Calculation of Thermodynamic Properties of Small Lennard-Jones Clusters Incorporating Anharmonicity Jonathan P

Calculation of thermodynamic properties of small Lennard-Jones clusters incorporating anharmonicity Jonathan P. K. Doye and David J. Wales University Chemical Laboratories, Lensﬁeld Road, Cambridge CB2 1EW, United Kingdom ͑Received 12 January 1995; accepted 13 March 1995͒ A method for calculating thermodynamic properties of clusters from knowledge of a sample of minima on the potential energy surface using a harmonic superposition approximation is extended to incorporate anharmonicity using Morse correction terms to the density of states. Anharmonicity parameters are found for different regions of the potential energy surface by ﬁtting to simulation results using the short-time averaged temperature as an order parameter. The resulting analytical expression for the density of states can be used to calculate many thermodynamic properties in a variety of ensembles, which accurately reproduce simulation results. This method is illustrated for 13-atom and 55-atom Lennard-Jones clusters. © 1995 American Institute of Physics.

I. INTRODUCTION figuration of a system can be mapped onto a minimum of the PES, an ‘‘inherent structure,’’ by a steepest-descent path or Before Monte Carlo ͑MC͒ and molecular dynamics ‘‘quench.’’ This mapping allows the partition function to be ͑MD͒ simulations became computationally feasible, consid- separated into a term due to the energy spectrum of the in- erable attention was given to the calculation of thermody- herent structures and a term due to thermal motion within the namic properties of clusters from knowledge of the vibrational spectrum of a low energy structure using a harmonic wells of the inherent structures. This approach can give much approximation.1–5 McGinty and Burton realized that if their greater physical insight into a process such as melting, be- results were to have relevance for more than low temperature cause the thermodynamics can be related to the structures of behavior other configurations needed to be included in their the minima on the PES. Similarly, an understanding of how partition function,2–4 but without the means to systematically the structure and topology of PES’s differ can be used to search the potential energy surface ͑PES͒ they were unable explain differences in thermodynamic properties. to implement this extension of their approach. The objective The success of Berry and co-workers in elucidating the behind these studies was to get free energies in order to study melting of small Lennard-Jones clusters was partly based on homogeneous nucleation. their emphasis on understanding the thermodynamic and dy- Once MC and MD simulations became feasible, work on namic properties in relation to the qualitative features of the 13,14 the thermodynamics of clusters concentrated on obtaining PES. In particular they used systematic quenching to find thermodynamic information from the simulations, as the in- the important low energy minima on the surface. This pro- formation gained is ‘‘exact’’ within the statistical errors of cedure has now become a standard tool in understanding the the simulation. In particular, it includes the thermodynamic thermodynamics and dynamics of clusters and has been ap- 15 16 17,18 19 20 effects of anharmonicity. This approach has been most suc- plied to LJ, argon, alkali halide, metal, water, and 6,7 21,22 cessful. The multihistogram method used by Labastie and C60 clusters. Similarly, a knowledge of the transition Whetten extracts the configurational density of states from states on the PES can provide a greater insight into the dy- 23 simulation.8 Convolution with the kinetic energy density of namics of a system. This information was first obtained by states gives the total energy density of states, ⍀(E), from the application of the eigenvector-following and ‘‘slowest 24 which many other thermodynamic functions can be calcu- slides’’ methods, and has been used in combination with 25 lated. Labastie and Whetten applied the method to the first quenching to understand the melting dynamics of small LJ, 8 17 19 26 21,22 three icosahedral Lennard-Jones ͑LJ͒ clusters, unequivo- alkali halide, metal, water, and C60 clusters. 27 cally showing that there are S-bends ͑or Van der Waals Bixon and Jortner considered the effect of model en- loops͒ in the microcanonical caloric curves of LJ55 and ergy spectra of the inherent structures on thermodynamic LJ147 . This method has since been used to calculate the com- properties, in particular on the microcanonical and canonical 9 plete phase diagram for LJ55 and to investigate the melting caloric curves. They showed that a large energy gap between 10 transition for LJ13 and ͑H2O͒8 . The multihistogram method the global minimum and a manifold consisting of a large obtains ⍀(E) to within an unknown multiplicative factor, number of higher energy minima was necessary to produce a and this is sufficient for most applications. Weerasinghe and significant feature in the caloric curves. Amar used an adiabatic switching method to fix the absolute Subsequently, we have developed an approach in which value of ⍀(E), and subsequently used ⍀(E) to calculate ⍀(E) is directly calculated from knowledge of the PES.28,29 rates of evaporation of LJ clusters using phase space A sample of minima is generated by systematic quenching theory.11 from a high energy MD run. ⍀(E) is then calculated by Although numerically successful, the above approach summing the harmonic density of states for each minima. gives little physical insight into the processes being simu- This approach has been called the harmonic superposition lated. Stillinger and Weber in their studies of bulk melting method. It has been applied to LJ,20,28 water,20 and model introduced the idea of ‘‘inherent structures.’’12 Every con- metal clusters19 to calculate the microcanonical caloric

