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

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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 1995! 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 ~MC! and molecular dynamics ‘‘quench.’’ This mapping allows the partition function to be ~MD! 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 vibra- tional 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 ~PES! 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, V(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 ~LJ! 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- loops! 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 ~H2O!8 . The multihistogram method the global minimum and a manifold consisting of a large obtains V(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 V(E), and subsequently used V(E) to calculate V(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. V(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 J. Chem. Phys. 102 (24), 22 June 1995 0021-9606/95/102(24)/9659/14/$6.00 © 1995 American Institute of Physics 9659 9660 J. P. K. Doye and D. J. Wales: Thermodynamic properties of clusters curve, the heat capacity, the Helmholtz free energy, the ca- TABLE I. Details of the two samples of minima used for LJ55 . E8 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 E8/e 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 k21 sions for the density of states to calculate a variety of ther- gsns ~E2Es ! V~E!5 ( k s , ~3! 0 G~k!P hn modynamic properties. LJ13 and LJ55 are used as examples to E ,E j51 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, g, 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., g(E8)s}gsV(E8)s . single minimum on the PES is, in the harmonic Hence, approximation,13 V~E!s ~E2E0!k21 V~E!} g~E8! ~4! 0 ( s V~E8! V~E!5 k u~E2E !, ~1! E0,E s G~k!P j51hn j s 0 0 k21 where E is the potential energy of the minimum, u is the E2Es Heaviside step function, and k, the number of vibrational } ( g~E8!s 0 , ~5! 0 S E82E D E ,E s degrees of freedom, is 3N26. To calculate the density of s states for the whole system, all the minima on the PES need where E8 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*~E2E !k21 s s paring the first terms of Eqs. ~2! and ~5! gives for the pro- V~E!5 ( k s , ~2! 0 G~k!P j51hn j portionality constant, c, Es ,E 0 k21 where the sum is over all the geometrically distinct minima n0*~E82E0! on the surface; n* , the number of permutational isomers of c5 k 0 . ~6! s g~E8!0G~k!P j51hn j minimum s, is given by ns* 5 2N!/hs , where hs is the order of the point group of s. A critical test of these formulas for V(E) is the pre- dicted microcanonical caloric curve, which for LJ55 has an S- The difficulty with Eq. ~2! 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.
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