Abstract |
2. Methods |
The study of atomic and molecular clusters is yielding a wealth of information about interatomic and intermolecular forces [1]. As we learn more about these forces we can in principle use what we learn to improve our understanding of bulk phases of materials through computational simulations of their dynamical and thermal properties [2,3].
For this reason, clusters have been the subject of many recent theoretical and experimental studies [1-3]. Such studies ultimately should shed light on which intermolecular potential models are reliable for computer simulations of the bulk materials. From our perspective an interesting question can be posed in the light of what we know so far about water in its bulk and cluster phases; if a potential or theoretical method works well for liquid water, will it work well for water clusters? And what about in the reverse direction? Will well-tuned cluster potentials yield accurate results for liquid water? Moreover, to what extent are these conclusions true for other substances, such as metals, inorganic salts and organics?
In order to answer the preceding questions one must have an energetic model for the cluster of interest. Once this is obtained, one must next somehow survey the potential energy surface of the cluster. One class of techniques available for this latter purpose are simulated annealing algorithms. These algorithms are basically the coupling of a temperature-controlled geometry fluctuation scheme, such as Monte Carlo, Brownian dynamics or molecular dynamics [4], to a scheduled temperature variation which slowly quenches the cluster down to an energy minimum. If quenching is rapid, local minima are generally located; if infinitely slow, the global minimum can be obtained (assuming that the highest temperature of the annealing schedule is high enough). These fluctuation schemes all have varying degrees of algorithmic complexity, numerical efficiency and ease of implementation. For example, molecular dynamics and Brownian dynamics both require the programmer to implement energy derivatives, usually explicitly, to obtain interatomic forces; Monte Carlo does not [3,4]. However, successful Monte Carlo work relies on the development of efficient and ergodic sampling strategies whereas in molecular and Brownian dynamics, all sampling is generated by the solution of the differential equations to be solved [4]. The numerical solution of these differential equations also poses certain numerical challenges which must be dealt with[4]. In addition, most fluctuation algorithms rely to some extent on the availability of high-quality random number generators with reasonably long recurrence times[4,5]. There are no simple guidelines for deciding which of the many available alternatives is most appropriate for a given application. Therefore, experimentation with new methods is of some interest.
A new fluctuation scheme, which we have coined the "Boltzmann simplex," was recently proposed by Press et.al. in the second edition of their well-received text Numerical Recipes [5]. This new algorithm caught our attention as we began our studies of cluster systems. Their algorithm, which is essentially a downhill simplex algorithm with thermally-smeared vertex energies, does not require one to code the derivatives of the potential energy function, which makes debugging new cluster systems relatively easy and makes it possible to study systems easily when analytical energy derivatives may not be available. In addition, the algorithm reduces to a simplex algorithm in the limit of zero temperature, which has well-known convergence properties in the vicinity of a local minimum. However, Press et.al. presented their algorithm without testing it in the context of computational molecular modeling. These facts, combined with our interest in clusters, led us to investigate the algorithm's properties extensively by testing it on three distinct types of cluster systems.
When testing new algorithms, benchmarks are essential. We began by considering argon clusters within a pairwise-additive Lennard-Jones potential. Hoare and Pal [6,7] have presented minimum-energy structures and vibrational frequencies for small Lennard-Jones clusters, and we use their results as a benchmark to test the efficacy of the BSSA algorithm and to assess the appropriate selection of annealing parameters.
We also consider small water clusters within an interaction potential recently proposed by Ferguson [8]. The Ferguson potential, a flexible-water model which is a modification of the well-known SPC rigid model [9], was shown by NPT-molecular dynamics simulations to reproduce a number of the experimental properties of liquid water at 298K and 1 atm. These properties include the radial distribution functions, the dielectric constant, the vibrational spectrum, the self-diffusion constant, and the free energies of solvation for neon atoms and water molecules [8]. We have obtained minimum-energy structures within this model for water clusters with up to 6 monomers, and we here compare the results obtained to recently obtained experimental values [10] as well as published ab initio, density-functional theory and model potential results. Ultimately we hope to begin to address the question of whether a potential which is well-parameterized for reproduction of the bulk's properties can predict the properties of clusters. Furthermore the results of the present study could be used to achieve further fine-tuning of the potential if necessary.