curve, the heat capacity, the Helmholtz free energy, the ca- TABLE I. Details of the two samples of minima used for LJ55 . EЈ is mea- nonical total energy distribution function,19,20,28 Landau free sured with respect to the global minimum icosahedron. energy barriers,30 and thermodynamic properties for different EЈ/⑀ Number of minima regions of the PES, defined by a suitable order parameter.31 Franke et al. have independently applied the same ideas to A 64.7485 989 small LJ clusters.32 B 70.2485 1153 In Sec. II we briefly review the harmonic superposition method. In Sec. III we consider possible ways of including anharmonicity, and in Sec. IV we apply the resulting expres- * 0 ␬Ϫ1 sions for the density of states to calculate a variety of ther- gsns ͑EϪEs ͒ ⍀͑E͒ϭ ͚ ␬ s , ͑3͒ 0 ⌫͑␬͒⌸ h␯ modynamic properties. LJ13 and LJ55 are used as examples to E ϽE jϭ1 j evaluate the effectiveness of the method. These are the two s smallest icosahedral clusters and have been much studied where the sum is now over a representative sample of theoretically because of their special stability. minima. The effect of gs can be incorporated by using the quench statistics.28 If the system is ergodic and the MD run II. THE HARMONIC SUPERPOSITION METHOD is performed at constant energy, the number of quenches to a minimum, ␥, is assumed to be proportional to the density of The total energy density of states associated with a states of the set of gs minima, i.e., ␥(EЈ)sϰgs⍀(EЈ)s . single minimum on the PES is, in the harmonic Hence, approximation,13 ⍀͑E͒s ͑EϪE0͒␬Ϫ1 ⍀͑E͒ϰ ␥͑EЈ͒ ͑4͒ 0 ͚ s ⍀͑EЈ͒ ⍀͑E͒ϭ ␬ ␪͑EϪE ͒, ͑1͒ E0ϽE s ⌫͑␬͒⌸ jϭ1h␯ j s 0 0 ␬Ϫ1 where E is the potential energy of the minimum, ␪ is the EϪEs Heaviside step function, and ␬, the number of vibrational ϰ ͚ ␥͑EЈ͒s 0 , ͑5͒ 0 ͩ EЈϪE ͪ E ϽE s degrees of freedom, is 3NϪ6. To calculate the density of s states for the whole system, all the minima on the PES need where EЈ is the energy of the MD run. to be considered. We make a superposition approximation If all the low energy minima are known, the n* formula and sum the density of states over all the minima low enough will be accurate at low energies. Therefore, the proportional- in energy to contribute. This approximation is equivalent to ity constant in the above equation can be found by matching assuming that the phase space hyperellipsoids associated it to the low energy form of the n* formula. For LJ13 and with each minimum do not overlap. This gives LJ55 , the term due to the icosahedron is dominant at low 0 energies, and other terms in the sum can be neglected. Com- n*͑EϪE ͒␬Ϫ1 s s paring the first terms of Eqs. ͑2͒ and ͑5͒ gives for the pro- ⍀͑E͒ϭ ͚ ␬ s , ͑2͒ 0 ⌫͑␬͒⌸ jϭ1h␯ j portionality constant, c, Es ϽE 0 ␬Ϫ1 where the sum is over all the geometrically distinct minima n0*͑EЈϪE0͒ on the surface; n* , the number of permutational isomers of cϭ ␬ 0 . ͑6͒ s ␥͑EЈ͒0⌫͑␬͒⌸ jϭ1h␯ j minimum s, is given by ns* ϭ 2N!/hs , where hs is the order of the point group of s. A critical test of these formulas for ⍀(E) is the predicted microcanonical caloric curve, which for LJ55 has an S- The difficulty with Eq. ͑2͒ is that for all but the very 8 smallest clusters this sum involves an impractically large bend. Using the thermodynamic definition of the microca- number of minima. Hoare and McInnes,33 and more recently nonical temperature, T␮ , 34 ⍀ץ ln ⍀ 1 ץ Tsai and Jordan have enumerated lower bounds to the num- 1 ϭ ϭ ; ͑7͒ ͪ Eץ ͩ E ͪ ⍀ץ ͩ ber of geometric isomers for LJ clusters from 6 to 13 atoms. kT This number rises exponentially with N. Extrapolating this ␮ N,V N,V 21 28 trend gives for LJ55 an estimate of 1ϫ10 geometric iso- an expression for T␮ can be derived. For LJ55 we have two mers. In such a case, as it is not possible to obtain a complete samples of minima produced by systematic quenching,28 de- set of minima, a representative sample is needed. A large set tails of which are given in Table I. Sample A is from a MD of minima can be obtained by systematic quenching from a run at an energy in the upper end of the coexistence region, high energy MD trajectory. However, this gives a greater and B at an energy just into the liquidlike region. The results proportion of the low energy minima than of the high energy for samples A and B using the n* and ␥ formulations28 are minima. Consequently, if the sample is used in Eq. ͑2͒ it is given in Fig. 1. From this it can be seen the n* formula fails likely to underestimate the density of states due to the high badly, predicting only an inflection in the caloric curve which energy minima, and so be inaccurate at high energies. is too high in energy, because it underestimates the contribu- A method is needed which corrects for the incomplete tion to ⍀(E) from the higher energy minima. The ␥ formula nature of the sample of minima. This correction can be is much more successful, reproducing the S-bend at the ob- achieved by weighting the density of states for each known served energy.35 That the harmonic superposition method 0 minimum by gs , the number of minima of energy Es for produces a caloric curve with the correct features shows, as which the minimum s is representative. Hence, Bixon and Jortner suggested,27 that the distribution of

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Comparison of the caloric curves from simulation and the harmonic superposition method ͑Fig. 1͒ shows that the harmonic superposition method does indeed underestimate ⍀(E) and so the harmonic approximation is likely to be the main source of error.

III. ANHARMONICITY

Most attempts to model anharmonicity have concentrated on small systems. As the size of the system increases the difficulty increases greatly. For example, it would be im- possible to do the necessary multidimensional phase space integrals in the definition of ⍀(E), FIG. 1. Microcanonical caloric curves for LJ55 . The solid lines were calculated from the ␥ formula, the dashed lines were calculated from the n* formula, and the dashed line with diamonds shows the simulation points. 1 ⍀͑E͒ϭ ␦͑HϪE͒dq␬dp␬. ͑8͒ The sample of minima used in the calculations is marked on the graph. For h␬ ͵͵ details of the simulation see Ref. 31. Most approaches either attempt to calculate the anharmonic element using known information from the PES, such as the minima is critical in determining the form of the caloric third and fourth derivatives of the potential at the minima curve. However, the S-bend is too shallow and lies at too and the dissociation energies36 or assume the PES has a cer- high a temperature. The temperature difference is due to the tain topology for which the partition function is known.37–38 harmonic approximation. The temperature rises linearly with A normal mode approximation is often used because the energy for a single harmonic well. The anharmonic wells of multidimensional partition function is then the product of the the cluster, however, are flatter than the harmonic case espe- one-dimensional normal mode partition functions. However, cially around the transition state regions. Consequently, the to obtain the density of states the partition function must be cluster spends more time in these high potential energy, low inverse Laplace transformed. This problem does not neces- temperature regions of the PES, and so the true temperature sarily have an analytic solution, but there are a number of is lower than that given by the harmonic approximation. numerical methods.39,40 For smaller clusters it was found that the performance of The only attempt that we know of to evaluate analyti- the n* formula improved.20 This improvement occurs be- cally the anharmonic density of states of clusters is due to cause there are fewer minima on the PES of a smaller cluster Chekmarev and Umirzakov.38 Their expression contained a and so the set of minima obtained from quenching is more number of unknown parameters, which they had to estimate. complete. This approximate approach was partly due to their lack of The harmonic superposition method has three main pos- information about the PES of LJ13 , the cluster they consid- sible sources of error. The first is from the systematic ered. They showed their form was able to produce the types quenching and the resulting sample of minima and quench of feature seen in the LJ13 caloric curve, if not to reproduce statistics. These errors can be mostly eliminated by having a it accurately. The approach we use here is similar. We are long enough quench run to ensure ergodicity and choosing looking for a relatively simple method that will provide an an appropriate energy for the run so that the relevant regions analytical expression for the anharmonic contribution to of phase space all have significant probabilities. When study- ⍀(E). We also want to examine how far it is possible to use ing the thermodynamics of melting it is most appropriate to information extracted from the PES in this task. We will choose EЈ to lie in the coexistence region, as in the case of focus on LJ55 as a test of the methods developed. For LJ55 we sample A, so that quenches to solidlike, liquidlike, and have a sample of 3481 transition states which were found surface-melted states are frequent. The second possible from a random selection of 402 of the minima in sample B source of error is the assumption that the phase volumes for defined above.22 We have also calculated the analytical third each minima can be summed independently, i.e., the hyper- derivatives for this potential. ellipsoids in phase space do not overlap. If they did the over- A. The effect of transition state valleys lap would cause ⍀(E) to be overestimated. Of course, above an energy threshold the true phase volumes of each mini- Transition states are crucial to the dynamics of a system, mum are interconnected, but this interconnection is normally but how much effect do they have on the thermodynamics? due to the extension of the phase volumes due to anharmo- Associated with a transition state is a flat valley on the PES nicity to form necks in phase space along the transition state which connects two minima. These transition state valleys valleys. The third possible source of error is the harmonic will make a contribution to ⍀(E). To consider their effect approximation. Near the bottom of the well this is a reason- we need an expression for the density of states of a transition able assumption, but as the energy is increased some parts of state valley. the well become increasingly flat. Consequently the har- We write the partition function for the transition state monic approximation causes ⍀(E) to be underestimated. valley as a product of a vibrational partition function for the

