body potential gives ∆Ecoh=1.21×10−3 kJ mol−1. It was recently shown that zero-point vibrational energy contributions including phonon disper- sion are required to stabilize the fcc over the hcp phase with a value of ∆Ecoh = −8.81×10−3 kJ mol−1.106 The (12,6)-LJ fcc/hcp phase sta- bilities have been investigated by Jackson et al. using a lattice-switch Monte Carlo method shifting the corresponding layers in the closed packed structures.108 Stillinger derived the re- duced pressure at which hcp and fcc become en- ergetically degenerate, P ∗=878.486, which con- verts to about 37GPa for solid argon.79 Concerning the low-temperature bcc phase, it is metastable or even unstable compared to the fcc structure for a (n,m)-LJ potential, even at high pressures.96,109,110 This is easily under- stood from the fact that a two-body poten- tial will maximize the number of nearest neigh- bor interactions which is 8 for bcc but 12 for fcc. Only at temperatures close to the melting point does the bcc phase dominant because of entropic contributions, in agreement with ba- sic Landau theory.103,111,112 It is interesting to note that the entropy difference between the two closed-packed phases fcc and hcp for a (12,6)-LJ potential with ∆S = Sfcc − Shcp = 0.0014754kB, in good agreement with the hard- sphere model simulations by Mau and Huse who reported ∆S = 0.001164kB.113 Hence fcc has a higher entropy than hcp, which explains the hcp→fcc phase transition at higher tempera- tures, as shown in Figure 7. 3.2 Surfaces There are a number of surface and adsorption studies for Lennard-Jones systems, a few only will be mentioned here. Already in 1932 Lennard-Jones studied ad- sorption and diffusion processes on surfaces using his potential.114 Broughton and Gilmer looked at the surface free energy and stress of a LJ crystal.115 Baidakov et al.116 showed that the temperature dependent surface energy σ(T ) for the solid-vapor interface for a fcc crystal is smallest for the [111] place, and we have the in- equality σ[111](T ) < σ[100](T ) < σ[110](T ). This preference for the [111] surface can be under- stood from the nearest neighbor interactions as the layer consists of a dense hexagonal packed sheet of atoms. However, for the surface-liquid interface one obtains σ[111](T ) < σ[110](T ) < σ[100](T ).117 As an example for adsorption stud- ies we mention the simulations of surfaces and interfaces of various fcc metals including steel by Kanhaiya et al.118 3.3 Glassy Landscapes Binary Lennard-Jones (BLJ) systems are pop- ular for simulations of model glassy sys- tems,119–146 since they usually avoid crystalli- sation on the timescales accessible in standard molecular dynamics runs. Early investigations compared simulations with the predictions of mode-coupling theory, considering quantities such as the incoherent part of the intermediate 9 scattering function.123–125 Alternative schemes have been used to treat the finite range cut- off.123,126 These alternatives are probably incon- sequential for liquid or solid simulations, but can affect results for the liquid-vapour coexis- tence curve.41 For a number density of 1.2σ−3 AA, the standard BLJ system that is usually con- sidered exhibits properties characteristic of a fragile glass.130 The relaxation dynamics have been associated with temperature ranges where different regimes are sampled in the under- lying energy landscape,126 and interpreted in terms of ergodicity metrics.121,122 The land- scape view has also been used to understand super-Arrhenius diffusion in terms of correla- tion effects147,148 and cage-breaking.149,150 In- sight into thermodynamic properties has been obtained from parallel tempering simulations151 and pinned systems, where subsets of the atoms are fixed.152 One particular BLJ parameterisation was introduced to represent the metallic glass Ni0.8P0.2.153 In fact, this mixture exhibits a phase separated crystal containing fcc and CsCl structures in contact.154 The lowest en- ergy crystal located in the absence of phase separation154,155 can be related to the struc- ture of Ir(UC)2. It is interesting to compare the landscape in a glassy region of configu- ration space149 (Figure 8) with the crystalline region characterised for the LJ solid at fixed vol- ume156 (Figure 9). The complexity of the glassy landscape, featuring a multitude of amorphous minima separated by high barriers,149,157,158 is clear, and explains how the system becomes trapped on cooling in one particular local fun- nel for the observation time scale of relevant experiments. In particular, the construction in Figure 8 highlights a higher order structure de- fined by cage-breaking rearrangements, which are required for atomic diffusion.156 Some fur- ther discussion of how global thermodynamic and dynamic properties are encoded in the en- ergy landscape is provided in section 6. 4 The Lennard-Jones Po- tential in Liquid-Phase Simulations The liquid state is far more complicated to simulate compared to both the (ordered) solid state or the (disordered) gaseous state, as the mean nearest neighbor distance (or mean free path) becomes rather small and comparable to the solid state. Hence, for most systems a simple two-body treatment to account for all the atomic interactions becomes insufficient.50 In addition, we lose the symmetry that repre- sents a computational advantage for any crystal structure treatment. Because of these difficul- ties, many aspects of the fluid part of the phase diagram (see Figure 6) are poorly understood. The liquid phase is usually referred to the state where a bulk material does not retain its shape similar to the gaseous state. However, already in 1903 Mie pointed out that, contrary to van der Waals’ original assumption, the ther- modynamic properties of the liquid state are closer to the solid state than the gas phase, es- pecially for metals.14 This observation is nicely illustrated for the radial distribution functions in the three different phases of argon, see Fig- ure 10. As a result, melting simulations become problematic as the free energy curves G(T ) of the liquid and the solid run almost parallel, hence it becomes difficult to exactly determine the crossing point between them. This issue does not arise for the liquid-vapour free ener- gies, so that boiling points are easier to deter- mine computationally.161 