Equilibrium properties of clusters in the harmonic superposition approximation
Introduction
The theoretical study of complex atomic and molecular systems often relies on extensive numerical experiments and simulations. Molecular dynamics (MD) and Monte Carlo (MC) methods can often provide accurate data for comparison with experiments or analytical theories. To circumvent the problems of slow convergence, barrier crossing, and broken ergodicity, a large set of specific tools have been developed [1], [2]. For example, one may cite the transition path sampling method [3], [4] for calculating reaction rates, or the recent Wang–Landau algorithm [5] aimed at computing the density of states without prior knowledge of the potential energy surface (PES).
The main interest in performing simulations is that any observable that is representable be cast as a time or ensemble average (as in real experiments), is readily accessible through the calculation of the same property for thousands of different configurations of the system. However, this averaging can also be a limitation of simulations. Firstly, the cost of performing a proper sampling can be excessive for complex potential energy surfaces, especially in the quantum regime where path integrals or centroid methods are commonly more demanding than classical simulations. Secondly, many properties may themselves be more difficult to calculate than the potential energy needed for sampling. This is the case for properties involving electronic excited states or multiple derivatives or gradients of the energy. A possible alternative to simulations is the superposition approach. In this method the dynamics of the system is decomposed into the various instantaneous basins of attraction corresponding to local minima on the PES, plus fluctuations inside each basin. This method, also known as the inherent structure approach, was first formalised by Stillinger and Weber for liquids [6], although similar ideas were previously proposed in cluster physics [7], but were not applied due to computational limitations. The superposition approach has since been used extensively for clusters [8], [9], and it has been exploited in the theory of liquids [10] and glasses [11]. The method is based on the knowledge of a statistical set of minima on the PES, conveniently obtained from quenches. The thermodynamic properties at equilibrium, in the microcanonical or canonical ensembles, are then calculated assuming simple approximations within each basin. Harmonic treatments are usually the starting point for more accurate predictions from systematic anharmonic perturbative expansions [12], [13].
Up to now, inherent structures have been employed mostly for calculating thermal and statistical quantities, partition functions or densities of states. However, any equilibrium property of interest can be computed, provided that a reliable estimate for each local minimum can be made.
The goal of this report is to illustrate how the superposition approximation can be employed to predict physical observables other than purely thermodynamic properties. By choosing to investigate very different observables, namely the geometry dependent polarizability of metal clusters and the photoabsorption spectrum of chromophore-doped van der Waals clusters, we show that inherent structures have a much wider range of application, and that they can be complementary to conventional simulations, especially in the quantum regime. After briefly describing the superposition approximation in the following section, we give our main results on the two above examples in Section 3. Finally some concluding remarks are given in Section 4.
Section snippets
Harmonic superposition approximation
We consider an N-atom system in thermal equilibrium at temperature T=1/kBβ. The total partition function Q is approximated as a sum over minima, or inherent structures, {α}:The terms nα are degeneracy factors that account for permutational isomers: nα=2N!/hα if all N atoms correspond to the same element, where hα is the order of the point group of structure α; Qα is the partition function of isomer α. The simplest expression for Qα corresponds to the harmonic approximation.
Electric polarizability of sodium clusters
The electric polarizability of alkali metal clusters has been measured for a variety of systems by the Broyer and Kresin groups [15], [16]. Ab initio theoretical studies on sodium clusters [17], [18] show reasonable agreement with experimental data, even though the observed values are underestimated. Recent dynamical investigations [17], [19] solved this discrepancy by taking the vibrational motion into account. It was then suggested [19] that the polarizability of metal clusters could be used
Conclusion
In this work we applied the harmonic superposition approximation to compute some physical and chemical observables at thermal equilibrium. We found good agreement with Monte Carlo simulations, however we noticed that the weighting procedure for correcting the incomplete sample of isomers should be used cautiously depending on the classical or quantum nature of the calculation. The main advantage of using the superposition method is the natural bridge that it provides between the estimated
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