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͑␬Ϫ1͒ modes orthogonal to the transition vector and a trans- 0 ␬Ϫ1 ns* ͑EϪEs ͒ ͱ2␲m lational partition function for motion along the transition ⍀͑E͒ϭ ͚ ␬ ␬ s ϩ 0 h ⌫͑␬͒⌸ ␯ ⌫͑␬Ϫ1/2͒ state valley. This separation gives E ϽE ͫ jϭ1 j s

Ϫ␤Ets 0 ␬Ϫ3/2 L 2␲m e ␴s,tLs,t͑EϪE Ϫ⌬s,t͒ Z ϭ , 9 s ts ͱ ␬Ϫ1 ts ͑ ͒ ϫ ͚ ␬Ϫ1 s,t , ͑11͒ h ␤ ⌸ jϭ1 ␤h␯ j 0 ⌸jϭ1 ␯j E ϩ⌬s,tϽE ͬ s where L is the length of the transition state valley. Inverse Laplace transforming this expression gives for the density of where ⌬s,t is the barrier height of transition state t from states of the transition state valley, minimum s, and the reaction path degeneracy ␴s,tϭhs/ht or ␬Ϫ3/2 2h /h for nondegenerate and degenerate rearrangement ͱ2␲mL͑EϪEts͒ s t ⍀͑E͒ ϭ ␪͑EϪE ͒. ͑10͒ mechanisms, respectively.41 A degenerate transition state ts h␬⌸␬Ϫ1␯ts⌫͑␬Ϫ1/2͒ ts jϭ1 j connects different permutational isomers of the same mini- Our expression for the total density of states is then mum. In the ␥ formulation we now have

0 ␬Ϫ1 ␬ s 0 ␬Ϫ3/2 ␬Ϫ1 s,t ͑EϪE ͒ /⌫͑␬͒⌸ ␯ ϩͱ2␲m/⌫͑␬Ϫ1/2͚͒E0ϩ ϽE␴s,tLs,t͑EϪE Ϫ⌬s,t͒ /⌸ ␯ s jϭ1 j s ⌬s,t s jϭ1 j ⍀͑E͒ϭ ͚ ␥͑EЈ͒s 0 ␬Ϫ1 ␬ s 0 ␬Ϫ3/2 ␬Ϫ1 s,t . 0 ͑EЈϪEs ͒ /⌫͑␬͒⌸jϭ1␯jϩͱ2␲m/⌫͑␬Ϫ1/2͚͒E0ϩ⌬ ϽE ␴s,tLs,t͑EЈϪEsϪ⌬s,t͒ /⌸jϭ1 ␯j Es ϽE s s,t Ј ͑12͒

Geometrically, the phase volume associated with the transi- the harmonic curve and the S-bend to be displaced down- tion state valley is an elliptical hypercylinder. There is also a wards, but these effects are small. Even if the number of term due to the phase space overlap of this hypercylinder transition states connected to a minimum is assumed to be with the hyperellipsoid associated with the minimum. It can greater than 20 the effects are only slightly increased. The be evaluated,42 however it is a small term and so we neglect addition of the density of states of the transition state valleys it, especially as initially we are only trying to determine the only accounts for a small part of the difference between the magnitude of the effect of the transition state valleys. harmonic and the true caloric curves. This result occurs not The set of transition states previously calculated is not because the contribution of the transition state valleys to the an exhaustive set for the sample of minima. Its incomplete- density of states is small compared to that of the minima—in ness could cause an underestimation of the density of states fact the transition state valleys make a larger contribution arising from the transition state valleys. We therefore per- than the minima above about 55⑀. Rather it occurs because formed a more exhaustive search for transition states from 13 the density of states for the transition state valleys is not very minima representing a wide range of energies. This search different from that of the minima; the valleys are modeled as indicated that a reasonable estimate of the average number of harmonic in all but one dimension. We therefore conclude low energy transition states per minimum was 20 ͑not count- that a method of modeling the anharmonicity of the wells on ing permutational isomers͒. For each minimum the density of the PES is needed to obtain an accurate ⍀(E). This agrees states of the known transition state valleys was multiplied by with Stillinger and Stillinger’s conclusion that intrawell an- s s 20/nts , where nts is the number of known transition states harmonicity is dominant for LJ55 because the caloric curve for minimum s. If the sample of transition states contained significantly deviates from the harmonic form at energies none connected to a particular minimum, it was assigned 20 where the cluster always resides in the icosahedral well.16 transition states with the average barrier height and average frequency. The length of the transition state valley was esti- B. The effect of well anharmonicity mated using Here we follow the method of Haarhoff37,39 to calculate 1 2⌬ an anharmonic correction term to the density of states. The Ls,tϭDs,tϪ , ͑13͒ energy levels for a Morse oscillator are given by 2␲␯s ͱ m 1 1 2 ͑h␯͒2 where D is the displacement in configuration space be- Eϭ nϩ h␯Ϫ nϩ . ͑14͒ s,t ͩ 2ͪ ͩ 2ͪ 4D e tween minimum s and transition state t, and ␯s is the geometric mean normal mode frequency of s. The second term This quadratic can be solved for n. The root corresponding to is the geometric mean radius of the harmonic well in con- a bound state is figuration space at an energy ⌬ above the minimum. 1 2De E From the expression for ⍀(E) in Eq. ͑12͒ the tempera- nϩ ϭ 1Ϫ 1Ϫ . ͑15͒ 2 h␯ ͱ D ture can be calculated using Eq. ͑7͒. It can be seen from the ͩ e ͪ microcanonical caloric curves given in Fig. 2͑a͒ that the tran- Assuming n is continuous and differentiating with respect to sition state valleys cause the caloric curve to bend away from E gives a classical density of states,

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Morse oscillator, but for the case of a cluster isomerization ⍀(E) remains ﬁnite as E ⌬ from below, where ⌬ is the barrier height. As we are seeking→ a correction term for well anharmonicity, it seems reasonable to truncate Eq. ͑16͒ and examine the effect of the ﬁrst term in this series. Laplace transformation then gives for the partition function 1 1 1 zϭ ϩ . ͑17͒ h␯ ͩ ␤ 2⌬␤2 ͪ The multidimensional partition function is then

␬ 1 1 Zϭ 1ϩ . ͑18͒ ͟ ␤h␯ ͩ 2⌬ ␤ͪ jϭ1 j j Making the approximation that an average value of 1/⌬ can be used then gives

␬ ␬ l 1 Cl a Zϭ , ͑19͒ ⌸␬ h␯ ͚ ␤␬ϩl jϭ1 j lϭ0 ␬ where aϭ͗1/2⌬͘ is an anharmonicity parameter and Cl is a binomial coefﬁcient. Inverse Laplace transforming gives