Despite all these difficulties, the LJ potential has become invaluable for liquid-phase dynam- ics simulations.40,41,162 Due to its importance in liquid state simulations, the LJ liquid is some- times called Lennard-Jonesium.163 An overview of the LJ equations of state for the liquid state has been given by Stephan et al.164 includ- ing a data collection.165 For a critical review on the Lennard-Jones-Devonshire (LJD) cell model for liquid phase simulations see Wood and Parker.166 For computational purposes it is often convenient to truncate and shift the LJ 10 Figure 8: Disconnectivity graph159,160 for a binary Lennard-Jones system of 60 atoms at fixed volume, with cage-breaking transition states removed. The connected minima are coloured in sets according to the potential energy at which they are disconnected from the full graph,149 revealing a higher order organisation in the landscape. Figure 9: Disconnectivity graph for 864 Lennard- Jones atoms in a cubic periodic box with number density 1.2σ−3 (low energy region). Four crys- tals fit precisely within the box: fcc, hcp, and the combinations (fcc+2hcp)/3 and (2fcc+hcp)/3. The high energy funnel corresponds to misaligned fcc and hcp layers with a layer of bcc atoms.156 potential (LJTS),40 ϕLJTS(r) = ϕLJ(r)− ϕLJ(rs) (14) for r ≤ rs and ϕLJTS(r) = 0 otherwise (with rs ≈ 2.5σ), or to truncate and add a simple long-range correction term. Either way, the truncation affects properties167 like the loca- tion of the critical point or the surface ten- sion.168 For a detailed discussion see for ex- ample Biscay et al.169 For example, Loach and Ackland170 demonstrated that for truncated LJ potentials the stability of Barlow dense pack- ings171 (mixture between fcc and hcp stackings) critically depend on the cut-off radius. The truncation procedure is also used for construct- ing the Weeks-Chandler-Andersen potential in thermodynamic perturbation theory.172 Figure 11 compares the gas-liquid coexistence line for the lighter rare gases with those ob- tained by a (12,6)-LJ potential.173 The Fig- ure shows that the (12,6)-LJ potential performs quite well for the coexistence line in the defined dimensionless units. Figure 10: The radial distribution functions g(r) for solid (T = 50 K), liquid (T = 80 K), and gaseous argon (T = 300 K). The radii are given in reduced units (σ = 3.822 Å). © Creative Com- mons. 11 https://commons.wikimedia.org/wiki/File:Simulated_Radial_Distribution_Functions_for_Solid,_Liquid,_and_Gaseous_Argon.svg https://commons.wikimedia.org/wiki/File:Simulated_Radial_Distribution_Functions_for_Solid,_Liquid,_and_Gaseous_Argon.svg Figure 11: Temperature T/Tc vs. density ρ/ρc gas-liquid coexistence line for He, Ne and Ar173 in comparison with the LJ model174 and the cut- and-shifted LJ model175 used in MC simulations. Graph taken from ref.173 ©American Institute of Physics. The liquid-gas phase equilibrium curve in the (T, P ) plane ends at the critical point Tc, Pc. The most accurate critical parameters (in re- duced units) reported so far for the (12,6)-LJ potential are: vapor–liquid–solid critical point T ∗ c = 1.321 ± 0.007, ρ∗c = 0.316 ± 0.005 and P ∗ c = 0.129 ± 0.005, and the triple point T ∗ tr = 0.690 ± 0.005.165 A list of literature references for thermodynamic parameters of the (12,6)-LJ potential can be found in ref. 165). Ahmed and Sadus176 looked at solid-liquid phase equi- libria and triple points for the general (n, 6)- LJ potentials. They found that the temper- ature for the triple point reduces to T ∗ tr = 0.482 if a hard-wall repulsive potential is cho- sen. However, concerning the performance of LJ potentials for liquid simulations, Pruteanu et al. pointed out that the (12,6)-LJ poten- tial produces highly over-structured liquids.177 The experimental data could only be repro- duced by a softer potential than implied by the repulsive r−12 term and a shallower mini- mum, which seems to be a general problem of the simple (12,6)-LJ potential, due to neglect of higher-order many-body effects. For exam- ple, Vrabec et al. improved on the LJ model for vapor-liquid equilibria by adding a point- quadrupole pair potential.178 They obtained very good agreement with thermodynamic data for a large range of atomic and molecular liq- uids. Here we note that supercritical liquid–gas boundaries around the critical point are lines in the (T, P ) diagram that distinguishes more liquid-like or more gas-like states of a supercrit- ical fluid.179,180 They comprise the Widom line, the Fisher–Widom line, and the Frenkel line. For a detailed discussion see refs.181–183 Concerning the use of certain algorithms in MD or MC melting simulations, one often re- quires corrections for overheating and for the limit to infinite particle number.184,185 The problem of overheating can be avoided if the solid and liquid phases are treated separately to obtain the free energies Gsolid(T ) and Gliquid(T ) at selected temperatures near the melting point, where the two free energy curves cross,161 or by algorithms such as the interface-pinning method,186 which is however computationally very demanding. A comparison of LJ melt- ing and boiling points with experimental val- ues for the rare gases has been provided by Sun et al.187 and Matsumoto,188 which shows gener- ally good agreement with experiment. Melting temperatures can also be estimated from clus- ter simulations189 and extrapolation of the clus- ter size to the bulk limit.184 This method gave T ∗ m = 0.676, close to the triple point value, as one should expect.190 For a recent discussion on this topic see Kataoka and Yamada.191 From a simple scaling relation using the em- pirical scaling parameter fm we obtain for a LJ potential, kBTm = fmϵ = fmfLJEcoh = λmEcoh, (15) i.e. the melting temperature is roughly propor- tional to the cohesive energy, as one naively expects. For example, for a (12,6)-LJ system we have λm = 12.16 obtained from parallel- tempering MC simulations190 and fLJ = 0.116 from the lattice sums of the fcc crystal.97 This linear relationship is approximately fulfilled, as shown in Figure 12. The rather large devia- tion from the ideal LJ behavior is due to the fact that many-body effects are not correctly accounted for in most of the elements in the periodic table. Nevertheless, the (12,6)-LJ po- tential gives a relatively good