Structural information about the ${\mathrm{Ar}}_6$ cluster from the frozen-Gaussian imaginary-time propagator

A numerically cheap way to obtain structural information about clusters of rare gas atoms at low temperatures is developed. The semiclassical frozen-Gaussian imaginary-time propagator is extended such that it can account for the mean values of all interatomic distances in the cluster and their variances. To reduce the required numerical effort an approximation for the mean values is developed which preserves the quality of the results offered by the semiclassical ansatz. The method is applied to the ${\text{Ar}}_6$ cluster. It is found that the cluster dissociates almost in one step to six free atoms when the temperature is increased. Precursors of the dissociation are only observable in the distances of the atoms via the appearance of a second isomer. The process is almost classical. However, the method is able to resolve small differences in the temperatures at which the dissociation takes place and in the mean distances of the bound configuration.

Figure 1

(a) Mean energies of the ${\mathrm{Ar}}_6$ cluster calculated with the two-parameter frozen-Gaussian propagator (2P-FG) and its classical counterpart. For the high-temperature limit both results agree well. At low temperatures the classical mean energy tends to the potential minimum and the frozen-Gaussian propagator approximates the quantum mechanical ground state. The inset shows a comparison of the 2P-FG method with the fully coupled thawed-Gaussian method (FC-TG). The energies differ by a few percent. (b) Specific heat around the dissociation, which seems to happen in one step.

Figure 2

(a) Comparison of the mean values of all 15 distances calculated with the frozen-Gaussian method (2P-FG) and a classical calculation at temperatures $T\le 20$. The distances appear in groups. Three distances converge for $T\to 0$ to values above $5\text{Å}$, whereas the remaining 12 are below $4\text{Å}$. (b) A comparison of the frozen-Gaussian and thawed-Gaussian method (FC-TG) shows that the mean distances agree very well.

Figure 3

(a) Mean values of all 15 distances calculated with the frozen-Gaussian method (2P-FG) around the dissociation. Two of the smaller distances join the group of the three larger distances starting at $T\approx 23\mathrm{K}$, but then the dissociation occurs at once. (b) The same behavior is observed for the classical calculation.

Figure 4

Distances of all six atoms from the center of mass in a classical calculation for $R_{\mathrm{c}}=35\text{Å}$. Shown is a thermal average of size-ordered distances as explained in Sec. 2c. With increasing temperature one distance becomes larger than the others before all raise drastically.

Figure 5

(a) Standard deviations ${\sigma }_{ij}$ of the distances in the frozen-Gaussian approximation (2P-FG) and (b) of the classical calculation. They increase drastically around the dissociation. (c) Quantum mechanical part (31) of the variances for the fully coupled thawed-Gaussian propagator. For low temperatures it increases due to an increasing width of the Gaussians.