␬ ␬ l ␬ϩlϪ1 1 Cl a E ⍀͑E͒ϭ . ͑20͒ ⌸␬ h␯ ͚ ⌫͑␬ϩl͒ jϭ1 j lϭ0 The total density of states is found by summing over all the FIG. 2. Microcanonical caloric curves for LJ55 using ͑a͒ the ␥ formula with ͑solid line͒ and without ͑dashed line͒ the contribution from transition state minima, giving valleys, and ͑b͒ using the ␥ formula including anharmonic terms with minima samples A ͑solid line͒ and B ͑dashed line͒. In both ͑a͒ and ͑b͒ the ␬ ␬ l 0 ␬ϩlϪ1 ns* Cl as͑EϪEs ͒ calculated caloric curves are compared to simulation ͑dashed line with dia- ⍀͑E͒ϭ ͚ ␬ s ͚ . ͑21͒ monds͒. 0 ⌸ jϭ1h␯ j ⌫͑␬ϩl͒ Es ϽE lϭ0 Converting the above equation to the ␥ formulation using dn 1 Eq. ͑4͒ gives ⍀͑E͒ϭ ϭ dE h␯ͱ1ϪE/De ␬ ␬ l 0 ␬ϩlϪ1 Cl as͑EϪEs ͒ ⍀͑E͒ϰ ␥͑EЈ͒ 1 E 3 E 2 ͚ s ͚ ⌫͑␬ϩl͒ E0ϽE ͫ lϭ0 ϭ 1ϩ ϩ ϩ••• , ͑16͒ s Ͳ h␯ ͫ 2D 8 ͩD ͪ ͬ e e ␬ ␬ l 0 ␬ϩlϪ1 where the square root has been expanded binomially. The Cl as͑EЈϪEs ͒ . ͑22͒ expansion is valid if E/De is small. The ﬁrst term in the ͚ ⌫͑␬ϩl͒ lϭ0 ͬ series is the harmonic term. The full series will diverge as E D . This divergence is the correct behavior for the The temperature follows from Eq. ͑7͒, → e

␬ ␬ l 0 ␬ϩlϪ1 ␬ ␬ l 0 ␬ϩlϪ1 ͚E0ϽE␥͑EЈ͒s ͚lϭ0͓Cl as͑EϪEs ͒ /⌫͑␬ϩl͔͒ ͚lϭ0͓Cl as͑EЈϪEs ͒ /⌫͑␬ϩl͔͒ s ͭ Ͳ ͮ T␮ϭ . ͑23͒ ␬ ␬ l 0 ␬ϩlϪ2 ␬ ␬ l 0 ␬ϩlϪ1 k͚E0ϽE␥͑EЈ͒s ͚lϭ0͓Cl as͑EϪEs ͒ /⌫͑␬ϩlϪ1͔͒ ͚lϭ0͓Cl as͑EЈϪEs ͒ /⌫͑␬ϩl͔͒ s ͭ Ͳ ͮ

This anharmonic correction term has a similar form to that of a 1D Morse oscillator can be found from the second and used by Chekmarev and Urmirzakov38 but has been derived third derivatives of the potential by in a different way. We consider two possible ways of calculating a from s 3 the potential energy surface of LJ . First, we consider using ͑VЉ͒ 55 D ϭ . ͑24͒ the third derivatives of the potential. The dissociation energy e ͑Vٞ͒2

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If off-diagonal elements in the multidimensional case are ig- good quantitative measure of the anharmonicity of an indi- nored, the above equation can be used to calculate the barrier vidual minimum. This method was therefore not considered heights associated with each normal mode. Using analytical further. third derivatives of the LJ potential, the Cartesian third de- Therefore, instead of trying to obtain as from the PES, rivatives for the LJ55 icosahedron were calculated and trans- we considered how it could be obtained by comparison with formed to obtain the diagonal elements in the Hessian eigen- the simulation results. If as is taken to be independent of the vector basis. This scheme underestimates the anharmonicity energy of the minimum, then a value of as can be found and gives values which when substituted into Eq. ͑22͒ have which reproduces the S-bend in the caloric curve at the ob- an insigniﬁcant effect on ⍀(E), because the off-diagonal el- served temperature and energy, but the temperature differ- ements, which far outnumber the diagonal elements, should ence between the two turning points is still too small. How- not be neglected. However, there is no obvious way to cal- ever, one would expect the anharmonicity to depend upon culate the effect of the off-diagonal elements and for a sys- the energy of the minimum; the higher energy minima are tem such as LJ55 the transformation of the third derivatives likely to have a greater anharmonicity than the solidlike into the Hessian eigenvector basis is too computationally ex- minima. pensive for the off-diagonal elements. In an accompanying paper31 we have shown how ther- The second method considered was to use the barrier modynamic properties can be calculated for different regions heights of our transition state sample. An average barrier of the PES deﬁned by a suitable order parameter by restrict- height was calculated for each minimum, but for some ing the sum of Eq. ͑22͒ to minima in these regions. This minima it led to a gross overestimation of the anharmonicity procedure gives for Ti , the temperature of region i, and unphysical caloric curves. The barrier heights are not a

␬ ␬ l 0 ␬ϩlϪ1 ␬ ␬ l 0 ␬ϩlϪ1 ͚E0ϽE,s෈i␥͑EЈ͒s ͚lϭ0͓Cl as͑EϪEs ͒ /⌫͑␬ϩl͔͒ ͚lϭ0͓Cl as͑EЈϪEs ͒ /⌫͑␬ϩl͔͒ s ͭ Ͳ ͮ Tiϭ . ͑25͒ ␬ ␬ l 0 ␬ϩlϪ2 ␬ ␬ l 0 ␬ϩlϪ1 k͚E0ϽE,s෈i␥͑EЈ͒s ͚lϭ0͓Cl as͑EϪEs ͒ /⌫͑␬ϩlϪ1͔͒ ͚lϭ0͓Cl as͑EЈϪEs ͒ /⌫͑␬ϩl͔͒ s ͭ Ͳ ͮ