description for the rare gas melting points, which occupy the lower left corner in Figure 12, as shown in more detail 12 Figure 12: Upper figure: Linear relationship of the cohesive energy with respect to the known melt- ing temperatures for 91 elements in the periodic table compared to the prediction of the (12,6)-LJ potential. Lower figure: Same as upper figure but for the rare gas elements only, which demonstrates the excellent agreement with the prediction of the LJ potential. See ref. 192 for details and further references. in the lower part of the figure. A similar empir- ical scaling law can be obtained approximately for the boiling points.190 Here we mention that in 1937, Lennard-Jones calculated the boiling point to be approximately T ∗ b = 0.72 using a (12,6)-LJ potential.193 For the performance of the LJ potential in reproducing bulk and shear viscosities of the rare gases, see for example Fer- nández et al.194 5 The Lennard-Jones Po- tential in Molecular Me- chanics Simulations Force fields used in molecular simulations consist of molecular geometric structure and interaction parameters, using both bonding (stretching, bending, torsional modes) and non-bonding (long range) interactions (ionic and dispersive terms). The use of the (12,6)- LJ potential in biomolecular force-field meth- ods195–197 is ubiquitous for the description of the intra- and intermolecular dispersive type interactions, from all-atom models of proteins and nucleic acids, to coarse-grained united atom models, such as UNRES,198 and even to minimal coarse-grained models of chro- matin.199 Many alternative modifications and paramterisations of such classical force fields have been discussed, including for example po- larization effects,200,201 designed for specific classes of molecules or materials. For exam- ple, for cyclic alkanes such as cyclopropane, Lustig202 proposed a three-center LJ model fitted to experimental values for vapor pres- sure and saturated liquid density. The number of different force fields available in the litera- ture is rather large, and reviews are provided, for example in refs.203–206 The Lennard-Jones form is a convenient way to account for disper- sion, and together with Coulomb (and polariza- tion) interactions, usually constitutes the long- range part of biomolecular force fields such as AMBER,207 CHARMM,208 OPLS,209 and the GROMACS program, which includes a variety of force fields.210 Exploring the multidimensional energy land- scape of a large biomolecule remains a ma- jor computational challenge. For example, the correct description of the tertiary structure in protein folding requires dispersive interac- tions.211 Hobza and coworkers212,213 also found that DNA structures collapsed when the disper- sion component of an empirical potential was switched off (Figure 13).213 Weak interaction forces such as dispersion may therefore have played an important role in the evolution of na- tive biomolecules and may be seen as a neces- sary condition for life to exist. One of the most active fields in force-field methods including interatomic/molecular inter- actions through the LJ potential is in protein folding. Even this more specialised literature is vast, and beyond the scope of this perspective. We refer instead to review articles such as refs. 214,215 and references therein. We only men- tion that different force fields can lead to rather different tertiary structures,216 and one has to carefully test the parametrization for specific biochemical applications. On a more funda- mental level, to find the global minimum for the biologically active protein or enzyme is by 13 Figure 13: Dynamical simulation of a DNA oligomer with molecular mechanics and dispersion interactions switched off, starting at the optimized structure including dispersion.213 Simulation time was 100 ns. Figure taken from ref. 213. © Amer- ican Chemical Society. no means a trivial problem, as highlighted by Levinthal’s paradox.217,218 For molecular sys- tems we generally expect the number of lo- cal minima to grow exponentially with the number of atoms.219,220 Nevertheless, evolved biomolecules can reliably locate their native state, otherwise life as we know would not ex- ist. Similarly, the observation of magic num- ber clusters (see discussion in the next section) and crystallisation is not compatible with a random search, even on the fastest vibrational timescale. Instead, the corresponding energy landscapes are organised such that the search is not random,221–225 with funneling properties corresponding to kinetically convergent path- ways.226 These properties have been analysed in detail for LJ clusters, as discussed below. In particular, the funnelling organisation can be visualised, and an example is presented in Figure 14. Clusters have played a fundamen- tal role in understanding how emergent kinetic and thermodynamic properties are encoded in the underlying landscape, and we elaborate in detail on this theme throughout the next sec- tion. Similar considerations hold for the nucleation problem.g However, significant progress has been made with the development of rare events techniques, and again the LJ potential has played a key role in the development of these gIn 1949 P. R. Rowland stated that The gap between theory and the experimental approaches to nucleation has been too wide. The subject is still in the alchemi- cal stage. Half a century on, there is still a large gap between experiment and theory.227 algorithms.228–237 6 The Lennard-Jones Po- tential in Cluster Simula- tions R. Stephen Berry once described clusters as mi- croscopic laboratories for testing new theory and simulation methodology.238 Clusters of LJ atoms have served this purpose well, and re- main a popular choice for such tests. Some of the earliest work was motivated by efforts to un- derstand nucleation from small clusters to the solid state,239–253 where noble gases modelled using the LJ potential presented more tractable targets than water. The finite size analogue of the first-order solid-liquid phase transition is in- teresting in its own right,254–256 and proved to be a useful testing ground for enhanced thermo- dynamic sampling, as outlined below in section 6.2. Short surveys of results for global optimisa- tion, enhanced thermodynamic sampling, and rare event dynamics are