Statistical measures applied to metal clusters: evidence of magic numbers

Email addresses: jsr@unex.es (Jaime Sañudo), rilopez@unizar.es (Ricardo López-Ruiz).

Metal clusters are useful quantum systems to understand how the physical properties evolve in the transition from atom to molecule to small particle to bulk solid [7,8]. They also present a shell structure where it can applied the statistical indicators before mentioned. As in the case of atoms and nuclei, the fractional occupation probabilities of valence electrons in the different orbitals can capture the shell structure. This set of probabilities can be used to evaluate the statistical quantifiers for metallic clusters as a function of the number of valence electrons. Similar calculations have been reported for the electronic atomic structure [4,5] and for nuclei [6]. In this work, by following this method, we undertake the calculation of statistical complexity and Fisher-Shannon information for metal clusters.

The jellium model provides an accurate description of some simple metal clusters. In this model, a valence electron is assumed to interact with the average potential generated by the other electrons and the ions [7,8]. The confinement potential in the Schrödinger equation leading to shell structure is taken as a potential intermediate between the three-dimensional harmonic oscillator and the three-dimensional square well. This yields a filling of shells with a number N of valence electrons given by the series: 2,8,18,20,34,40,58,68,70,92,106,112,138,156,166,168,198, 220 and so on. Each shell is given by (nl) w , where l denotes the orbital angular momentum (l = 0, 1, 2, . . .), n counts the number of levels with that l value, and w is the number of valence electrons in the shell, 0 ≤ w ≤ 2(2l + 1).

As an example, we explicitly give the shell configuration of a metal cluster formed by N = 58 valence electrons. It is obtained:

The fractional occupation probability distribution of electron orbitals {p k }, k = 1, 2, . . . , Π, being Π the number of shells, can be defined in the same way as it has been done for calculations in atoms and nuclei [4,5,6]. This normalized probability distribution {p k } ( p k = 1) is easily found by dividing the superscripts w by the total number N of electrons. Then, from this probability distribution, the different statistical magnitudes (Shannon entropy, disequilibrium, Fisher information, statistical complexity and Fisher-Shannon entropy) can be obtained.

Here, we undertake the calculation of entropic products, a statistical measure of complexity C and the Fisher-Shannon entropy P , that result from the product of two statistical quantities, one of them representing the information content of the system, and the other one giving an idea of how far the system is from the equilibrium. The classical indicator of information is the Shannon entropy, that in the discrete version is expressed as

Any monotonous function of S can also be used for this purpose, such as the exponential Shannon entropy [9], H = e S or J = 1 2πe e 2S/3 . The indicator of how much concentrated is the probability distribution of the system can be related with some kind of distance to the equilibrium distribution, that in our case is the equiprobability. The disequilibrium D given by

and the Fisher information I by

where p Π+1 = 0, are two useful parameters in this direction. Then, the statistical measure of complexity, C, the so-called LMC complexity [10,11], is defined as

and the Fisher-Shannon information [12,13,14], P , is given by

The statistical complexity, C, of metal clusters as a function of the number of valence electrons, N, is given in Fig. 1. We can observe in this figure that this magnitude fluctuates around an slightly increasing average value N. This trend is also found for the electronic structure of atoms [5] and for the shell structure of nuclei [6], reinforcing the idea that in general complexity increases with the number of units forming a system. However, the shell model supposes that the system encounters certain ordered rearrangements for some specific number of units (electrons or nucleons) that coincide with closed shells. In the present case, this fact is reflected by the notable increase of C in the metal clusters with one valence electron more than those with closed shells, which a

…(Full text truncated)…