In the other paper the short-time averaged ͑STA͒ temperature V were chosen that were intermediate between the values for is used as an order parameter to distinguish regions of the regions III and VI, but as these regions only make a small LJ55 PES, and it is shown that these regions are associated contribution to ⍀(E) the exact value chosen will only have a with minima in the different potential energy ranges given in very small effect on the overall caloric curve. The values of Table II. Region I corresponds to the solidlike state, regions ai are given in Table II. As would be expected the anharmo- II and III to surface-melted states, and region VI to the liq- nicity increases with the potential energy of the minima. uidlike state. Consequently, the temperatures associated with The microcanonical caloric curve was calculated using the minima in the energy ranges I, II, III, and VI are known. these values of ai . Figure 2͑b͒ shows that for sample A the Different values of the anharmonicity parameter, ai , were calculated curve is in remarkable agreement with the simu- therefore assigned to minima in the six different energy lation results. The calculated S-bend now has the correct ranges. Values were chosen for regions I, II, III, and VI depth, because the effect of the greater anharmonicity of the which reproduced the simulation results in Fig. 13 of Ref. liquidlike state is to increase the difference in temperature 31. The caloric curves for each region accurately fitted the between the two branches of the caloric curve. The success simulation curves showing that our anharmonic correction of this method can be understood from the equation term has an appropriate form. Values of ai for regions IV and 1 pi ϭ , ͑26͒ T ͚ T i i TABLE II. Partition of the minima of LJ55 into energy ranges and values of the anharmonicity parameter, a , for each range. Values for I, II, III, and VI i where pi is the probability that the cluster is in region i. The were found by fitting the temperatures for these regions to the simulation probabilities at EЈ, p (EЈ), are fixed by the quench frequen- results. i cies, Ϫ1 Region Lower energy bound/⑀ Higher energy bound/⑀ ai/⑀ ͚s෈i␥s͑EЈ͒ p ͑EЈ͒ϭ . ͑27͒ I Ϫ279.248 47 Ϫ279.248 47 0.50 i ͚ ␥ ͑EЈ͒ II Ϫ276.604 29 Ϫ276.199 35 0.51 s s III Ϫ274.500 00 Ϫ273.000 00 0.53 Furthermore, we show elsewhere that p calculated from Ϫ Ϫ i IV 273.000 00 271.500 00 0.65 sample A are in excellent agreement with simulation over a V Ϫ271.500 00 Ϫ268.840 00 0.70 31 VI Ϫ268.840 00 ••• 0.73 wide range of energy. The values of ai have been chosen to reproduce the simulation temperatures, Ti , and so the overall

J. Chem. Phys., Vol. 102, No. 24, 22 June 1995 J. P. K. Doye and D. J. Wales: Thermodynamic properties of clusters 9665

temperature, T, is bound to be very accurate. Sample B pro- TABLE III. Details of the two samples of minima for LJ13 . EЈ is measured duces worse results because there are fewer quenches to the with respect to the global minimum icosahedron. solidlike and the surface-melted states, and so their quench EЈ/⑀ Number of minima frequencies are subject to larger statistical errors than for sample A. C 13.7768 95 D 18.3268 295 For LJ13 , the STA temperature distribution is bimodal. The high temperature peak corresponds to solidlike clusters associated with the icosahedral global minimum, and the low temperature peak to liquidlike clusters. The residence times in the solidlike and liquidlike states are much shorter than for 2͗EK͘ T ϭ . ͑30͒ LJ55 and so the information provided by short-time averaging K k␬ for LJ13 is less well-resolved. We partitioned the minima This expression is exact in the canonical ensemble, but TK distribution into these two regions, and assigned values of ai Ϫ1 to each. The values of a were higher than for LJ and the differs by O (N ) from the thermodynamic definition of i 55 temperature, Eq. ͑7͒, in the microcanonical ensemble.43 For curvature of the Ti curves differed from the simulation results. The apparently greater anharmonicity can be explained LJ55 the difference between the thermodynamic and kinetic temperatures is negligible. For LJ13 it is still small but not by the larger number of surface atoms for LJ13 . The incorrect curvature suggests that the energy dependence of the density insignificant, and so TK rather than T␮ is fitted to the simu- of states is inappropriate and so the second-order Morse cor- lation results. Bixon and Jortner found both temperatures produced very similar results in their model calculations of rection term to the density of states was included. The result- 27 ing T curves fitted the simulation results much more accu- the microcanonical caloric curve. TK has been obtained for i the superposition method by Franke et al. using the equipar- rately. The partition function for a single minimum including 32 this term is tition theorem, however this method cannot be applied when anharmonicity is included. We obtain T through a ␬ K 1 a a 2 ␬ different method. First, we note that Zϭ 1ϩ ϩ3 ͟ ␤h␯ ͫ ␤ ͩ ␤ͪ ͬ E jϭ1 j ͗EK͘ϭ p͑EK͒EK dEK ͵0 1 ␬ ␬Ϫl 3malϩ2m ␬ 1 E ϭ ␬ ͚ ͚ Dl,m ␬ϩlϩ2m , ͑28͒ ⌸ jϭ1h␯ j ␤ ϭ EK⍀c͑EϪEK͒⍀K͑EK͒dEK , ͑31͒ lϭ0 mϭ0 ⍀͑E͒ ͵0 where where ⍀ (E ) is the kinetic density of states, ⍀ (E ) is the ␬! K K c c D␬ ϭ . configurational density of states, and the potential energy Ec l,m l!m!͑␬ϪlϪm͒! is given by EcϭEϪEK . Deconvoluting ⍀K(EK) from ⍀(E) Inverse Laplace transforming and summing over all minima gives gives for the total density of states, 1 ␬ ␬Ϫl ⍀c͑Ec͒ϭ /2 ␬ ns* ͑2␲m͒␬ ⌸ ␯ ⍀͑E͒ϭ jϭ1 j ͚ ⌸␬ h␯s ͚ ͚ E0ϽE jϭ1 j lϭ0 mϭ0 s ␬ ␬Ϫl ␬ m lϩ2m 0 ␬/2ϩlϩ2mϪ1 Dl,m3 as ͑EϪEs ͒ ␬ m lϩ2m 0 ␬ϩlϩ2mϪ1 ϫ . Dl,m3 as ͑EϪEs ͒ ͚ ͚ ⌫͑␬/2ϩlϩ2m͒ ϫ . ͑29͒ lϭ0 mϭ0 ⌫͑␬ϩlϩ2m͒ In a MD simulation one calculates the kinetic tempera- ͑32͒ ture, TK , from the kinetic energy, EK , via the generalized Integrating Eq. ͑31͒, summing over all minima and substitut- equipartition theorem, ing into Eq. ͑30͒ gives

␬ s ␬ ␬Ϫl ␬ m lϩ2m 0 ␬ϩlϩ2m ͚E0ϽE n*/⌸ ␯ ͚ ͚ D 3 a ͑EϪE ͒ /⌫͑␬ϩlϩ2mϩ1͒ s s jϭ1 j lϭ0 mϭ0 l,m s s TKϭ ␬ s ␬ ␬Ϫl ␬ m lϩ2m 0 ␬ϩlϩ2mϪ1 . ͑33͒ k͚E0ϽE n*/⌸ ␯ ͚ ͚ D 3 a ͑EϪE ͒ /⌫͑␬ϩlϩ2m͒ s s jϭ1 j lϭ0 mϭ0 l,m s s

Sample C ͑Table III͒ was used to calculate the caloric minima. The values of ai assigned to solidlike and liquidlike curve of LJ13 for the ␥ formulation because it was produced clusters are given in Table IV. From Fig. 3 it can be seen that from a MD run in the coexistence region and so should have the caloric curve given by the n* formulation agrees very more accurate quench statistics for the low energy minima, well with simulation, because it is possible to obtain a near- and sample D was used for the n* formulation because it is complete set of minima for LJ13 . The ␥ formulation, how- from the liquidlike region and so has a larger sample of ever, has too high a transition temperature. This error arises