presented in the three subsections below. Depending on the number of atoms, N , clusters bound by the LJ poten- tial, denoted LJN , can exhibit landscapes that cover a wide range of complexity in terms of or- ganisation, which is reflected in the difficulties associated with exploring and sampling them. The emergent observable thermodynamic and kinetic properties of a system are encoded in the underlying energy landscape, and extract- ing one or more these observables is the pri- mary aim of most simulations. Certain LJ clus- ters exhibit landscapes that produce efficient relaxation to the global minimum, while oth- ers support broken ergodicity, distinct heat ca- pacity features, and rare events. The univer- sal properties associated with these landscapes enable clusters to provide fundamental insight into phenomena ranging from protein folding to molecular switches, as suggested by Berry in his microscopic laboratory view. Determining the global potential energy min- imum is usually the first part of an energy land- scape exploration, since this structure corre- 14 sponds to the free energy minimum at low tem- perature in equilibrium. Global optimisation approaches are summarised in section 6.1. Pre- dicting equilibrium thermodynamic properties can be attempted once the global minimum is known. In particular, calculating the heat ca- pacity and relating peaks to the structure of the underlying landscape can provide detailed insight into how the equilibrium populations of alternative morphologies shift with tempera- ture (see thermodynamics section 6.2). Multi- funnel landscapes often produce characteristic signatures in the heat capacity, and also sup- port multiple relaxation time scales. Extracting the corresponding rates may require rare events methodology, as discussed in 6.3. Magic number clusters, corresponding to sizes that exhibit pronounced intensity in molecular beams,257–272 occur where complete Mackay icosahedra273 are the global poten- tial energy minimum (N = 13, 55, 147, 308, . . . ,(10n3 + 15n2 + 11n+ 3) /3 for n = 1, 2,. . .). Structures based on close-packed or decahedral geometries are not competitive at these sizes.274–276 The potential energy land- scape277–279 for magic sizes with particularly stable structures has self-organising proper- ties, where relaxation to the global minimum is guided downhill over relatively low barri- ers. This organisation is referred to as a single funnel, following analysis of protein folding in terms of kinetically convergent pathways,226 which provides a resolution of the Levinthal paradox.221–225 These single funnel landscapes are associated with efficient relaxation to the global minimum. An example is shown for LJ13 in Figure 14, as discussed below. It is multifunnel landscapes, with compet- ing alternative morphologies separated by a high barrier, that produce challenges for sam- pling, with broken ergodicity and rare event dynamics, associated with multiple relaxation time scales.280,281 The contrast between single and multifunnel landscapes becomes particu- larly clear when we visualise the landscape us- ing disconnectivity graphs,159,160 which provide a natural way to translate results from calcu- lations based on the underlying energy land- scape.277 In this approach the potential en- Figure 14: Disconnectivity graph159,160 for the LJ13 cluster and the global minimum Mackay icosa- hedron.273 This landscape corresponds to a single funnel. The scale bar in red is one ϵ. ergy surface is coarse-grained into local min- ima and the transition states282 that con- nect them,277,283–285 producing a kinetic tran- sition network.278,286–288 A single funnel land- scape suggests that the system will support effi- cient self-organisation, in terms of relaxation to the global minimum, while a molecular switch would probably correspond to a double fun- nel. Examples are illustrated for the LJ13 and LJ38 clusters in Figures 14 and 15. In these graphs the branches terminate at the energies of local minima, and the vertical scale is the potential (or free) energy. At regular energy thresholds we test whether minima can inter- convert without exceeding the threshold, and merge the corresponding branches. The seg- regation into disjoint sets of minima at each threshold is based upon a database of minima and transition states, which enables us to lo- cate the lowest energy path between any pair of minima.159,160 Returning to Berry’s picture of clusters as 15 Figure 15: Top: Disconnectivity graph159,160 for the LJ38 cluster, which features a double funnel en- ergy landscape. Minima associated with the fcc and icosahedral regions of configuration space are coloured red and blue, respectively and the lowest energy structures of each family are shown below the graph. Bottom: first passage time distribu- tions for relaxation to the fcc and icosahedral re- gions from a higher energy minimum, P(ln t). The mean first passage time as a function of the obser- vation time scale, T (tobs), exhibits steps that line up with the peaks in P(ln t).289 microlaboratories, the LJ potential has formed the basis for a diverse range of studies, rang- ing from dimensionality reduction strategies for global optimisation,290 to fundamental ques- tions about the organisation of the landscape in terms of small world291 properties.292,293 The corresponding networks have also been anal- ysed in terms of scale-free behaviour,294,295 but the results are inconclusive.296 Structural phase diagrams for homogeneous297,298 and bi- nary299–301 LJ clusters have been characterised, along with the effects of anharmonicity,302–307 rotation,308 three-body terms,309 and quantum corrections.307,310,311 The role of the underly- ing potential energy surface in chaotic dynamics has revealed interesting effects associated with passage over transition states.312–320 In the fol- lowing sections we provide a further overview of research that has benefited from experiments that employ LJ clusters, focusing on structure, thermodynamics, and kinetics. 