The bulk consists of atoms and molecules, so the properties of the atoms and molecules should decide the properties of the bulk. This is a challenging problem for scientists and only a little progress has been made in this direction. One would like to see how things change as one goes from atoms and molecules to small assemblies of them, to large assemblies, to larger assemblies, and finally to the bulk. One would like to see how various atoms, which do not appear very different individually, work together to form metals, insulators and semiconductors.

The study of clusters is aimed at looking at this transition from the atomic state to the bulk.

It turns out that in this primeval soup of atoms, clusters of some size
are much more abundant than others. These clusters containing certain
special number of atoms seem to be much more stable compared to
others. These numbers are referred to as **magic numbers**. In this
primeval soup clusters collide with each other and lose atoms. So, only
the clusters which are more stable will be found to be abundant. Thus,
the magic numbers indicate the stability of certain clusters. The task
of a theoretian would be to explain this stability of some clusters,
or in other words, explain the magic numbers.

So, the special stability of rare-gas clusters is believed to be due to this geometric close-packing of atoms in the clusters.

This can be understood with the help of what is called the Jellium model. In the Jellium model, one assumes that the outer electrons of the metal atoms are loosly bound so that when a cluster is formed, they can move around everywhere in the clusters, and are not bound to a particular atom. The atoms without their outer electrons are positively charged ions. So, the free electrons "see" a jellium of positive ions in the cluster. The electrons move around in the positive jellium. If one tries to theoretically solve the problem of free electrons in a positive jellium, one indeed observes that the clusters are energetically most stable when the total number of electrons in them are 8, 20, 34, 40, 58 etc! This simple picture can be applied to many metals and works very well.

Let us have a look at the abundance spectra for Barium and Ytterbium. Surprisingly, Ba and Yb also show magic numbers close to those of rare-gases! So, what is common between these three elements? Their electronic configuration looks like this:

- Sr ------------ [Krypton] 5s
^{2}4d^{0} - Ba ------------ [Xenon] 6s
^{2}5d^{0} - Yb ------------ [Xenon] 4f
^{14}6s^{2}5d^{0}

The tool one uses to calculate the electronic structure of the cluster is the density functional theory introduced by Walter Kohn. Using this one can calculate the ground state energy of the cluster. M. Parrinello and R. Car came up with an innovative method which combines the density functional calculation for electrons and classical molecular dynamics for atoms, in a single unified formalism. This method has come to be known as the Car-Parrinello molecular dynamics (CPMD).

---Inclusion of d-electrons leads to---->