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TABLE IV. Values of the anharmonicity parameter, ai , for the solidlike ͑I͒ Harmonic and anharmonic results for the caloric curves, and liquidlike ͑II͒ regions of the PES. Region I is the potential well of the the heat capacity Cv , the Helmholtz free energy A, the tran- icosahedron and region II consists of all other minima. sition temperature T1/2, and the latent heat Lm are compared Ϫ1 in Figs. 4 and 5 and Table V. We define T as the tempera- Region ai/⑀ 1/2 ture for which the two states have an equal Landau free I 0.470 energy, F(E ). L is the internal energy difference between II 0.625 c m the two states at T1/2 and was obtained by extrapolating the caloric curves for each state to T1/2 using Eq. ͑25͒. For LJ13 , this procedure simply gives LmϭUIIϪUI , where Ui is the because the probability that the cluster is in the solidlike internal energy of region i. For LJ55 , we have used state derived from quenching is higher than the correspond- LmϭUV–VIϪUI–IV , where UI–IV is the internal energy for ing probability derived from the STA temperature distribu- the region of the PES formed from the combination of re- tions. In the ␥ formulation the probabilities, pi ,atEЈare gions I–IV, and so Lm does not include the latent heat of fixed by the quench statistics ͓Eq. ͑27͔͒, and so this con- surface-melting. Just integrating Cv over the transition re- straint leads to the higher transition temperature. The differ- gion would overestimate Lm because it would also include ence between the probabilities derived from quenching and the energy needed to raise the temperature of the cluster. the STA temperature distributions may be because there are The anharmonic caloric curves are displaced downward regions of the PES which are in the basin of attraction of the and away from their harmonic equivalents because of the icosahedron but which have thermodynamic properties simi- increased densities of states associated with higher potential lar to a liquidlike well, or because the short-time averaging energy, lower temperature regions of the PES. The anhar- does not distinguish between the solidlike and liquidlike monic heat capacity curve of LJ55 is in very good agreement 8 states with the same resolution as for LJ55 . This difference with results from the multihistogram MC method. The peak does not stem from the quench method since we obtained in the heat capacity curve is larger and sharper when anhar- similar quench statistics with steepest-descent,44 conjugate monicity is included. This change can be understood by con- gradient,45 and eigenvector-following46 methods. sidering a two level system, which is a reasonable model to describe the equilibrium between solidlike and liquidlike states of the cluster. The partition function can be written as IV. THERMODYNAMIC PROPERTIES Zϭ͚iZi , where the sum is over the two distinct regions of The accurate expressions for ⍀(E) developed in the pre- phase space. It follows that vious section can be used to give a wide range of thermody- Ziץ ln Z 1 ץ namic functions, in fact, all except those that depend on de- UϭϪ ϭϪ ϭ p U , ͚ i i ͪ ␤ץ ͩ Z ͚ ͪ ␤ץ ͩ rivatives of N or V. The exceptions arise because the N,V i N,V i thermodynamic properties of small clusters are discontinu- ͑34͒ ously dependent on N and because the volume of a cluster is ,Hence . (␤ץ/ ln Z ץwhere p ϭZ /Z and U ϭϪ͑ hard to define.9 The formulas for the functions illustrated in i i i i N,V piץ Uץ this section are given in the Appendix. They have, for the C ϭ ϭ p C ϩU most part, been derived in previous work within the har- v T ͚ i v,i i T ͬ ͪ ץ ͩ ͫ ͪ ץ ͩ 28,30 monic approximation; the extension to incorporate an- N,V i N,V harmonicity follows simply from the expressions given in p2 1ץ 1 Sec. III. The results are given for the most accurate partition ϭ ͑C ϩC ͒ϩL when p ϭp ϭ , T ͪ 1 2 2ץ v,1 v,2 mͩ 2 functions; the first-order corrected ␥ formula for LJ55 and the N,V second-order corrected n* formula for LJ13 . ͑35͒

T)N,V .Asץ/Uiץ)where LmϭU2(T1/2)ϪU1(T1/2) and Cv,iϭ

Zi piץ Z 1ץ pi 1 Ziץ ϭ Ϫ ϭ ͑U ϪU͒, kT2 i ͬ ͪ ␤ץ Zͩ ͪ␤ץTͪ kT2ͫZ2ͩץͩ N,V N,V N,V ͑36͒ p2 p1p2 Lm 1ץ ϭ ͑U ϪU ͒ϭ when p ϭp ϭ . T ͪ kT2 2 1 4kT2 1 2 2ץ ͩ N,V 1/2 ͑37͒ Substituting this result into Eq. ͑35͒ gives 2 1 Lm Cvϭ ͑Cv,1ϩCv,2͒ϩ 2 . ͑38͒ 2 4kT1/2

The greater anharmonicity of liquidlike minima causes Lm to FIG. 3. Microcanonical caloric curves for LJ13 using the n* formula ͑solid line͒, the ␥ formula ͑dashed line͒, and from simulation ͑dashed line with be larger when anharmonicity is included. This change has diamonds͒. Both the n* and ␥ formula curves include anharmonic terms. two effects; it increases the area under the heat capacity

J. Chem. Phys., Vol. 102, No. 24, 22 June 1995 J. P. K. Doye and D. J. Wales: Thermodynamic properties of clusters 9667

FIG. 4. Comparison of harmonic and anharmonic thermodynamic functions of LJ55 ; ͑a͒ the microcanonical ͑solid line͒ and canonical ͑dashed line͒ caloric curves; ͑b͒ the isopotential caloric curve; ͑c͒ Cv(T); and ͑d͒ A(T). In all except ͑a͒ the anharmonic curve is denoted by a solid line and the harmonic by a dashed line. Energies are measured with respect to the global minimum icosahedron.

peak, and causes the cluster to change between the two states for a narrower range of the independent variable, i.e., the more rapidly with temperature, decreasing the width of the contribution of the liquidlike states overtakes the surface- peak. melted states at an earlier stage in the surface-melting tran- From Fig. 6 it can be seen that the probability of LJ55 sition. being in a surface-melted state is lowest in the canonical The Landau free energy, F(Q), is the free energy of a ensemble and highest in the isopotential ensemble. This re- system for a particular value of an order parameter Q.Itis sult is due to the dependence of the partition function on the deﬁned by independent variables, T, E, and Ec , of the three ensembles. In the canonical ensemble, Z is exponentially dependent on T, and is the most steeply varying of the three partition func- F͑Q͒ϭA ϪkT ln p ͑Q͒, ͑39͒ tions. In the microcanonical and isopotential ensembles, ⍀ c Q and ⍀c are dependent on powers of E and Ec , respectively. As the exponent of Ec is lower than that for E, ⍀c is the slowest varying of the partition functions. For LJ55 where pQ(Q) is the canonical probability distribution of the nl/nsmӷnsm/ns ͑Table VI͒, where ns , nsm , and nl are the order parameter. The presence of two minima in F(Q) indi- number of minima in the solidlike, surface-melted, and liq- cates that there are two thermodynamically stable states at uidlike states. Consequently, as the partition function be- this temperature. When Ec is used as an order parameter, comes more steeply varying, the surface-melted state is seen simulations have shown that LJ55 has two Landau free energy minima which correspond to the solidlike and liquidlike states.30 Bimodality in the canonical energy distribution TABLE V. Transition temperatures and latent heats for LJ13 and LJ55 . function, f (E), implies that there are two Landau free energy minima as a function of the order parameter, E. Figures 7 LJ13 LJ55 and 8 show that both LJ and LJ have a range of tempera- Ϫ1 13 55 T1/2/⑀k ͑anharmonic͒ 0.2869 0.2985 Ϫ1 ture for which two Landau free energy minima are observed. T1/2/⑀k ͑harmonic͒ 0.3512 0.3427 The predicted free energy curves for LJ are in very good L /⑀ ͑anharmonic͒ 3.696 15.821 55 m agreement with simulation.30 The solidlike and surface- Lm/⑀ ͑harmonic͒ 2.887 12.837 melted states both contribute to the low energy minimum in