6.1 Structure and Global Optimi- sation The low-lying, energetically favourable struc- tures of LJ clusters are interesting in their own right: as discussed in the preceeding sections, the LJ potential is not an accurate represen- tation of any particular system, but many of the physiochemical properties it supports are widely reflected in diverse applications. Iden- tifying these structures requires global optimi- sation procedures, and the simplicity of the LJ potential makes the corresponding clusters an ideal testing ground. Various databases of pu- tative global minima are available online, as surveyed by Kiessling,321 who applied an op- timality condition for clusters in the size range 2 ≤ N ≤ 1610 and identified a misprint. The mean first encounter time for the global minimum provides an important benchmark for alternative global optimisation schemes. Clus- ters with double-funnel landscapes are good test cases; some benchmarks can be found at the optbench web site.324,325 The basin-hopping approach326–328 has proved to be very useful, with applications extending from clusters to biomolecules, condensed matter, and neural 16 MINIMA TRANSITION STATES TS1 linear TS2 linear equilateral triangle linear TS3 Figure 16: Visualisation of the solutions obtained using an eigenvector-following geometry optimisa- tion technique,322 searching for minima (left) and transition states (right).323 Each pixel corresponds to a starting geometry for a three-atom cluster in a cut through configuration space, and the colour reports on the particular stationary point that is located. For the chosen potential, which has an ex- tra three-body term, there are three distinct linear minima, each linked to the unique equilateral trian- gular minimum via an isosceles triangle transition state.323 network loss function landscapes.279 Thousands of applications have been presented, and nu- merous variants of the basic algorithm have been developed.329–332 The basic idea of basin- hopping is to propose moves between local min- ima and accept or reject them according to the property we wish to optimise,328 which is often the total potential energy. However, the free energy333,334 or a grand potential can also be used,335 and a generalised basin-hopping frame- work has been developed for optimisation in different metric spaces.336,337 Lamarckian ge- netic algorithms338,339 that feature populations of local minima can be considered in terms of basin-hopping with mutational moves. It is the formulation in terms of steps between lo- cal minima that appears to be the crucial fea- ture, and incorporation of approximate symme- try has proved to be particularly effective.340 To produce a more efficient algorithm for structure prediction in LJ clusters a variety of additional modifications to the basic algorithm are usually employed, as featured in the public domain GMIN program.341 For example, incor- porating some memory into the accept/reject procedure,342 and treating weakly bound sur- face atoms differently327 accelerate exploration, and were incorporated early on in GMIN. Formulating the problem in terms of steps be- tween local minima is advantageous for at least two reasons. First, the minimisation effectively removes downhill barriers on the landscape, ex- ploiting any funneling organisation. Atoms can pass through each other in the corresponding moves, because the structures that are proposed by initial perturbations are optimised before the energies are compared. There is a second, more subtle, effect: analysis of the thermodynamic properties for the transformed landscape of the LJ38 cluster reveals broadened thermodynamic transitions,343 where the occupation probabil- ities for the global minimum and competing structures overlap significantly over a wider temperature range. LJ38 features a double- funnel landscape (Figure 15),160,280,344–346 with competition between the global minimum trun- cated octahedron, with a face centred cubic (fcc) Oh symmetry, and incomplete Mackay icosahedra, where the lowest energy minimum is C5v. The contrast between the transformed and the original landscape can be understood by writing the partition function as a sum over local minima and comparing the terms that contribute to the stability of the fcc region, the defective icosahedra, and the higher en- ergy minima, which correspond to the liquid- like phase on melting. For the original land- scape, the equilibrium occupation probability exhibits two phase-like transitions from fcc to icosahedra, and then from icosahedra to liquid- like structures. These transitions have associ- ated heat capacity features and relaxation time scales, as discussed further below. In both cases the system progresses to a higher entropy region of configuration space as the tempera- ture increases, driven by both landscape en- tropy137,347–350 (the potential energy density of local minima) and vibrational entropy. Here, the mean vibrational frequencies decrease in the same order that the landscape entropy in- creases, reinforcing the transitions.343 In con- trast, for the transformed landscape the vibra- tional partition function for each minimum is replaced by the relative volumes of the basin of attraction, defined as the configuration space 17 for which steepest-descent paths lead to the minimum in question.277,351 These volumes de- crease from fcc to icosahedral to liquid-like min- ima, opposing the landscape entropy, broaden- ing the transitions. The landscape entropy still dominates at higher temperature, so that the melting temperature shifts systematically.343 The basins of attraction themselves are ac- tually interesting in their own right, both for analysis of configurational entropy,352,353 and in visualising the performance of minimisation al- gorithms.354 The basins corresponding to the steepest-descent equations are well-defined be- cause the uniqueness theorem applies to these linear differential equations.355 In practice, the true steepest-descent paths are relatively ex- pensive to calculate, and more efficient meth- ods are generally used, such as L-BFGS356 (lim- ited memory Broyden,357 Fletcher,358 Gold- farb,359 Shanno360) and FIRE.361 The LJ po- tential, augmented by an Axilrod-Teller three- body term,68 has been used to investigate how well these alternative methods reproduce the basins of attraction.354 The same approach was used in earlier investigations of algorithms to locate transition states, as illustrated in Figure 16.323,362 6.2 Cluster Thermodynamics and Enhanced Sampling Early investigations from Berry and coworkers considered coexistence of liquid-like and solid- like forms