Introduction

Much of the interest in the thermodynamics of clusters has centred on the melting transition, particularly its character as the finite-size analogue of the bulk first-order phase transition. Cluster melting differs from bulk in a number of ways. First, the finite size of clusters causes `rounding' of thermodynamic properties[124,125,126]. The cluster melting transition occurs over a temperature range, not a single temperature; heat capacity peaks are not $\delta$-functions, but instead remain finite.

Secondly, phase separation does not occur in most small clusters (potassium chloride clusters may provide an exception[127]), because the energetic cost of the interface is too large. This leads to certain thermodynamic features which are peculiar to clusters. One is that an ensemble of clusters in the melting region is a mixture of solid-like and liquid-like clusters with the ratio fixed by the relative stabilities of the two forms. If one was to observe a single cluster, it would be seen to occasionally change form between solid-like and liquid-like[128,129]. There is a dynamic coexistence of the two states. These phase-like forms have different characteristic properties. For example, the liquid-like state has a higher potential energy, floppier vibrations, a more disordered structure, and a larger diffusion constant[130]. If these differences between states can be quantified using an order parameter, the presence of more than one state will give rise to multimodal probability distributions of that order parameter. Such multimodality can be related to the thermodynamic stability of the states through the concept of Landau functions[131,132], which describe the thermodynamic potential, i.e. the entropy (microcanonical ensemble) or the Helmholtz free energy (canonical ensemble), in terms of an order parameter. The presence of a first-order phase transition is indicated by two stable states of the Landau function, e.g. two minima in the Landau free energy separated by a barrier.

The order parameters that have been most successful in elucidating the nature of cluster melting have taken advantage of the differences in potential energy between the solid-like and liquid-like states[133]. The ability of the order parameter to distinguish between states can sometimes be enhanced by averaging the parameter over short times to reduce the vibrational broadening[112]. The short-time averaged (STA) temperature has proved to be a particular useful microcanonical order parameter[130]. For this time averaging to succeed the length of time the cluster resides in each state needs to be much longer than the time scale for vibrational motion.

Another unusual feature that results from the absence of phase separation for clusters is the possibility of Van der Waals loops (or S-bends) in the microcanonical caloric curve[134]; there can be a range of energy for which the cluster has a negative specific heat capacity[135,136,137]. The loops are a result of the transition from the high temperature, low potential energy solid-like state to the low temperature, high potential energy liquid-like state. In §3.7 we use an analytical model to investigate the conditions under which they are most likely to occur, examining in particular the effect of system size.

The standard way to calculate the thermodynamic properties of clusters is by some form of simulation, normally based on the Monte Carlo (MC) or molecular dynamics (MD) methods, perhaps augmented by more sophisticated techniques to help achieve ergodicity, such as jump-walking[138] or umbrella sampling[139], or to extract densities of states, such as the multi-histogram methods[140,141,142]. Such simulations give `exact' results for the thermodynamics within the statistical errors of the simulation. An alternative route to calculate accurate densities of states is provided by Reinhardt's adiabatic switching method[143]; it has the advantage over histogram MC that it can extract absolute (not relative) values for the densities of states.

However, before such simulations became computationally feasible, considerable attention was given to the calculation of thermodynamic properties of clusters from the PES. Initially, calculations were performed in the harmonic approximation using only knowledge of the vibrational spectrum of a single low energy structure[10,11,144,145,146]. McGinty and Burton realized that if their results were to have relevance for more than low temperature behaviour, other configurations needed to be included in their partition function[10,145], but without the means to systematically search the PES they were unable to implement this extension of their approach. In particular, if this method is to model the melting transition, the large number of high potential energy minima associated with the liquid-like state need to be taken into account. Hoare and McInnes did obtain a suitable database of minima for small LJ clusters[64,65], and although they outlined the method by which the thermodynamic properties could be calculated from this sample[147], the results were never published. However, with the advent of reliable simulation methods and adequate computer power to carry them out, such calculations were neglected.

Interest in the relationship of the thermodynamics of clusters to the PES was rekindled by the application of Stillinger and Weber's inherent structure approach[148] to clusters. The basic idea is that every configuration of a system can be mapped onto a minimum of the PES (an `inherent structure'), by a steepest-descent path or `quench'. This mapping allows one to consider the thermodynamic effects of the energy spectrum of minima separately from thermal motion within the wells, and gives a clearer view of structure, especially for liquids, by removing vibrational noise. In the application of this method to clusters, the thermodynamic and dynamic properties were elucidated by relating them to the qualitative features of the PES, especially the low energy minima and saddle points, thus allowing an intimate connection to be made between structure and thermodynamics[108,130,149,150,151].

This relationship between cluster thermodynamics and the PES was further developed by Bixon and Jortner[152] who considered the thermodynamics that resulted from different model energy spectra of minima. They showed that a large energy gap between the global minimum and a manifold consisting of a large number of higher energy minima was necessary to produce a significant feature in the caloric curves.

The wheel finally returned to the ideas of Burton, McGinty and Hoare, when Wales[153] and Franke [154] independently developed a method to quantitatively determine the thermodynamic properties from the PES. A sample of minima is generated by systematic quenching from a high energy MD run. $\Omega(E)$ is then calculated by summing the harmonic density of states for each minima. This approach has been called the harmonic superposition method, and has been applied to a number of cluster systems to obtain a wide variety of thermodynamic properties[133,155].

The main focus of this chapter is to extend the harmonic superposition method to incorporate anharmonicity (§3.5) and to show how this method can give added physical insights into the thermodynamics by allowing the roles of different regions of the PES to be quantitatively examined (§3.6). First though, we briefly outline the simulation methods that we have used both for 55 in §3.3 and for the Morse systems studied in Chapter 4. The results for 55 (§3.3) provide both an example of some of the typical features seen in cluster thermodynamics, and also some of the results necessary for the analysis in §3.5. We then briefly review the harmonic superposition method (§3.4), before addressing the extension of the method to include anharmonicity. Finally, we investigate the effect of size on the possibility of Van der Waals loops appearing in the microcanonical caloric curves (§3.7).