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FIG. 5. Comparison of harmonic and anharmonic thermodynamic functions of LJ13 ; ͑a͒ the microcanonical ͑solid line͒ and canonical ͑dashed line͒ caloric curves; ͑b͒ the isopotential caloric curve; ͑c͒ Cv(T); and ͑d͒ A(T). In all except ͑a͒ the anharmonic curve is denoted by a solid line and the harmonic by a dashed line. Energies are measured with respect to the global minimum icosahedron.

F(Ec). For LJ13 the free energy barrier is much smaller and of the PES can be obtained from the quench frequencies. has not been observed in simulations. This may be because Using Eqs. ͑3͒, ͑4͒, and ͑6͒ an expression for gs can be only three temperatures were tested in the simulations, the derived, barrier is too small to be detected by simulation or the barrier is an artifact of inaccuracies in our analytical partition func- ␥ E h ⌸␬ ␯s ␬ C␬al E ϪE0 ␬ϩlϪ1 tion. ͑ Ј͒s s jϭ1 j l 0͑ Ј 0͒ gsϭ ␬ 0 ͚ ␥͑EЈ͒0h0⌸ ␯ ⌫͑␬ϩl͒ Ͳ The turning points in F(Ec) and f (E) correspond to jϭ1 j lϭ0 points on the isopotential and the microcanonical caloric ␬ curves, respectively. This can be demonstrated for F(Ec)by ␬ l 0 ␬ϩlϪ1 Cl as͑EЈϪEs ͒ E ) ϭ0. The solution is . ͑40͒ץ/Fץ) solving the equation c N,V ͚ ⌫͑␬ϩl͒ Ec)N,V , which is simply the deﬁnition of lϭ0ץ/ln ⍀c ץkTϭ͑/1 the isopotential temperature. The maxima in F(Ec) correspond to the segment of the caloric curve with negative From gs , the sum [Gm(E)] and associated energy density of slope, and the minima to segments with positive slope. states [⍀m(E)] of geometrically distinct minima can be cal- Therefore the temperature range for which F(Ec) has two culated using minima is the same as the depth of the S-bend in the isopotential caloric curve, as can be seen from Figs. 4 b and 7 c , ͑ ͒ ͑ ͒ dGm G ͑E͒ϭ g and ⍀ ͑E͒ϭ . ͑41͒ and 5͑b͒ and 8͑b͒. m ͚ s m dE E0ϽE Comparing the results for LJ13 and LJ55 we see that the s effects of size are apparent. For LJ55 the melting transition is much more pronounced; it has a sharper peak in Cv , more This approach has been applied to LJ55 . From Fig. 9 it can be pronounced S-bends in the microcanonical and isopotential seen that the number of minima for LJ55 rises exponentially caloric curves, a larger Landau Free energy barrier, a larger with the energy. The total number of minima in the energy temperature range for which solidlike and liquidlike clusters range probed by the MD quenching ͑potential energies up to coexist, a larger latent heat per atom, and a higher melting Ϫ259⑀͒ is 8.3ϫ1011. This is much less than the total number temperature. This behavior is closer to the ﬁrst-order phase of minima predicted by extrapolating Tsai and Jordan’s re- transition of bulk matter. sults and so suggests that the number of minima will con- Estimates of the number of minima in different regions tinue to rise exponentially above Ϫ259⑀. The present method

J. Chem. Phys., Vol. 102, No. 24, 22 June 1995 J. P. K. Doye and D. J. Wales: Thermodynamic properties of clusters 9669

FIG. 7. Plots for LJ55 of ͑a͒ f (E)atTϭ0.2985, ͑b͒ F(Ec)atTϭ0.2985, FIG. 6. Probabilities of LJ55 being in regions I, II, III, IV, and VI of the PES and ͑c͒ the free energy barrier heights ͑solid lines͒ and the free energy for ͑a͒ canonical, ͑b͒ microcanonical, and ͑c͒ isopotential ensembles. difference between solidlike and liquidlike states ͑dashed line͒ for F(Ec). In ͑a͒ and ͑b͒ the contributions of the different regions of the PES are also indicated.

can also be used to estimate the number of minima in the energy ranges I–VI ͑Table VI͒. The number of minima in 16 TABLE VI. Estimated numbers of geometrically distinct minima of LJ55 in range II is known to be 11. The results from the quench the six energy ranges given in Table II calculated from the quench frequen- frequencies agree well with this ﬁgure. cies for sample A.

Number of minima

Region anharmonic harmonic V. CONCLUSIONS

I1 1Accurate analytic expressions for the density of states II 11.3 6.3 that include anharmonicity have been produced for LJ13 and III 994.6 457.9 LJ55 , which primarily use information obtained from the po- IV 81.5 2871.2 tential energy surface. From these expressions, many ther- V 1.26ϫ105 1.22ϫ106 VI 8.30ϫ1011 3.11ϫ1012 modynamic properties can be calculated. The analytical results accurately reproduce values obtained from simulation,

J. Chem. Phys., Vol. 102, No. 24, 22 June 1995 9670 J. P. K. Doye and D. J. Wales: Thermodynamic properties of clusters

could also, in principle, be applied to periodic models of bulk matter. However, there would then be the added difﬁ- culty that the PES, and consequently the minima on the PES, will depend on the pressure. The expressions for the density of states could also be used to calculate accurate rate con- stants using RRKM theory,17,39 and so aid quantitative eluci- dation of dynamic properties from a knowledge of the transition states on the PES.

ACKNOWLEDGMENTS

J. P. K. D. is grateful to the EPSRC for ﬁnancial support and D. J. W. is a Royal Society Research Fellow.

APPENDIX

In this Appendix, the formulas for some of the thermodynamic quantities that can be calculated by the superposition method are given. The thermodynamic functions are derived from the following deﬁnitions. In the isopotential ensemble, ln ⍀c ץ ⍀c,i͑Ec͒ 1 pi͑Ec͒ϭ , ϭ . ͑A1͒ ͪ Eץ ͩ ͒ FIG. 8. Plots for LJ13 of ͑a͒ F(Ec)atTϭ0.2869, and ͑b͒ the free energy ⍀ ͑E ͒ kT ͑E c c c c c barrier heights ͑solid lines͒ and the free energy difference between solidlike N,V and liquidlike states ͑dashed line͒ for F(Ec). In ͑a͒ the contributions of In the microcanonical ensemble, liquidlike and solidlike states are shown.