for small clusters.253,363–369 Short- time averages of molecular dynamics simula- tions revealed well-defined segments of the tra- jectory associated with higher or lower ki- netic energy, which correspond to solid-like and liquid-like regions of the potential energy sur- face in the microcanonical ensemble. Detailed analysis of simulations for small LJ clusters re- vealed van der Waals loops,370 and was con- firmed by accurate calculations of the total en- ergy density of states.371–373 Such features are forbidden in the bulk limit by the van Hove the- orem,374 where all ensembles agree, but not for finite systems, which includes astrophysics.375 The possibility of an S-bend in the microcanoni- cal caloric curve, ⟨T (E)⟩, predicted for LJ clus- ters, with an associated negative heat capac- ity, is realised in experiments for sodium clus- ters.376 The computational results also helped to guide theoretical efforts, which identified equivalent conditions for an S-bend to occur: a bimodal distribution in the canonical prob- ability distribution for the total energy in the range between the freezing (f) and melting (m) temperature, Tf < T < Tm; two inflection points for the entropy S(E); a double min- imum for the corresponding Landau free en- ergy; and three energies where ⟨T (E)⟩ = T , again for Tf < T < Tm. The microcanonical caloric curve then has turning points at Tf and Tm.277,377–379 Calculations based on weighted histogram analysis,380,381 jump-walking,382,383 and parti- tion functions constructed from databases of local minima,304,305 played an important role in understanding the thermodynamics of finite systems in terms of LJ clusters.371–373 A well- defined density of states is obtained for clus- ters confined to a container, or by restricting the partition function to the bound configura- tion space.377 The effect of container size has also been investigated systematically;373 the re- sults are consistent so long as the container is large enough to accommodate the solid-like and liquid-like minima, and not so large that atoms can evaporate. Double-funnel landscapes, exemplified by LJ31, LJ38, and LJ75, have provided an im- portant testing ground for thermodynamic sampling,277,298,384–391 because the competing morphologies associated with the two funnels are separated by a high barrier compared to the thermal energy kT where they have equal occupation probabilities. Enhanced sampling techniques are needed to tackle such broken er- godicity problems. The corresponding thermo- dynamic signature is a low temperature heat capacity feature, marking a finite size solid- solid transition.277,280,298,384,385,387,388,390–393 The smaller low-temperature Cv peak identified for harmonic densities of states343,394 becomes a shoulder in the much larger peak correspond- ing to melting when anharmonic effects are accounted for.384,385 The competing structures in these double funnels are themselves of in- 18 terest, complementing the global optimisation results described above. For LJ31 the com- petition is between Mackay and anti-Mackay overlayers.268,395 As mentioned above, for LJ38 the two funnels correspond to fcc and incom- plete icosahedral structures,160,280,344–346 while for LJ75 the global minimum is a Marks deca- hedron.396 The LJ31, LJ38, and LJ75 clusters have be- come particularly important benchmarks for enhanced sampling methods. Parallel temper- ing calculations can converge the density of states and corresponding thermodynamic prop- erties, such as Cv, accurately for the two smaller sizes, providing a reference for new algorithms, such as Wang-Landau sampling,397 simulated tempering,398 nested sampling,399,400 basin- sampling,350 superposition enhanced nested sampling,401 nested basin-sampling,402 and funnel hopping.403 Some particularly powerful techniques are obtained by combining thermo- dynamic sampling schemes with knowledge of the low-lying minima obtained from global opti- misation.350,401,403,404 These hybrid approaches exploit techniques that are efficient in surveying both the high and low energy regimes, where entropy and potential energy dominate the free energy, respectively. For example, this combi- nation of complementary approaches solves the lockout problem in nested sampling, by main- taining a live point in disconnected regions.401 Obtaining ergodic sampling for LJ75 is sig- nificantly harder than for the smaller clusters, because the barrier that separates the com- peting funnels is larger at the temperature where the corresponding structures are in equi- librium. The low temperature heat capacity feature proved difficult to converge even for 1011 Monte Carlo steps in an adaptive exchange parallel tempering scheme.391,405 However, this cluster has been treated successfully using an auxiliary harmonic superposition reference,390 as well as the basin-sampling350 and funnel hop- ping403 techniques. Basin-sampling also pro- vides the potential energy density of minima, which can be identified as the landscape en- tropy.350 Converged thermodynamic properties can generally be obtained with much less compu- tational effort for clusters with a single funnel landscape. However, parallel tempering has revealed interesting structure when applied to a larger magic number cluster, LJ309, with a premelting feature in Cv identified in terms of a combination of structural transformations.406 Figure 17: Heat capacity analysis for LJ31. The top figure shows the heat capacity calculated using basin-sampling parallel tempering (BSPT) com- pared to a reference calculation with the den- sity of states converged using parallel tempering (REF).350 The low temperature feature, corre- sponding to a transition between minima with anti- Mackay to Mackay overlayers, is magnified in the inset panel. HSA is the result for the harmonic su- perposition approximation, which is accurate at low temperature. The bottom panels show disconnec- tivity graphs159,160 for LJ31 highlighting the min- ima that account for 99% of the positive (in blue) and negative (in red) contributions to the heat ca- pacity features at kBT/ϵ = 0.0268 (left) and (b) kBT/ϵ = 0.329 (right).407 We conclude this section with a brief ac- 19 count of how LJ clusters have provided a test- ing ground for theory that enables heat capac- ity features to be assigned to transitions