Silicon Clusters

The following is a description of our work on Sin clusters (n=2-20), J.C. Grossman, L. Mitas, Phys. Rev. Lett. 74, 1323 (1995) .

One of our fundamental goals in this work has been to expand the applicability of QMC methods to clusters and large molecules. As a first application of QMC to clusters, we were drawn to silicon since there have been some discrepancies between various methods and recent conjectures about ground state properties. Silicon clusters are still the subject of much discussion and research, especially when looking for low energy geometries. For example, one interesting case is the ground state structure of Si13.

Above are shown two possible candidates: on the left is a perfect icosahedron, in which 12 atoms lie on the surface of a sphere, and on the right is a 3-fold symmetric structure which was found to be about 5 eV more stable (Rothlisberger, Andreoni and Gianozzi, JCP 92, 1248 ('92)). There was then a conjecture that electron correlation had not been treated properly and that if it were, the icosahedron would be found to be more stable. We therefore thought it would be interesting and useful to answer simple questions such as (1) how much correlation is there in two competing structures, and (2) can correlation be the decisive factor in stabilizing a geometry.

There are several examples of similar situations. For example, the two possible ground state geometries for Si10 are shown below.

The structure on the left is a tetracapped octahedron and the one on the right is a tetracapped trigonal prism. Different mean-field methods have given a different energetical ordering for these two clusters, so clearly higher accuracy calculations are needed.

For N=20 we calculated the perfect dodecahedron and a new 3-fold symmetric structure which we have found through a structural search.

This cluster was at the time of publications the most stable form for Si20, and it is part of a family of elongated (``stacked triangles'') silicon clusters.

We first studied the binding energies, and in particular we wanted to compare our diffusion Monte Carlo (DMC) results with those from Hartree-Fock (HF) and the Local Density Approximation (LDA).

We included smaller clusters in order to compare with available experimental data, and for this size range (n=2-7) there is a five-fold improvement over LDA. The LDA energies are about 20% overbound, while DMC is about 2-4% underbound. Another important point is that the error in DMC does not increase as the cluster size increases.

In addition to binding energies, the QMC method also enables us to answer our original questions concerning the impact of correlation energy on these clusters. The following is a plot of correlation energy per atom vs. number of atoms in the cluster.

The dashed line connects the most stable clusters. For 13 atoms the icosahedron does indeed have more correlation energy, however it the difference is so small that it could not overcome unfavorable Coulomb and exchange terms, and the lower symmetry structure is more stable. In the case of 20 atoms the opposite is true: here the correlation energy in our new structure is substantially more than that of the dodecahedron, enough to make it the more stable cluster. It is worth pointing out that such a graph is very difficult to get from any other method, especially when we consider that in these calculations we get more than 90% of the total correlation energy.

There are two additional contributions in this work: (1) We calculated the electron affinity for the closed-shell Si13 cluster with C3v symmetry and found a different ground state than that predicted by LDA, and (2) We were able to decrease the 2-4% discrepancy between our results and experiment by using Natural orbitals instead of the Hartree-Fock orbitals, which improved the quality of our trial wavefunction for several small cases. The details and results of these calculations can be found in our paper, so they will not be repeated here.

J.C. Grossman 1995