⍀i͑E͒ p ͑E͒ϭ . ͑A2͒ i ⍀͑E͒ In the canonical ensemble, and give greater physical insight into the thermodynamics by allowing the roles of different parts of the PES to be ana- Zi p ͑T͒ϭ , lyzed. An alternative approach in which the partition func- i Z tion was corrected by allowing for transition state valleys ln Z 1 ץ -showed little improvement from the original harmonic super U͑T͒ϭϪ ϭ ͑Z ϩZ ͒, Z 1,0 0,1 ͪ ␤ץ ͩ position approximation. The methods developed here should 0,0 be applicable to other types of clusters, although, as in our N,V Uץ examples, the form of the anharmonic terms would probably depend on the type and size of cluster considered. They C ͑T͒ϭ , ͪ Tץ ͩ v N,V 2 1 1 Z1,0ϩZ0,1 ϭ ͑Z ϩ2Z ϩZ ͒Ϫ , kT2 ͫZ 2,0 1,1 0,2 ͩ Z ͪ ͬ 0,0 0,0 ͑A3͒

0 A͑T͒ϭE0ϪkT ln Z,

f ͑E͒ϭ⍀͑E͒exp͑Ϫ␤E͒,

F͑Ec͒ϭEcϪkT ln ⍀c͑Ec͒,

where in Z␣,␦ the derivative of the exponential function of ␤ in Z has been taken ␣ times and the derivative of the inverse power of ␤ in Z ␦ times. Therefore, ZϭZ0,0. Below, we give the thermodynamic functions for the ␥ formulation with a ﬁrst-order anharmonic correction which FIG. 9. Plot of Gm(E) for LJ55 as a function of the potential energy of the minimum calculated from the quench frequencies of sample A. was used for LJ55 ,

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␬ ␬ l 0 ␬/2ϩlϪ1 ␬ ␬ l 0 ␬ϩlϪ1 ͚E0ϽE ␥͑EЈ͒s͚lϭ0͓Cl as͑EcϪEs ͒ /⌫͑␬/2ϩl͔͒ ͚lϭ0͓Cl as͑EЈϪEs ͒ /⌫͑␬ϩl͔͒ s c Ͳ Tc͑Ec͒ϭ , ␬ ␬ l 0 ␬/2ϩlϪ2 ␬ ␬ l 0 ␬ϩlϪ1 k͚E0ϽE ␥͑EЈ͒s͚lϭ0͓Cl as͑EcϪEs ͒ /⌫͑␬/2ϩlϪ1͔͒ ͚lϭ0͓Cl as͑EЈϪEs ͒ /⌫͑␬ϩl͔͒ s c Ͳ

␬ ␬ l ␬ ␬ l 0 ␬ϩlϪ1 0 ϩlϩ C a C a ͑E ϪE ͒ 0 Ϫ E ⌫͑␬ ␦͒ l s l s Ј s Z ϭc ␥͑EЈ͒ ͑E ͒␣e ␤ s , ͑A4͒ ␣,␦ ͚ s s ͚ ⌫͑␬ϩl͒ ␤␬ϩlϩ␦ Ͳ͚ ⌫͑␬ϩl͒ s lϭ0 lϭ0

␬ ␬ l 0 ␬/2ϩlϪ1 ␬ ␬ l 0 ␬ϩlϪ1 1 c Cl as͑EcϪEs ͒ Cl as͑EЈϪEs ͒ F͑Ec͒ϭEcϪ ln ␬/2 ͚ ␥͑EЈ͒s͚ ͚ , ␤ ␤ 0 ⌫͑␬/2ϩl͒ Ͳ ⌫͑␬ϩl͒ EsϽEc lϭ0 lϭ0 where

␬ ␬ l 0 ␬ϩlϪ1 n0* Cl a0͑EЈϪE0͒ cϭ . ␥͑E ͒ ⌸␬ h␯0 ͚ ⌫͑␬ϩl͒ Ј 0 jϭ1 j lϭ0 The harmonic forms can be recovered by taking the lϭ0 term. We obtain the n* formulation by replacing

␬ ␬ l 0 ␬ϩlϪ1 Cl as͑EЈϪEs ͒ ns* c␥͑EЈ͒ with . ͑A5͒ s Ͳ͚ ⌫͑␬ϩl͒ ⌸␬ h␯s lϭ0 jϭ1 j Below, we give the thermodynamic functions for the n* formulation with second-order anharmonic corrections which was used for LJ13 , ␬ s ␬ ␬Ϫl ␬ m lϩ2m 0 ␬/2ϩlϩ2mϪ1 ͚E0ϽE ͑n*/⌸ h␯ ͒ ͚ ͚ ͓D 3 a ͑EcϪE ͒ /⌫͑␬/2ϩlϩ2m͔͒ s c s jϭ1 j lϭ0 mϭ0 l,m s s Tc͑Ec͒ϭ ␬ s ␬ ␬Ϫl ␬ m lϩ2m 0 ␬/2ϩlϩ2mϪ2 , k͚E0ϽE ͑n*/⌸ h␯ ͒ ͚ ͚ ͓D 3 a ͑EcϪE ͒ /⌫͑␬/2ϩlϩ2mϪ1͔͒ s c s jϭ1 j lϭ0 mϭ0 l,m s s

0 ␬ ␬Ϫl ϩ Ϫ␤Es 0 ␣ ␬ m l 2m ns*e ͑Es ͒ ⌫͑␬ϩlϩ2mϩ␦͒ Dl,m3 as Z ϭ , ͑A6͒ ␣,␦ ͚ ⌸␬ h␯s ͚ ͚ ⌫͑␬ϩlϩ2m͒ ␤␬ϩlϩ2mϩ␦ s jϭ1 j lϭ0 mϭ0

␬ ␬Ϫl ␬ m lϩ2m 0 ␬/2ϩlϩ2mϪ1 1 ns* Dl,m3 as ͑EcϪEs ͒ F͑Ec͒ϭEcϪ ln ͚ ␬/2 ␬ s͚ ͚ . ␤ 0 ␤ ⌸jϭ1h␯j ⌫͑␬/2ϩlϩ2m͒ Es ϽEc lϭ0 mϭ0 The harmonic forms can be recovered by taking the lϭ0, mϭ0 term, and the ﬁrst order correction forms by taking the mϭ0 terms. We obtain the ␥ formulation by replacing

␬ ␬Ϫl ␬ m lϩ2m 0 ␬ϩlϩ2mϪ1 ns* Dl,m3 as ͑EЈϪEs ͒ with c␥͑EЈ͒ , ⌸␬ h␯s s Ͳ͚ ͚ ⌫͑␬ϩlϩ2m͒ jϭ1 j lϭ0 mϭ0 where

␬ ␬Ϫl ␬ m lϩ2m 0 ␬ϩlϩ2mϪ1 n0* Dl,m3 a0 ͑EЈϪE0͒ cϭ . ͑A7͒ ␥͑E ͒ ⌸␬ h␯0 ͚ ͚ ⌫͑␬ϩlϩ2m͒ Ј 0 jϭ1 j lϭ0 mϭ0

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