be- tween specific local minima.407 This thermo- dynamic assignment of competing structures in multifunnel landscapes complements the ki- netic analysis in the following section, where the multifunnel structure is associated with differ- ent relaxation time scales.289 Figure 17 shows the heat capacity for the LJ31 cluster, compar- ing basin-sampling and parallel tempering re- sults,350 which are in good agreement. The low temperature peak in this system is well sepa- rated from the melting peak, and is accurately reproduced by a harmonic vibrational density of states. In this harmonic approximation CV can be decomposed into sums over local min- ima, CV = κkB + kBT 2 gγ(T )>0∑ γ gγ(T ) 2/pγ(T ) + kBT 2 gγ(T )<0∑ γ gγ(T ) 2/pγ(T ), (16) with positive (first sum) and negative (second sum) occupation probability gradients, gγ(T ) = ∂pγ(T )/∂T , where pγ(T ) is the occupation probability of minimum γ at temperature T . κ is the number of vibrational degrees of freedom in the leading harmonic term. Ranking the con- tributions in each sum according to their mag- nitude provides a direct assignment of the heat capacity peaks in terms of occupation proba- bility transfer from minima with negative gra- dients to minima with positive gradients. Trun- cating the sums at a fixed percentage of the to- tal highlights the key minima involved in the transition; no structural order parameter is re- quired, which enables us to apply this approach beyond molecular science to machine learning loss landscapes.408 6.3 Dynamics and Rare Events Applications to analysis of chaotic dynamics were mentioned above. In this section we fo- cus on applications of LJ clusters in efforts to -12.7 -12.4 -12.2 -11.9 -11.7 -11.4 -11.2 -10.9 -10.7 -10.4 Figure 18: Disconnectivity graphs for LJ2D7 . Left: permutation-inversion isomers of the four local minima are collected together. Right: one of the atoms is tagged, lowering the permutational degen- eracy according to the number of distinct sites the tagged atom can occupy. The energy is in ϵ. −11.037 −12.535−12.535 −11.037 −11.037 −11.501−11.501 −11.037 −12.535−12.535 −11.501 −10.087 −12.535 −12.535 −11.040 −10.841 −11.037 −11.501 −12.535 −11.040 −11.477 −11.501 −11.037 −11.501 −10.807 −11.477 −12.535 −10.799 Figure 19: Minima and transition states for the four fastest discrete paths at kT/ϵ = 0.05 for per- mutational isomerisation of the global minimum of LJ2D7 .409 The tagged atom is coloured green and the energies of each minimum and transition state are given in ϵ. 20 characterise pathways and dynamics for cases where the relevant time scales are beyond the reach of conventional simulations. The two ex- amples we consider are a two-dimensional clus- ter of seven LJ atoms, denoted LJ2D 7 , and the LJ38 cluster, which we discussed above in terms of global optimisation and enhanced thermody- namic sampling. LJ2D 7 possesses only four distinct local min- ima and nineteen transition states, excluding permutational isomers. The landscape is illus- trated in Figure 18, where the effect of dis- tinguishing a single atom is highlighted. All the pathways can be characterised in a few seconds of computer time using geometry op- timisation techniques within the discrete path sampling framework.409 The corresponding rate constants are immediately obtained within a harmonic transition state theory approxima- tion. There are four key paths for migration of the centre atom to a peripheral position, each one consisting of three diamond-square- diamond410 rearrangements (Figure 19, and a movie is provided in the SI).409 Since the landscape and rearrangements are well understood, LJ2D 7 is a useful test system for benchmarking rare event methods based on explicit dynamics.411–416 These dynamical path sampling approaches are complementary to the discrete path sampling framework,409,417 incorporating anharmonic and recrossing ef- fects directly. The simplest harmonic transi- tion state theory rates are found to agree to within an order of magnitude.409 LJ2D 7 has also be used to test methods based on reaction coor- dinates and dimensionality reduction,414,418,419 as well as new transition state optimisation al- gorithms.420 In contrast, calculating interconversion rates between the fcc and incomplete icosahedral morphologies in LJ38 is significantly harder. The methodology that employs geometry op- timisation to construct a kinetic transition net- work,278,286–288 and then solves the correspond- ing master equation421,422 for the dynamics, produced quantitative analysis of the rates in 1999.345 This approach evolved into the dis- crete path sampling framework,409,417 and it was first suggested in the context of cluster dy- namics by Kunz and Berry.283,284,423,424 Locat- ing kinetically relevant pathways using alterna- tive methods proved difficult.425 However, the pathways and rates were reproduced using di- rect transition current sampling,426 and by a hybrid scheme (Voter and coworkers, personal communication). Since then the pathways have been analysed in detail by several groups,427–430 and extended to LJ75.431 Figure 20: Energy as a function of integrated path length for the union of steepest-descent paths that connect the global minimum fcc structure in LJ38 with the lowest member of the icosahedral set. There are ten minima on this nine-step path.432 LJ38 has also been used to test sampling schemes for free energy profiles,389,432 and ex- traction of kinetics from ill-conditioned net- works.433,433,434 Assuming that all the kinet- ically relevant paths are contained within a database of minima and transition states, we can employ standard tools, such as Dijkstra’s shortest path algorithm,435 for analysis. It is then straightforward to construct an overall pathway between any two minima via the union of approximate steepest-descent paths that con- nect them from knowledge of the sequence of transition states. We can then use this path to sample a free energy profile, using the inte- grated path length, s, as a progress coordinate for umbrella-type sampling.389 This approach retains all degrees of freedom, and does not re- quire assumptions about a reaction coordinate. Since the steepest-descent paths constitute lo- 21 Figure 21: Alternative views of the two- dimensional free energy surface F (s,Q6) obtained by a Monte Carlo scheme that samples the land- scape based up the integrated path length, s.432 cal minima with respect to all perpendicular de- grees of freedom, there should be no projection error from averaging over barriers in orthogo- nal coordinates,278,287,419,436–440 in this construc- tion. For discriminating fcc and icosahedral struc- tures in LJ38 the geometrical order parame- ter441,442 Q6 actually works reasonably well. However, there is a barrier to surface reorgan- isation that is not resolved by this parame- ter, and the corresponding free energy profile, F (Q6) is missing this feature.432 In contrast, F (s) faithfully reproduces the barriers associ- ated with all the transition states. The differ- ence between these representations can be vi- sualised from view of the two-dimensional free energy landscape F (s,Q6) shown in Figure 21. Here we see that the last barrier before reach- ing the target C5v minimum is averaged in the Q6 coordinate, but by construction, it is paral- lel to the direction defined by the path length, s. One-dimensional profiles based on s and Q6 show how all the barriers are clearly resolved in F (s), but not F (Q6).432 More recent work has focused on analysis of the first passage time distribution for the change in morphology.289,443 Some results were shown in Figure 15. Plotting this probability distribution, P(y), as a function of y = lnt, highlights peaks that correspond to relaxation on different time scales.444,445 Figure 15 shows the distributions for relaxation to either the fcc or the icosahedral funnel from a high en- ergy minimum. They are both bimodal, with a faster time scale for direct pathways, and a slower timescale for trajectories that first visit the other morphology, and must then escape over a high barrier. Hence the first passage time distribution provides a kinetic signature that reports on kinetic traps in a multifunnel landscape, which is complementary to the heat capacity peaks for finite size first order phase transitions, discussed in the thermodynamics section. It may also be possible to diagnose such features by scanning the mean first pas- sage time as a function of the experimental ob- servation time scale, T (tobs). This function is also shown in Figure 15, revealing that plateaux in T (tobs) line up quite well with the peaks in P(ln t).289,443 Once again, we see how the LJ potential has provided new and important insight with potential impact throughout the physical sciences. 7 Conclusions The literature on LJ simulations on atomic and molecular systems is so vast that we can only mention a fraction of the applications in this account, which is a personal perspective, moti- vated by the influence of the LJ potential on our research over many years. A full review would easily fill a book, and instead we have focused on our own experience in attempting to illus- trate the diversity of research that now exists in this field. We certainly expect the LJ po- tential to continue providing a cornerstone for advancing theory and simulation in the future, 22 throughout the physical sciences. There are many modifications of the LJ po- tential that improve on accuracy and perfor- mance, such as the extended LJ potential (ELJ) as an inverse power expansion97 (useful for lat- tice sums) or truncated and shifted versions compatible with finite range cutoffs,40,115,446 or more sophisticated generalizations leading to phase diagrams quite different from the usual LJ potentials,447 useful for example for pro- tein crystal nucleation in solutions. Here we note that cutting off the LJ potential at a cer- tain radius can alter thermodynamic proper- ties such as the triple point substantially.116 One can also replace the rather inaccurate re- pulsive potential, which for example creates problems in high pressure simulations, by a more appropriate analytical function, such as in the Buckingham (exp-6) potential64 or Aziz- type potentials.43,73,74,172 The Buckingham po- tential or modified forms of it such as the Beest-Kramer-van Santen (BKS) potential448 has been used extensively in molecular dynam- ics simulations.449,450 For recent discussions of different interaction potentials see for example Lim,451 Araújo and Ballester45 and Müser et al.452 Where many-body forces become important, such as the Axilrod-Teller-Muto term and be- yond,68,69,453 as for example in metallic systems where the many-body expansion is slowly con- verging,50 one has to apply different potentials incorporating many-body forces in an effective way. Glue potentials454 or the Sutton-Chen po- tential455 are more appropriate than LJ, and ideally we would use more accurate quantum theoretical methods, which are computationally demanding. For a detailed discussion when and when not to use the LJ potential see the arti- cle by Frenkel and coworkers.456 In terms of re- cent developments, it is noteworthy that some empirical potentials fitted using machine learn- ing also make use of a r−6 long range term in a similar spirit to the Lennard-Jones construc- tion.457,458 In summary, it is evident that after 100 years the LJ potential is alive and well, and used in many applications, providing deep insight into molecular structures and bulk properties including phase transitions, despite its obvious limitations as a two-body potential. In this an- niversary year, it seems therefore highly appro- priate to reflect on the many achievements of Lennard-Jones, and celebrate the insight that his eponymous potential has provided in the physical sciences. Supplementary Information A movie 2DLJ7.new.mpg that accompanies Figure 19 is provided in the supporting infor- mation. Acknowledgment DJW thanks the Alexander von Humboldt Foundation for financial support. We are very grateful to Florent Calvo, Gabor Csanyi, Jonathan Doye, Daan Frenkel and Axel Traves- set and to the referees for helpful comments on the original draft. Notes The authors declare no competing financial in- terest. References (1) Jones, J. E. On the Determination of Molecular Fields. I. From the Variation of the Viscosity of a Gas with Tempera- ture. Proc. R. Soc. Lond. A 1924, 106, 441–462. (2) Jones, J. E. On the Determination of Molecular Fields. II. From the Equation of State of a Gas. Proc. R. Soc. Lond. A 1924, 106, 463–477. (3) Jones, J. E. 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